diff --git "a/-NE3T4oBgHgl3EQfSglq/content/tmp_files/2301.04433v1.pdf.txt" "b/-NE3T4oBgHgl3EQfSglq/content/tmp_files/2301.04433v1.pdf.txt" new file mode 100644--- /dev/null +++ "b/-NE3T4oBgHgl3EQfSglq/content/tmp_files/2301.04433v1.pdf.txt" @@ -0,0 +1,3242 @@ +DIAMETRAL NOTIONS FOR ELEMENTS OF THE UNIT BALL OF A +BANACH SPACE +MIGUEL MART´IN, YO¨EL PERREAU, AND ABRAHAM RUEDA ZOCA +Abstract. We introduce extensions of ∆-points and Daugavet points in which slices are replaced by +relative weakly open subsets (super ∆-points and super Daugavet points) or by convex combinations of +slices (ccs ∆-points and ccs Daugavet points). These notions represent the extreme opposite to denting +points, points of continuity, and strongly regular points. We first give a general overview on these +new concepts and provide some isometric consequences on the spaces. As examples: if a Banach space +contains a super ∆-point, then it does not admit an unconditional FDD (in particular, unconditional +basis) with suppression constant smaller than two; if a real Banach space contains a ccs ∆-point, then +it does not admit a one-unconditional basis; if a Banach space contains a ccs Daugavet point, then +every convex combination of slices of its unit ball has diameter two. We next characterize the notions in +some classes of Banach spaces showing, for instance, that all the notions coincide in L1-predual spaces +and that all the notions but ccs Daugavet points coincide in L1-spaces. We next remark on some +examples which have previously appeared in the literature and provide some new intriguing examples: +examples of super ∆-points which are as closed as desired to strongly exposed points (hence failing +to be Daugavet points in an extreme way); an example of a super ∆-point which is strongly regular +(hence failing to be a ccs ∆-point in the strongest way); a super Daugavet point which fails to be a ccs +∆-point. The extensions of the diametral notions to point in the open unit ball and the consequences +on the spaces are also studied. Last, we investigate the Kuratowski measure of relative weakly open +subsets and of convex combinations of slices in the presence of super ∆-points or ccs ∆-points, as well +as for spaces enjoying diameter 2 properties. We conclude the paper with a section on open problems. +Contents +1. +Introduction +2 +2. +Notation and preliminary results +5 +3. +Characterisations of diametral-notions and implications on the geometry of the ambient +space +9 +3.1. +Spaces with a one-unconditional basis and beyond +12 +3.2. +Absolute sums +16 +4. +Examples and counterexamples of diametral elements +19 +4.1. +Characterization in C(K)-spaces, L1-preduals, and M¨untz spaces +19 +4.2. +Characterization in L1-spaces +22 +4.3. +Remarks on some examples from the literature +24 +4.4. +A super ∆-point which fails to be a Daugavet point in an extreme way +26 +4.5. +A super ∆-point which is a strongly regular point +27 +4.6. +A super Daugavet point which is not ccs ∆-point +28 +Date: January 11th, 2023. +The first and third named authors were supported by grant PID2021-122126NB-C31 funded by MCIN/AEI/ +10.13039/501100011033 and “ERDF A way of making Europe”, by Junta de Andaluc´ıa I+D+i grants P20 00255 +and FQM-185, +and by “Maria de Maeztu” Excellence Unit IMAG, reference CEX2020-001105-M funded by +MCIN/AEI/10.13039/501100011033. The second named author was supported by the Estonian Research Council grant +SJD58. +1 +arXiv:2301.04433v1 [math.FA] 11 Jan 2023 + +2 +MART´IN, PERREAU, AND RUEDA ZOCA +4.7. +A summary of relations between the properties +35 +5. +Diametral-properties for elements of the open unit ball +35 +6. +Kuratowski measure and large diameters +38 +6.1. +Kuratowski measure and diameter two properties. +39 +6.2. +Kuratowski measure and ∆-notions +41 +7. +Commented open questions +42 +Acknowledgments +45 +References +45 +1. Introduction +It is fair to say that one of the most studied properties of Banach spaces is the Radon-Nikod´ym +property (RNP) because it has shown to be very useful; due to the large amount of its geometric, +analytic, and measure theoretic characterisations; in several fields of Banach space theory such as +representation of bounded linear operators, representation of dual spaces or representation of certain +tensor product spaces (see [18, 21]). +A famous geometric characterization of the Radon-Nikod´ym property is related to the size of slices. +Recall that a slice of a bounded non-empty subset C of a Banach space X is simply the (nonempty) +intersection of C with a half-space. A Banach space X has the RNP if and only if every non-empty +closed and bounded subset of X admits slices of arbitrarily small diameter (see e.g. [18]). +A closely related and equally important geometric property of Banach spaces is the point of con- +tinuity property. Recall that a Banach space X has the point of continuity property (PCP) if every +non-empty closed and bounded subset of X admits non-empty relatively weakly open subsets of ar- +bitrarily small diameter. Let us emphasize here as an example the striking equivalence between the +Radon-Nikod´ym property and the weak∗ version of the point of continuity property for dual spaces, +and the related characterization of Asplund spaces as preduals of RNP spaces (see e.g. [19]). In his +proof of the determination of the Radon-Nikod´ym property by subspaces with a finite dimensional +decomposition (FDD) in [17], Bourgain also introduced an important weakening of the point of conti- +nuity property, that he called property “(∗)”, and that is nowadays referred to as the convex point of +continuity property. Recall that a Banach space X has the convex point of continuity property (CPCP) +if every non-empty closed, convex and bounded subset of X admits non-empty relatively weakly open +subsets of arbitrarily small diameter. +In fact, Bourgain implicitly used in his work the notion of strong regularity which, as he showed, +is implied by the CPCP. Recall that a Banach space X is strongly regular (SR) if every non-empty +closed, convex and bounded subset of X contains convex combinations of slices of arbitrarily small +diameter. Observe that the convexity of the subset is required in this definition in order to guarantee +that it contains all the convex combinations of its slices. It later turned out that strong regularity +had important applications to the famous (still open) question of the equivalence between the Radon- +Nikod´ym property and the Krein-Milman property. Recall that a Banach space X has the Krein- +Milman property (KMP) if every non-empty closed, convex and bounded subset C of X admits an +extreme point. The RNP implies the KMP (see e.g. [18, Theorem 3.3.6]), and it follows from [48] that +every strongly regular space with the KMP has the RNP. Also recall that it was proved in [29] the +RNP and the KMP are equivalent in dual spaces. + +DIAMETRAL NOTIONS FOR ELEMENTS OF THE UNIT BALL OF A BANACH SPACE +3 +From the definitions it follows that RNP⇒PCP⇒CPCP and it is also known that CPCP⇒SR. +None of the above implications reverse (see e.g [49] and references therein). In order to show that +strong regularity is implied by the CPCP, Bourgain made an important geometric observation, namely +that in every non-empty bounded and convex subset of a Banach space X, every non-empty relatively +weakly open subset contains a convex combination of slices. We will discuss this “Bougain Lemma” +and its applications to the subject of the present paper in more details in Section 2. +Another classical refinement of the above characterization of the Radon-Nikod´ym property is related +to the notion of denting points. Recall that a point x0 of a bounded subset C of X is a denting point +of C if there are slices of C containing x0 of arbitrarily small diameter. A Banach space X has the +RNP if and only if every closed, convex and bounded subset contains a denting point. Actually, every +nonempty closed, convex and bounded subset C of a Banach space X with the RNP is equal to the +closure of the convex hull of the set of its denting points (see e.g. [18, Corollary 3.5.7]). +For the PCP and the CPCP, a similar role is played by points of weak-to-norm continuity. Given +a bounded subset C of X, we say that a point x0 ∈ C is a point of weak-to-norm continuity (point +of continuity in short) if the identity mapping i: (C, w) −→ (C, τ) is continuous at the point x0 or, +equivalently, if x0 belongs to relatively weakly open subsets of C of arbitrarily small diameter. Note +that a classical result by Lin-Lin-Troyanski [39] establishes that a point x0 ∈ C is a denting point if, +and only if, x0 is simultaneously a point of continuity and a extreme point of C. In a space with the +PCP every non-empty closed and bounded subset contains a point of continuity; and the set of all +points of continuity of a given closed, convex and bounded subset C of a Banach space X with the +CPCP is weakly dense in C (see e.g. [22, Theorem 1.13]). +In relation to strong regularity, a point x0 of a bounded, convex subset C of X is a point of strong +regularity if there are convex combinations of slices of C containing x0 of arbitrarily small diameter. +Then the set of all points of strong regularity of a given closed, convex and bounded subset C of a +strongly regular Banach space X is norm dense in C (see [25, Theorem 3.6]). Let us observe that +points of strong regularity may be in the interior of a set, while denting points (and points of continuity +in the infinite-dimensional case) belong always to the border of the set. +In [3] the extreme opposite notion to denting point of the unit ball was introduced in the following +sense: an element x in the unit sphere of a Banach space X is a ∆-point if we can find in every slice +of BX containing x points which are at distance from x as close as we wish to the maximal possible +distance in the ball (distance 2). A similar yet stronger notion appeared simultaneously in relation to +another quite famous property of Banach spaces, the Daugavet property. Recall that a Banach space +X has the Daugavet property (DPr) if the Daugavet equation +(DE) +∥Id + T∥ = 1 + ∥T∥ +holds for every rank-one operator T : X −→ X, where Id denotes the identity operator. +In this +case, all weakly compact operators also satisfy (DE). We refer the reader to the seminal paper [35] +for background. Recent results can be found in [42] and references therein. The Daugavet property +admits a beautiful geometric characterization involving slices related to the notion of Daugavet points: +an element x on the unit sphere of a Banach space X is a Daugavet point if in every slice of BX (not +necessarily containing the point x) there are points which are at distance from x as close as we wish to +2. With this definition in mind, [35, Lemma 2.1] states that X has the DPr if and only if all elements +in SX are Daugavet points. Let us comment that the Daugavet property imposes severe restriction +on the Banach space: if X is a Banach space with the DPr, then it fails the RNP and it has no +unconditional basis (actually, it cannot be embedded into a Banach space with unconditional basis). + +4 +MART´IN, PERREAU, AND RUEDA ZOCA +On the other hand, ∆- and Daugavet points have proved to be far more flexible than the global +properties that they define. For example, there exists a Banach space with the RNP and a Daugavet +point [51] (see paragraph 4.3.1), there exists a Banach space with a one-unconditional basis and a large +subset of Daugavet points [6] (see paragraph 4.3.3), and there is an MLUR Banach space for which all +elements in its unit sphere are ∆-points, which contains convex combinations of slices of arbitrarily +small diameter, but satisfying that every convex combination of slices intersecting its unit sphere has +diameter two [2] (see paragraph 4.3.2). Nonetheless, it has been recently proved that ∆-points have +some influence on the isometric structure of the space. For example, it is shown in [5] that uniformly +non-square spaces do not contain ∆-points; actually, it has been very recently proved in [37] that +a ∆-point cannot be a locally uniformly non-square point. Also, combining the results from [5] and +[52], asymptotic uniformly smooth spaces and their duals do not contain ∆-points. However, it is still +an important open problem to understand whether ∆- or Daugavet point have any influence on the +isomorphic structure of the space. +In this paper, our main aim is to study natural strengthening of the notions of Daugavet- and +∆-points obtained by replacing slices by non-empty relatively weakly open subsets (“super points”) +or convex combination of slices (“ccs points”) in order to provide new diametral notions which are +extreme opposites to points of continuity and to strongly regular points, respectively. See Definitions +2.5 and 2.4 for details. Our main goal will be to understand the influence, for a given Banach space, +of the existence of such points on its geometry, and to study the different diametral notions in several +families of Banach spaces. A particular emphasis will be put on trying to distinguish between all the +various formally different notions. +Let us end this section by giving a brief description about the organization of the paper and the +main results obtained. Section 2 contains the necessary notation (which is standard, anyway), needed +definitions, and some preliminary results. We include in Section 3 some characterizations of the newer +diametral point notions and some necessary conditions on the existence of such points. In particular, +we study the existence of super ∆-points and ccs ∆-points in spaces with a one-unconditional basis. +We first give an analogue for ccs ∆-points to a result from [6] which implicitly states that such spaces +contain no super ∆-points. Second, we provide sharper and improved versions of this super ∆ result +in the context of unconditional FDDs with a small unconditional constant, and more generally in the +context of spaces in which special families of operators are available. The section finishes with the +study of the behaviour of super ∆-points and super Daugavet points with respect to absolute sums +somehow analogous to the known one for ∆-points and Daugavet points; however, not all the results +extend to ccs ∆-points and ccs Daugavet points, but we also give some partial results. Section 4 is +devoted to examples and counterexamples. We first characterize the diametral notions in some families +of classical Banach spaces: we show that all notions are equivalent in L1-preduals and M¨untz spaces +(Subsection 4.1); all notions but ccs Daugavet points also coincide in L1-spaces (Subsection 4.2). +We next give in Subsection 4.3 some remarks on examples which have previously appeared in the +journal literature, discussing the new diametral notions on them, and showing that they may help to +distinguish between the diametral notions. The most complicated and tricky examples are produced +in the last three subsection of this section: super ∆-points which are as closed as desired to strongly +exposed points (hence failing to be Daugavet points in an extreme way) in Subsection 4.4; a super ∆- +point which is strongly regular (hence failing to be ccs ∆-point in an extreme way) in Subsection 4.5; +super Daugavet points which belong to convex combinations of slices of diameter as small as desired +(hence failing to be ccs ∆-points in an extreme way). We finish this subsection with a summary of +relations between all the diametral notions. The idea in Section 5 is to generalize the diametral notions +to elements of the open unit ball, and use these notions to characterize some geometric properties. In + +DIAMETRAL NOTIONS FOR ELEMENTS OF THE UNIT BALL OF A BANACH SPACE +5 +particular, we properly localize the result by Kadets that the DSD2P is equivalent to the Daugavet +property. Section 6 deals with Kuratowki index of non-compactness of slices, relative weakly open +subsets, and convex combinations of slices. We get that every relative weakly open subset (respectively, +every convex combination of slices) in a space with the diameter 2 property (respectively, with the +strong diameter 2 property) has Kuratowski measure 2; these results extends the analogous result for +slices and the the local diameter 2 property proved in [20, Proposition 3.1]. Also, we show that every +relative weakly open subset that contains a super ∆-point has Kuratowski measure 2, and a similar +result is obtained with convex combinations of relative weakly open subsets containing a ccw ∆-point; +these results extend [52, Corollary 2.2]. Finally, Section 7 is devoted to collect some interesting open +questions and some remarks on them. +2. Notation and preliminary results +We will use standard notation as in the books [8], [23], and [24], for instance. Given a Banach +space X, BX (respectively, SX) stands for the closed unit ball (respectively, the unit sphere) of +X. +We denote by X∗ the topological dual of X and we write JX : X −→ X∗∗ for the canonical +injection. +We denote by dent (BX) and ext (BX) the sets of all denting points of BX and of all +extreme points of BX, respectively. The set of preserved extreme points of BX (i.e. those x ∈ BX such +that JX(x) ∈ ext (BX∗∗)) is denoted by pre-ext (BX). For Banach spaces X and Y , L(X, Y ), F(X, Y ), +K(X, Y ) denote, respectively, the set of all (bounded linear) operator, the finite-rank operators, and +the compact operators. The properties in which we are interested only deal with the real structure of +the involved Banach spaces, but we do not restrict the study to real spaces in order to consider real or +complex examples. We will use the notation K to denote either R or C, Re(z) to denote the real part +of z (which is just the identity when dealing with a real space), and T to represent the set of scalars +of modulus one. +Given a non-empty subset C of X, we will denote by co(C) the convex hull of C and by span(C) +the linear hull of C. Also we denote by co(C) (respectively, span(C)) the norm closure of the convex +hull (respectively, of the linear hull) of C. By a slice of C we will mean any subset of C of the form +S(x∗, δ; C) := {x ∈ C : Re x∗(x) > M − δ} +where x∗ ∈ X∗ is a continuous linear functional on X, δ > 0 is a positive real number, and M := +supx∈C Re x∗(x). +For slices of the unit ball we will simply write S(x∗, δ) := S(x∗, δ; BX). +By a +relatively weakly open subset of C we mean as usual any subset of C obtained as the (non-empty) +intersection of C with an open set of X in the weak topology. +If C is assumed to be convex we will mean by a convex combination of slices of C (ccs of C in +short) any subset of C of the form +�n +i=1 λiSi, +where λ1, . . . , λn ∈ (0, 1] are such that �n +i=1 λi = 1 and Si is a slice of C for every i ∈ {1, . . . , n}. +Observe that convex combinations of slices are convex sets. We define in the same way convex com- +binations of relatively weakly open subsets of C (ccw of C in short). +The following lemma from [30] is a very useful tool when working with ∆-points. +Lemma 2.1 ([30, Lemma 2.1]). Let X be a Banach space, and let x∗ ∈ SX∗ and α > 0. For every +x ∈ S(x∗, α) and every 0 < β < α there exists y∗ ∈ SX∗ such that +x ∈ S(y∗, β) ⊆ S(x∗, α). + +6 +MART´IN, PERREAU, AND RUEDA ZOCA +We also often rely on the following result, due to Bourgain, and that we already mentioned in the +introduction. We provide a proof below, following the one from [25, Lemma II.1], for the sake of +completeness and for further discussions. +Lemma 2.2 (Bourgain). Let X be a Banach space and let C be a bounded convex closed subset of X. +Then, every non-empty relatively weakly open subset W of C contains a convex combination of slices +of C. +Proof. Assume with no loss of generality that W := +m� +i=1 +S(fi, αi, C), write �C = JX(C) +w∗ +⊂ X∗∗, and +W ∗∗ := +m +� +i=1 +S +� +JX∗(fi), αi; �C +� +, +which is a non-empty relatively weak∗ open subset of �C. By the Krein-Milman theorem (see e.g. [24, +Theorem 3.37]), it follows that �C = co(ext (BX∗∗)) +w∗ +, so co(ext (BX∗∗)) ∩ W ∗∗ ̸= ∅. Pick a convex +combination of extreme points �n +i=1 λie∗∗ +i +contained in W ∗∗. By the continuity of the sum we can find, +for every 1 ⩽ i ⩽ n, a weak-star open subset W ��∗ +i +with e∗∗ +i +∈ W ∗∗ +i +and such that �n +i=1 λiW ∗∗ +i +⊂ W ∗∗. +Now, since each e∗∗ +i +is an extreme point of �C, we have by Choquet’s lemma (see [24, Lemma 3.40], +for instance) that there are weak-star slices S +� +JX∗(gi), βi; �C +� +with e∗∗ +i +∈ S +� +JX∗(gi), βi; �C +� +⊆ W ∗∗ +i +for +every i ∈ {1, . . . , n}. Henceforth, �n +i=1 λiS +� +JX∗(gi), βi; �C +� +⊆ �n +i=1 λiW ∗∗ +i +⊆ W ∗∗. Now, if we take +U := +n +� +i=1 +λiS(gi, βi, C) +it is not difficult to prove that U ⊆ W, as desired. +□ +Remark 2.3. Observe that, in general, it is unclear from the above proof whether or not, if we fix +x ∈ W, we can guarantee that there exists a convex combination of slices U of C such that x ∈ U ⊆ W. +On the other hand, the result holds true if x ∈ W ∩ co(pre-ext (C)) in view of the above proof. +Indeed, if such situation, if we write x = �n +i=1 λixi ∈ W satisfying that x1, . . . , xn ∈ pre-ext (C) +and λ1, . . . , λn ∈ (0, 1] with �n +i=1 λi = 1, by the weak continuity of the sum, we can find, for every +1 ⩽ i ⩽ n, a non-empty relatively weakly open subset Vi with xi ∈ Vi for every i and such that +x = �n +i=1 λixi ∈ �n +i=1 λiVi ⊆ W. Now, observe that, since each xi is a preserved extreme point of C, +slices of C containing xi are a neighbourhood basis for xi in the weak topology. Hence, we can find, +for 1 ⩽ i ⩽ n, a slice Si of C with xi ∈ Si ⊆ Vi, and so x = �n +i=1 λixi ∈ �n +i=1 λiSi ⊆ �n +i=1 λiVi ⊆ W, +so U := �n +i=1 λiSi is the desired convex combination of slices. +Throughout the text, we will often be discussing various “diameter 2 properties”. +We use the +notation introduced in [7]. A Banach space X has the local or slice diameter 2 property (LD2P) if +every slice of BX has diameter 2; X has the diameter two property (D2P) if every non-empty relatively +weakly open subset of BX has diameter 2; finally, X has the strong diameter 2 property (SD2P) +whenever every ccs of BX has diameter 2 (and then, every ccw has diameter 2 due to Lemma 2.2). +For definitions and for examples concerning those properties, we refer to [2, 14, 15, 45]. In particular, +let us comment that the three properties are different, a result which was not easy to show, see [14]. +Our paper is closely related to the diametral versions of those properties which have been implicitly +studied for a long time in the literature, but whose formal definitions and names where fixed in [13]. +A Banach space X has the diametral local diameter 2 property (DLD2P) if for every slice S of BX +and every x ∈ S ∩ SX, supy∈S ∥x − y∥ = 2; if slices are replaced by non-empty relatively weakly + +DIAMETRAL NOTIONS FOR ELEMENTS OF THE UNIT BALL OF A BANACH SPACE +7 +open subsets of BX, we obtain the diametral diameter 2 property (DD2P). It is immediate that these +properties are not satisfied by any finite-dimensional space. Clearly, DLD2P implies LD2P, DD2P +implies D2P (and none of these implications reverses, e.g. X = c0), and DD2P implies DLD2P. It is +unknown whether the DLD2P and the DD2P are equivalent; in fact it is even unknown whether the +DLD2P implies the D2P. For the analogous definition using ccs, we have to discuss a little bit. Even +for an infinite-dimensional space X, it is not true that every ccs of BX intersects SX; actually, this +happens if and only if X has a property stronger than the SD2P (see [41, Theorem 3.4]). Thus, the +definition of the diametral strong diameter 2 property (DSD2P) given in [13] deals with all points in +BX as follows: for every ccs C and every x ∈ C, supy∈C ∥x − y∥ = ∥x∥ + 1. This definition allows to +show that DSD2P implies the SD2P. But, actually, it has been recently shown by V. Kadets [34] that +the DSD2P is equivalent to the Daugavet property. We will discuss this in detail in Section 5. On the +other hand, we will use the following property which is weaker than the DSD2P: a Banach space X +has the restricted DSD2P if for every ccs C and every x ∈ C ∩ SX, supy∈C ∥x − y∥ = 2. This property +is strictly weaker than the DSD2P, see Paragraph 4.3.2. +Let us now introduce all the notions of diametral points that we will consider in the text. Let us +start with the more closely related ones to the definitions above. +Definition 2.4. Let X be a Banach space and let x ∈ SX. We say that +(1) [3] x is a ∆-point if supy∈S ∥x − y∥ = 2 for every slice S of BX containing x, +(2) x is a super ∆-point if supy∈V ∥x − y∥ = 2 for every non-empty relatively weakly open subset +V of BX containing x, +(3) x is a ccs ∆-point if supy∈C ∥x − y∥ = 2 for every slice ccs C of BX containing x. +∆-points were introduced in [3] as a natural localization of the DLD2P (i.e. X has the DLD2P if +and only if every element of SX is a ∆-point). The other two definitions are new. Clearly, super +∆-points are the natural localization of the DD2P: X has the DD2P if and only if every element of +SX is a super ∆-point. Besides, ccs ∆-points are the localization of the restricted DSD2P: X has the +restricted DSD2P if and only if every element of SX is a ccs ∆-point. +In relation with the Daugavet property, we have the following notions for points. +Definition 2.5. Let X be a Banach space and let x ∈ SX. We say that +(1) [3] x is a Daugavet point if supy∈S ∥x − y∥ = 2 for every slice S of BX, +(2) x is a super Daugavet point if supy∈V ∥x − y∥ = 2 for every non-empty relatively weakly open +subset V of BX, +(3) x is a ccs Daugavet point if supy∈C ∥x − y∥ = 2 for every ccs C of BX. +Let us recall that Daugavet points were introduced in [3] as a natural localization of the Daugavet +property in the sense that a Banach space X has the Daugavet property if and only if every point in SX +is a Daugavet point ([35, Lemma 2.1]). From the geometric characterization given in [50, Lemma 3] +and the implicit result contained in its proof, it follows that super Daugavet points as well as ccs +Daugavet points are also natural localizations of the Daugavet property. +Since every slice of BX is relatively weakly open, and since by Bourgain’s lemma (see Lemma 2.2) +every non-empty relatively weakly open subset of BX contains a ccs of BX, we clearly have the diagram +of Figure 1. +We will show throughout the text that none of the above implications reverses, see Subsection 4.7 +for a description of all the relations and the counterexamples. However, let us point out right away + +8 +MART´IN, PERREAU, AND RUEDA ZOCA +ccs Daugavet +ccs ∆ +super Daugavet +super ∆ +∆ +Daugavet +Figure 1. Relations between the diametral notions +that we do not know whether there exists ccs ∆-points which are not super ∆. In view of Remark 2.3 +such examples may exist since Bourgain’s lemma is not localizable. Also let us point out that it follows +again from Bourgain’s lemma that a ccs Daugavet point x ∈ SX also satisfies supy∈D ∥x − y∥ = 2 for +every ccw D of BX. Again this is not clear for ccs ∆-points and we could thus naturally distinguish +between ccs ∆-points and “ccw ∆-points”. Since we do not have concrete examples at hand, we will +focus on convex combination of slices and specifically point out any available ccw behavior throughout +the text. +Let us also comment that it is clear that if every ccs of the unit ball of a given Banach space is weakly +open (respectively, has non-empty relative weak interior), then every super ∆-point (respectively, every +super Daugavet point) in this space is a ccs ∆-point (respectively, a ccs Daugavet point). Several +properties of this kind where introduced and studied in [1], [4], and [41]. We refer to those papers for +some background and for examples. +Remark 2.6. There are natural weak∗ versions in dual spaces of all the notions of diametral-points +introduced in the present section where slices and relatively weakly open subsets are respectively +replaced with weak∗ slices (i.e. slices defined by elements of the predual) and relatively weak∗ open +subsets. With obvious terminology, it then follows from [35, Lemma 2.1] and from [50, Lemma 3] that +a Banach space X has the Daugavet property if and only if every element in SX∗ is a weak∗ Daugavet +point if and only if every element in SX∗ is a weak∗ ccs Daugavet point. It also follows from [2, +Theorem 3.6] that X has the DLD2P if and only if every point in SX∗ is a weak∗ ∆-point. However, +the relationship between the DD2P in X and weak∗ super ∆-points in SX∗ is currently unknown. +Observe that a direct consequence of those results is that weak∗ diametral points and their weak +counterparts might differ in a very strong way since, for instance, the unit ball of the space C[0, 1]∗ +admits denting points. Yet clearly all the results from the following sections concerning the different +notions of diametral-points admit obvious analogues for their weak∗ counterparts. We leave the details +to the reader to avoid unnecessary repetitions, but let us still point out that it follows from Goldstine’s +theorem and from the lower weak∗ semicontinuity of the norm in dual spaces that there is a natural +correspondence between diametral-properties of points in SX and weak∗ properties of their image in +the bidual under the canonical embedding JX. Namely: +(1) x ∈ SX is a Daugavet point (respectively, a ccs Daugavet point) if and only if JX(x) is a weak∗ +Daugavet point (respectively, a weak∗ ccs Daugavet point). + +DIAMETRAL NOTIONS FOR ELEMENTS OF THE UNIT BALL OF A BANACH SPACE +9 +(2) x ∈ SX is a super Daugavet point if and only if JX(x) is a weak∗ super Daugavet point if +and only if for every y ∈ BX there exists a net (y∗∗ +s ) in BX∗∗ which converges to JX(y) in the +weak∗ topology and such that ∥πX(x) − y∗∗ +s ∥ −→ 2 (see Section 3). +(3) x ∈ SX is a ∆-point (respectively, a super ∆-point) if and only if JX(x) is a ∆-point (respec- +tively, a super ∆-point). +(4) x ∈ SX is a ccs ∆-point if and only if JX(x) is a weak∗ ccs ∆-point. +Let us point out that (3) essentially follows from the obvious fact that ∆-points and super ∆-points +naturally pass to superspaces, that is if Y is a subspace of X and if x ∈ SY if a ∆-point (respectively, +a super ∆-point) in Y , then x is a ∆-point (respectively, a super ∆-point) in X . This property is +unclear for ccs ∆-points, so the assertion (4) is not analogous to assertion (3). +3. Characterisations of diametral-notions and implications on the geometry of the +ambient space +In view of the definitions of diametral-points, it is natural to expect that the presence of any kind +of Daugavet- or ∆-element in a given Banach space will affect, by the severe restrictions it inflicts +on the nature of the considered point, its global isometric geometry or even its topological structure. +However, previous studies in the context have shown that the situation is much more complicated +than one could expect at first sight. +For example, let us comment that a Banach space X with +the RNP and admitting a Daugavet point, and a Banach space with a one-unconditional basis and +admitting a weakly dense subset of Daugavet points, were respectively constructed in [51] and in [6]. +In this section, we provide useful characterizations of the new diametral notions, and investigate the +immediate effect of the presence of such points on the geometry of the considered space. +We start by an intuitive but not completely trivial observation. +Observation 3.1. By definition, it is clear that super ∆-points do not exist in finite dimensional +spaces because the weak and norm topology coincide in this context. +Also, it was proved in [5, +Theorem 4.4] that finite dimensional spaces do also fail to contain ∆-points (hence ccs ∆-points). +In fact they fail to contain them in a stronger way, see [5, Corollary 6.10]. Consequently, the study +of diametral-notions only makes sense in infinite dimension, and from now on we will assume unless +otherwise stated that all the Banach spaces we consider are infinite dimensional. +Let us next prove a bunch of characterisations for super Daugavet- and super ∆-points. +Let X be a Banach space. For every x ∈ SX and for every ε > 0, let us define +∆ε(x) := {y ∈ BX : ∥x − y∥ > 2 − ε}. +We recall the following characterization of Daugavet- and ∆-points from [3]. +Lemma 3.2 ([3, Lemma 2.1 and 2.2]). Let X be a Banach space. +(1) An element x ∈ SX is a Daugavet point if and only if BX = co ∆ε(x) for every ε > 0. +(2) An element x ∈ SX is a ∆-point if and only if x ∈ co ∆ε(x) for every ε > 0. +We have similar characterisations for super points. +Lemma 3.3. Let X be a Banach space. +(1) An element x ∈ SX is a super Daugavet point if and only if BX = ∆ε(x) +w for every ε > 0. +(2) An element x ∈ SX is a super ∆-point if and only if x ∈ ∆ε(x) +w for every ε > 0. + +10 +MART´IN, PERREAU, AND RUEDA ZOCA +Proof. Observe that for given x ∈ SX, y ∈ BX, and ε > 0, we have that y belongs to the weak +closure of the set ∆ε(x) if and only if ∆ε(x) has non-empty intersection with any neighborhood of +y in the relative weak topology of BX. Thus y belongs to ∆ε(x) +w for every ε > 0 if and only if +supz∈V ∥x − z∥ = 2 for every relatively weakly open subset V of BX containing y. The conclusion +easily follows. +□ +For any given x ∈ BX, we denote by V(x) the set of all neighborhoods of x for the relative weak +topology of BX. We can provide characterizations of super points using nets which is just a localization +of [13, Proposition 2.5]. +Proposition 3.4. Let X be an infinite-dimensional Banach space. +(1) An element x ∈ SX is a super Daugavet point if and only if for every y ∈ BX there exists a +net (ys) in BX which converges weakly to y and such that ∥x − ys∥ −→ 2. +(2) An element x ∈ SX is a super ∆-point if and only if there exists a net (xs) in BX which +converges weakly to x and such that ∥x − xs∥ −→ 2. +In both cases we can moreover force the nets to be in SX. +Proof. Let us fix x ∈ SX. Given any y ∈ BX, it is clear that if there exists a net (ys) in BX which +converges weakly to y and such that ∥x − ys∥ −→ 2, then y belongs to the weak closure of ∆ε(x) for +every ε > 0. Conversely let us pick y ∈ BX satisfying this property. We turn S := V(y) × (0, ∞) into +a directed set by (V, ε) ⩽ (V ′, ε′) if and only if V ′ ⊂ V and ε′ ⩽ ε. By the assumptions we have that +V ∩ ∆ε(x) is a non-empty subset of BX for every couple s := (V, ε) in S. Picking any ys in this set +will then provide the desired net. +Finally observe that for x ∈ BX and ε > 0, we have that BX\∆ε(x) = {y ∈ BX : ∥x − y∥ ⩽ 2 − ε} +is weakly closed by the lower semi-continuity of the norm, so that ∆ε(x) is a relatively weakly open +subset of BX. Thus we have that V ∩ ∆ε(x) is a non-empty relatively weakly open subset of BX for +every couple s := (V, ε) in S. Since X is infinite dimensional, this set has to intersect SX, and we can +actually pick ys in V ∩ ∆ε(x) ∩ SX. +□ +Remark 3.5. In [36] an example of a Banach space satisfying simultaneously the Daugavet property +and the Schur property was provided. Such example shows that there is no hope to get a version of +the above result involving sequences. +Observe that the following result, similar to [32, Lemma 2.1 and 2.2], is included in the preceding +proof. +Proposition 3.6. Let X be a Banach space and let x ∈ SX. +(1) If x is a super Daugavet point, then for every ε > 0 and every non-empty relatively weakly +open subset V of BX we can find a non-empty relatively weakly open subset U of BX which is +contained in V and such that ∥x − y∥ > 2 − ε for every y ∈ U. +(2) If x is a super ∆-point, then for every ε > 0 and every non-empty relatively weakly open subset +V of BX containing x we can find a non-empty relatively weakly open subset U of BX which +is contained in V and such that ∥x − y∥ > 2 − ε for every y ∈ U. +Proof. Fix any x ∈ SX and any y ∈ BX which belongs to the weak closure of ∆ε(x) for every ε > 0. +Then, for every V ∈ V(y) and every ε > 0, we have that U := V ∩ ∆ε(x) is a non-empty relatively +weakly open subset of BX. +□ + +DIAMETRAL NOTIONS FOR ELEMENTS OF THE UNIT BALL OF A BANACH SPACE +11 +It is clear from the definition that denting points of BX cannot be ∆-points. Also it was first +observed in [32, Proposition 3.1] that every Daugavet point in a Banach space X has to be at distance +2 from every denting point of the unit ball of X. This elementary observation turned out to play +an important role in the study of Daugavet points in Lipschitz-free spaces in [32] and [51]. We have +similar observations for super points. +Lemma 3.7. Let X be a Banach space and let x ∈ SX. If x is a super ∆-point, then x cannot be +a point of continuity. If, moreover, x is a super Daugavet point, then x has to be at distance 2 from +every point of continuity of BX. +Proof. If and element y of BX is a point of continuity, then it is contained in relatively weakly open +subsets of BX of arbitrarily small diameter. Clearly no super ∆-point can have this property, and any +super Daugavet point has to be at distance 2 from any such points. +□ +This lemma provides quite a few examples of Banach spaces which fail to contain super points. +Following [26] let us recall that X has the Kadets property if the norm topology and the weak topology +coincide on SX, and that X has the Kadets-Klee property if weakly convergent sequences in SX are +norm convergent. Let us also recall that any LUR space has the Kadets-Klee property, and that any +space with the Kadets-Klee property which fails to contain ℓ1 has the Kadets property. By proposition +3.4 we clearly have the following result. +Proposition 3.8. If X has the Kadets property, then X fails to contain super ∆-points. +As a corollary we obtain the following. Recall that a Banach space is asymptotic uniformly convex +(AUC in short) [31] if its modulus of asymptotic uniform convexity +δX(t) := inf +x∈SX +sup +dim X/Y <∞ +inf +y∈SY ∥x + ty∥ − 1 +is strictly positive for every t > 0. +Corollary 3.9. Let X be AUC. Then, X fails to contain super ∆-points. +Proof. In an AUC space, every element of the unit sphere is a point of continuity of BX, see [31, +Proposition 2.6]. +□ +Remark 3.10. It was proved in [5, Theorem 3.4] that any reflexive AUC space fails to contain ∆- +points. Also, combining the observations from [5, End of Section 4] about weak∗ quasi-denting points +in the unit ball of AUC∗ duals and [52, Corollary 2.4] about the maximality of the Kuratowski index +of weak∗ slices containing weak∗ ∆-points, we have that every AUC∗ dual space fails to contain weak∗ +∆-points. However, note that it is currently unknown whether non-reflexive AUC spaces (and, in +particular, whether the dual of the James tree spaces JT∗) may contain Daugavet- or ∆-points. +It turns out that Daugavet points are characterized by this distance to denting points in RNP +spaces (because the unit ball of an RNP space X can be written as the closed convex hull of the set +of its denting points) as well as in Lipschitz-free spaces ([32, Theorem 3.2] for compact metric spaces +and [51, Theorem 2.1] for a general statement). In the same way we can characterize super Daugavet +points in terms of this distance to points of continuity of BX is spaces with the CPCP. +Proposition 3.11. If a Banach space X has the CPCP, then a point x ∈ SX is a super Daugavet +point if and only if it is at distance 2 from any point of continuity of BX. + +12 +MART´IN, PERREAU, AND RUEDA ZOCA +Proof. If X has the CPCP, then the set of all points of continuity of BX is weakly dense in BX (see for +example [22, Proposition 3.9]), that is, every non-empty relatively weakly open subset of BX contains +a point of continuity. The conclusion follows easily. +□ +For ccs points, the situation is quite different. Indeed, although ccs ∆-points can clearly not be +points of strong regularity, we have by [41, Theorem 3.1] that X has the SD2P if and only if every +convex combination of slices of BX contains elements of norm arbitrarily close to 1. It readily follows +that any space X which contains a ccs Daugavet point satisfies the SD2P, so it is very far from being +strongly regular. We will provide more details on this topic in Section 5, but for later reference let us +state the following. +Proposition 3.12. Let X be a Banach space. If X contains a ccs Daugavet point, then it has the +SD2P (it fails to be strongly regular). +Next, we show that extreme points have a nice behaviour with respect to diametral notions. +Proposition 3.13. Let X be a Banach space and let x ∈ SX. +(1) If x ∈ pre-ext (BX) and it is a ∆-point, then x is a super ∆-point. +(2) If x ∈ ext (BX) and it is a super ∆-point, then x is a ccs ∆-point. +(3) In particular, if x ∈ pre-ext (BX) is a ∆-point, then x is a super ∆-point as well as a ccs +∆-point. +Proof. It follows from Choquet’s lemma (see for example [23, Lemma 3.69]) that slices form neighbor- +hood bases in the relative weak topology of the unit ball of a Banach space for its preserved extreme +points, so (1) immediately follows. For (2), if x is extreme and belongs to a ccs C := �n +i=1 λiSi of BX +then x ∈ �n +i=1 Si, which is a relatively weakly open subset of BX. +□ +Remark 3.14. Observe that, in fact, any extreme super ∆-point is “ccw ∆-point” as we discussed in +Section 2. Also, Choquet’s lemma implies that every extreme weak∗ ∆-point in a dual space is weak∗ +ccw ∆-point. +3.1. Spaces with a one-unconditional basis and beyond. In [6], it was proved that no real +Banach space with a subsymmetric basis contains a ∆-point. On the other hand, an example of a +Banach space with a one-unconditional basis that contains a ∆-point was provided, and a more involved +example of a Banach space with a one-unconditional basis that contains many Daugavet-points was +constructed. We will discuss this second example in detail in Subsection 4.3.3. +In the process, it was also implicitly shown that real Banach spaces with a one-unconditional basis +cannot contain super ∆-points. In the present subsection, we prove that the same goes for ccs ∆- +points. Also, we provide sharper and more general versions of [6, Proposition 2.12]. In the first part +of this section, we follow [6] and restrict ourselves to real Banach spaces. +Let X be a real Banach space with a Schauder basis (ei)i⩾1. We denote by (e∗ +i )i⩾1 the corresponding +sequence of biorthogonal functionals. +Recall that (ei)i⩾1 is said to be unconditional if the series +� +i⩾1 e∗ +i (x)ei converges unconditionally for every x ∈ X. +Also, recall that an unconditional basis +(ei)i⩾1 is said to be one-unconditional if +����� +� +i⩾1 +θie∗ +i (x)ei +����� = +����� +� +i⩾1 +e∗ +i (x)ei +����� + +DIAMETRAL NOTIONS FOR ELEMENTS OF THE UNIT BALL OF A BANACH SPACE +13 +for every (θi)i⩾1 ∈ {−1, 1}N and for every x ∈ X. Moreover, if +����� +� +i⩾1 +θie∗ +i (x)eni +����� = +����� +� +i⩾1 +e∗ +i (x)ei +����� +for every (θi)i⩾1 ∈ {−1, 1}N, for every x ∈ X, and for every strictly increasing sequence (ni)i⩾1 in N, +then the basis is called subsymmetric. +Observe that for spaces with a one-unconditional basis, it is enough, in order to study the various +Daugavet- and ∆-notions, to work in the positive sphere +S+ +X := {x ∈ SX : e∗ +i (x) ⩾ 0 ∀i} +of the space X. Also, the following result is well known. +Lemma 3.15. Let X be a real Banach space with a one-unconditional basis (ei)i⩾1, and let (ai)i⩾1 +and (bi)i⩾1 be sequences of real numbers. If the series � +i⩾1 biei converges, and if |ai| ⩽ |bi| for every +i, then � +i⩾1 aiei converges as well, and we have +����� +� +i⩾1 +aiei +����� ⩽ +����� +� +i⩾1 +biei +����� . +Let us now recall a few notation and preliminary results from [6]. Let X be a real Banach space +with a normalized one-unconditional basis (ei)i⩾1. For every subset A of N, we denote by PA the +projection on span{ei, i ∈ A}. Then for every x ∈ X, we define +M(x) := {A ⊂ N: ∥PA(x)∥ = ∥x∥ , and +��PA(x) − e∗ +j(x)ej +�� < ∥x∥ ∀j ∈ A}. +The set M(x) can be seen as the set of all minimal norm-giving subsets of the support of x. We denote +respectively by MF(x) and M∞(x) the subsets of all finite and infinite elements of M(x). It follows +from [6, Lemma 2.7] that the set M(x) is never empty, and from [6, Proposition 2.15] that no element +x ∈ SX satisfying M∞(x) = ∅ can be a ∆-point. +For every non-empty ordered subset A := {a1 < a2 < . . . } of N, and for every n ∈ N smaller than +or equal to |A|, we denote by A(n) := {a1, . . . , an} the subset consisting of the n first elements of A. +We will implicitly assume in the following that the elements of M(x) are ordered subsets of N. The +next two results were proved in [6, Lemma 2.8 and Lemma 2.11]. +Lemma 3.16. Let X be a real Banach space with a normalized one-unconditional basis (ei)i⩾1 and +let x ∈ SX. For every n ∈ N, the sets +� +A ∈ M(x): |A| ⩽ n +� +and +� +A(n): A ∈ M(x), and |A| > n +� +are both finite. +Lemma 3.17. Let X be a real Banach space with a normalized one-unconditional basis and let x ∈ SX. +For every subset E of N such that E ∩ A ̸= ∅ for every A ∈ M(x), we have ∥x − PE(x)∥ < 1. +With those tools at hand, we can now prove an analogue to [6, Proposition 2.13] for convex combi- +nation of slices. +Proposition 3.18. Let X be a Banach space with a normalized one-unconditional basis and x ∈ S+ +X. +Then, there exists δ > 0 and a ccs C of BX containing x such that supy∈C ∥x − y∥ ⩽ 2 − δ. + +14 +MART´IN, PERREAU, AND RUEDA ZOCA +Proof. Let x ∈ S+ +X, and define E = � +A∈M(x) A(1). From Lemma 3.16 and Lemma 3.17, we have +that E is a finite subset of N and that ∥x − PE(x)∥ < 1. In particular, there exists γ > 0 such that +∥x − PE(x)∥ ⩽ 1 − γ. For every i ∈ E, we define +Si := S +� +e∗ +i , 1 − e∗ +i (x) +2 +� +. +Then we consider the ccs +C := +1 +|E| +� +i∈E +Si. +Since x ∈ S+ +X, we clearly have that x ∈ � +i∈E Si and, in particular, that x ∈ C. +So let us pick y := +1 +|E| +� +i∈E yi in C. +Then we have e∗ +i (yi) > +e∗ +i (x) +2 +for every i. +In particular, +e∗ +i (yi) ⩾ 0, and +��e∗ +i (yi) − e∗ +i (x) +�� ⩽ e∗ +i (yi). Indeed, for any given non-negative real numbers α and β +with β ⩾ α +2 , we have +|β − α| = β − α ⩽ β +if β ⩾ α, and +|β − α| = α − β ⩽ α − α +2 = α +2 ⩽ β +if β ⩽ α. So in either case, |β − α| ⩽ β as desired. +It then follows from Lemma 3.15 that +��yi − e∗ +i (x)ei +�� ⩽ +��yi�� ⩽ 1 and, finally, +∥x − y∥ ⩽ +����x − x +|E| +���� + +���� +x +|E| − PE(x) +|E| +���� + +���� +PE(x) +|E| +− y +���� +⩽ 1 − 1 +|E| + 1 − γ +|E| ++ 1 +|E| +� +i∈E +��e∗ +i (x)ei − yi�� ⩽ 2 − γ +|E|. +The conclusion follows with δ := +γ +|E|. In particular, note that since x belongs to the relative weakly +open set � +i∈E Si ⊂ C, we also get that x is not super ∆, recovering the result from [6]. +□ +So combining [6, Proposition 2.13] and Proposition 3.18, we immediately get that spaces with a +normalized one-unconditional basis fail to contain super ∆-points and ccs ∆-points. So let us state +the following here for future reference. +Theorem 3.19. Let X be a real Banach space with a normalized one-unconditional basis. Then X +does not contain super ∆-points, and X does not contain ccs ∆-points. +In the rest of the subsection, we aim at providing sharper and improved versions of [6, Proposi- +tion 2.13]. In particular we will go back to working with either real or complex Banach spaces. The +main result of this study is the following proposition. +Proposition 3.20. Let X be a Banach space, and let us assume that there exists a subset A ⊆ F(X, X) +satisfying that sup +� +∥Id − T∥ : T ∈ A +� +< 2 and that for every ε > 0 and every x ∈ X, there exists +T ∈ A such that ∥x − Tx∥ < ε. Then, X contains no super ∆-point. +Let us provide a lemma which is a localization of the above result from which its proof is immediate. +Lemma 3.21. Let X be a Banach space, and let x ∈ SX. If there exists a finite-rank operator T on +X such that ∥x − Tx∥ + ∥Id − T∥ < 2, then x is not a super ∆-point. + +DIAMETRAL NOTIONS FOR ELEMENTS OF THE UNIT BALL OF A BANACH SPACE +15 +Proof. Consider ε > 0 such that K := ∥x − Tx∥ + ∥Id − T∥ + ε < 2. Since T has finite rank, we can +find N ⩾ 1, w1, . . . , wN ∈ SX and f1, . . . , fN ∈ X∗ such that T(z) = �N +n=1 fn(z)wn for every z ∈ X. +Let us consider +W := +� +y ∈ BX : |fn(x − y)| < +ε +2n+1 ∀n ∈ {1, . . . , N} +� +. +W is a neighborhood of x in the relative weak topology of BX, and for every y ∈ W, we have +∥x − y∥ ⩽ ∥x − Tx∥ + ∥Tx − Ty∥ + ∥y − Ty∥ +⩽ ∥x − Tx∥ + ∥Id − T∥ + +N +� +n=1 +|fn(x − y)| ∥wn∥ +⩽ ∥x − Tx∥ + ∥Id − T∥ + ε +N +� +n=1 +1 +2n+1 ⩽ K < 2. +□ +N.B. It is unclear whether an analogue to Lemma 3.21 can be given for ccs ∆-points. So we do not +know whether Proposition 3.20 extends to this notion. +As particular cases of Proposition 3.20, we have the following ones. Recall that a sequence (En)n⩾1 +of finite dimensional subspaces of a given Banach space X is called a finite dimensional decomposition +(FDD) for X if every element x ∈ X can be represented in a unique way as a series x := � +n⩾1 xn +with xn ∈ En for every n ⩾ 1. Such an FDD is said to be unconditional if the above series converges +unconditionally for every x ∈ X. In this case, it is well known that the family (PA)A⊂N, where PA is the +projection given by PA(x) := � +n∈A xn, is uniformly bounded, and the constant KS := supA⊂N ∥PA∥ is +called the suppression-unconditional constant of the FDD. We refer to [40, Section 1.g] for the details +and to [8, Section 3.1] for the particular case of unconditional bases. +Corollary 3.22. A Banach space X fails to have super ∆-points provided one of the following condi- +tions is satisfied. +(1) There exists a family A ⊆ F(X, X) satisfying that sup +� +∥Id − T∥ : T ∈ A +� +< 2 and that the +identity mapping belongs to its strong operator topology (SOT) closure. +(2) There exists a family {Pλ}λ∈Λ of finite rank projections on X such that X = � +λ∈Λ Pλ(X), +and such that supλ∈Λ ∥Id − Pλ∥ < 2. +(3) The space X admits a FDD with suppression-unconditional constant less than 2. In particular, +if X admits an unconditional basis with suppression-unconditional constant less than 2. +Let us observe that the value 2 in the above results is sharp in several ways. +Remark 3.23. +(1) The space C[0, 1] admits a monotone Schauder basis, so there exists a sequence +{Pn}n⩾1 of norm one finite rank projections on this space which converges to Id in SOT +topology. As C[0, 1] has the Daugavet property, all elements in SX are super Daugavet points. +Observe that ∥Id − Pn∥ = 2 for every n ⩾ 1 by the DPr. +(2) Let X be an arbitrary Banach space. For every x ∈ SX choose fx ∈ SX∗ such that fx(x) = 1, +and define Px(z) = fx(z)x for every z ∈ X. Then {Px : x ∈ SX} is a family of norm one +rank-one projections on X, X = � +x∈SX Px(X), and ∥Id − Px∥ ⩽ 2 for every x ∈ SX. +(3) The space c admits ccs Daugavet points (hence super Daugavet points), see Theorem 4.2, but +it is easy to check that its usual basis is 3-unconditional and 2-suppression unconditional. +(4) It is shown in [30] that a Banach space has the DLD2P if and only if ∥Id − P∥ ⩾ 2 for every +rank-one projection P. It follows that the suppression constant of an unconditional basis on +a Banach space with the DLD2P has to be greater than or equal to 2. Let us mention here + +16 +MART´IN, PERREAU, AND RUEDA ZOCA +that there is no local version of this result, as there are Banach spaces with one-unconditional +basis and containing many Daugavet points [6] (see Paragraph 4.3.3). +3.2. Absolute sums. In this subsection we look at the transfer of the diametral points through +absolute sums of Banach spaces. Let us first recall the following definition. +Definition 3.24. A norm N on R2 is absolute if N(a, b) = N(|a| , |b|) for every (a, b) ∈ R2 and +normalized if N(0, 1) = N(1, 0) = 1. +If X and Y are Banach spaces, and if N is an absolute normalized norm on R2, we denote by +X ⊕N Y the product space X × Y endowed with the norm ∥(x, y)∥ = N(∥x∥ , ∥y∥). It is easy to check +that X ⊕N Y is a Banach space, and that its dual can be expressed as (X ⊕N Y )∗ ≡ X∗ ⊕N∗ Y ∗ where +N∗ is the absolute norm given by the formula N∗(c, d) = maxN(a,b)=1 |ac|+|bd|. Classical examples of +absolute normalized norms on R2 are the ℓp norms for p ∈ [1, ∞]. Information on absolute norms can +be found in [16, §21] and [43] and references therein, for instance. Let us recall that for every absolute +normalized sum N, given non-negative a, b, c, d in R with a ⩽ b and c ⩽ d we have N(a, b) ⩽ N(c, d). +In particular, ∥·∥∞ ⩽ N ⩽ ∥·∥1. +Similar to the DD2P (see [13, Theorem 2.11]) and to ∆-points [28], super ∆-points transfer very +well through absolute sums. +Proposition 3.25. Let X and Y be Banach spaces, and let N be an absolute normalized norm. +(1) If x ∈ SX and y ∈ SY are super ∆-points, then (ax, by) is a super ∆-point in X ⊕N Y for +every (a, b) ∈ R2 with N(a, b) = 1. +(2) If x ∈ SX is a super ∆-point, then (x, 0) is a super ∆-point in X ⊕N Y . If y ∈ SY is a super +∆-point, then (0, y) is a super ∆-point in X ⊕N Y . +Proof. (1). We can find two nets (xs)s∈S and (yt)t∈T respectively in SX and SY such that xs +w +−→ x, +yt +w +−→ y, and ∥x − xs∥ , ∥y − yt∥ −→ 2. Now, if we take (a, b) ∈ R2 with N(a, b) = 1 we clearly have +(axs, byt) +w +−→ +(s,t)∈S×T (ax, by) and ∥(ax, by) − (axs, byt)∥ = N (a ∥x − xs∥ , b ∥y − yt∥) −→ 2N(a, b) = 2, +so (ax, by) is a super ∆-point in X ⊕N Y . For (2), we just repeat the previous proof with a = 1 and +b = 0 or with a = 0 and b = 1 and so we only need one of the points to be super ∆-point. +□ +For super Daugavet points the situation is more complicated and we need to distinguish between +different kinds of absolute norms. The following definitions can be found, for instance, in [28]. +Definition 3.26. Let N be an absolute normalized norm on R2. +(1) N has property (α) if for every a, b ∈ R+ with N(a, b) = 1 we can find a neighborhood W of +(a, b) in R2 with sup(c,d)∈W c < 1 or sup(c,d)∈W d < 1 and such that any couple (c, d) ∈ R2 ++ +satisfying N(c, d) = 1 and N ((a, b) + (c, d)) = 2 belongs to W. +(2) N is A-octahedral if there exists a, b ∈ R+ such that N(a, b) = 1 and N ((a, b) + (c, d)) = 2 for +c = max{e ∈ R+ : N(e, 1) = 1} +and +d = max{f ∈ R+ : N(1, f) = 1}. +(3) N is positively octahedral if there exists a, b ∈ R+ such that N(a, b) = 1 and +N ((a, b) + (0, 1)) = N ((a, b) + (1, 0)) = 2. +Positively octahedral norms where introduced in [27] in order to characterize the absolute norms for +which the corresponding absolute sum is octahedral. It is clear that property (α) and A-octhaedrality + +DIAMETRAL NOTIONS FOR ELEMENTS OF THE UNIT BALL OF A BANACH SPACE +17 +exclude each other and that every positively octahedral absolute normalized norm is A-octahedral +(while there clearly exists absolute A-octahedral norms which are not positively octahedral). Moreover +it was proved in [28, Proposition 2.5] that every absolute normalized norm on R2 must either satisfy +property (α) or be A-octahedral. +For ℓp-norms, we have that ∥·∥1 and ∥·∥∞ are both positively +octahedral, and that ∥·∥p satisfies property (α) for every p ∈ (1, ∞). +Observe that if an absolute normalized norm N on R2 is positively octahedral, and if (a, b) is as in +the above definition, then the intersection of the unit sphere of N with the positive quadrant of R2 is +equal to the union of the segments [(1, 0), (a, b)] and [(0, 1), (a, b)] (see [45, Section 3.3.1] for pictures). +In particular, it follows that N((a, b)+(c, d)) = 2 for every non-negative c, d with N(c, d) = 1. Similar +to the results from [3, Section 4] concerning Daugavet points, we have the following. +Proposition 3.27. Let X and Y be Banach spaces, and let N be an absolute normalized norm. +(1) [3, Proposition 4.6] If N has property (α), then X ⊕N Y has no Daugavet point (hence, in +particular, no super Daugavet points). +(2) If N is positively octahedral and if x ∈ SX and y ∈ SY are super Daugavet points, then (ax, by) +is a super Daugavet point in X ⊕N Y for every (a, b) ∈ R2 ++ as in the above definition. +Proof. (2). Assume that N is positively octahedral, take (a, b) ∈ R2 ++ as in the definition, and let +x ∈ SX and y ∈ SY be super Daugavet points. For any given (u, v) ∈ X ⊕N Y of norm ∥(u, v)∥ = 1 we +can find two nets (us)s∈S and (vt)t∈T respectively in SX and SY such that ∥u∥us +w +−→ u, ∥v∥vt +w +−→ v, +and ∥x − us∥ , ∥y − vt∥ −→ 2. Then (∥u∥ us, ∥v∥ vt) +w +−→ +(s,t)∈S×T (u, v). Since +∥ax − ∥u∥ us∥ = ∥(x − us) − [(1 − a)x − (1 − ∥u∥)us]∥ +⩾ ∥x − us∥ − (1 − a + 1 − ∥u∥), += a + ∥u∥ − (2 − ∥x − us∥), +and, in the same way, +∥by − ∥v∥ vt∥ ⩾ b + ∥v∥ − (2 − ∥y − vt∥), +we have +∥(ax − ∥u∥ us, by − ∥v∥ vt)∥ = N (∥ax − ∥u∥ us∥ , ∥by − ∥v∥ vt∥) +⩾ N (a + ∥u∥ − (2 − ∥x − us∥), b + ∥v∥ − (2 − ∥y − vt∥)) +−→ N ((a + ∥u∥ , b + ∥v∥) = 2. +This shows that (ax, by) is a super Daugavet point in X ⊕N Y . +□ +Remark 3.28. Note that if (a, b) = (1, 0) (respectively, (a, b) = (0, 1)) in the previous statement (for +example, when N = ∥·∥1), then we only need to assume that x (respectively, y) is super Daugavet +in order to get that (x, 0) (respectively, (0, y)) is super Daugavet in X ⊕N Y . Also, if N = ∥·∥∞, +then we only need to assume that x (respectively, y) is super Daugavet in order to obtain that (x, βy) +(respectively, (αx, y)) is super Daugavet in X ⊕N Y for every β ∈ [0, 1] (respectively, α ∈ [0, 1]). +In [28, Theorem 2.2] it is proved that regular Daugavet points do also transfer through A-octahedral +sums. We do not know if a similar result can be obtained for super Daugavet points. Indeed, observe +that if N is an A-octahedral norm, and if c, d, and (a, b) are as in the above definition, then the +intersection of the unit sphere of N with the positive quadrant of R2 is equal to the union of the +segments [(1, 0), (1, d)], [(1, d), (a, b)], [(0, 1), (c, 1)] and [(c, 1), (a, b)]. In particular, N((a, b)+(e, f)) = +2 for every couple (e, f) on the segments [(1, d), (a, b)] and [(c, 1), (a, b)], but this is no longer true + +18 +MART´IN, PERREAU, AND RUEDA ZOCA +on the segments [(1, 0), (1, d)] and [(0, 1), (c, 1)] and the argument in the above proof does not work +anymore. +The situation for ccs ∆-points and ccs Daugavet point is not clear and the proofs of the above results +do not seem to admit easy extensions. For instance, it follows from the next result that Remark 3.28 +is not valid for ccs Daugavet points. +Proposition 3.29. Let X be an arbitrary Banach space, let Y be a Banach space containing an +strongly exposed point y0 ∈ SY , and let E := X ⊕1 Y . Then, there are convex combinations of slices of +BE around 0 of arbitrarily small diameter. In particular, E fails to contain ccs Daugavet points and +also fails to have the SD2P. +Proof. Let y∗ +0 ∈ SY ∗ strongly exposes y0. Given ε > 0, there is 0 < δ < ε such that ∥y − y0∥ < ε +whenever y ∈ BY satisfies Re y∗ +0(y) > 1 − δ. Consider f = (0, y∗ +0) ∈ SE∗ and write +C := 1 +2 (S(f, δ; BE) + S(−f, δ, BE)) +Take u := 1 +2(u1 + u2) ∈ C with u1 ∈ S(f, δ; BE) and u2 ∈ S(−f, δ; BE). So if write u1 := (x1, y1) and +u2 := (x2, y2), we have +Re y∗ +0(y1) = Re f(x1, y1) > 1 − δ and +Re y∗ +0(y2) = Re f(x2, y2) < −1 + δ. +On the one hand, it follows that ∥y1−y0∥ < ε and ∥y2+y0∥ < ε. On the other hand, ∥y1∥, ∥y2∥ > 1−δ, +hence ∥x1∥ < δ < ε and ∥x2∥ < δ < ε. Summarizing, we have +∥u∥ = 1 +2 +� +∥x1 + x2∥ + ∥y1 + y2∥ +� +⩽ 1 +2(2ε + 2ε) = 2ε. +□ +Remark 3.30. It is straightforward to adapt the previous proof to ℓp-sums for 1 < p < ∞. +However, note that the situation is very different for ℓ∞-sums. +Theorem 3.31. Let X and Y be Banach spaces, and let E := X ⊕∞ Y . If x ∈ SX is a ccs Daugavet +point, then (x, y) ∈ SE is a ccs Daugavet point for every y ∈ BY . +Proof. Let C := �n +i=1 λiSi be a ccs of BE. For every i ∈ {1, . . . , n}, we can write Si := S(fi, δi) with +fi := (x∗ +i , y∗ +i ) ∈ SE∗ satisfying 1 = ∥fi∥ = ∥x∗ +i ∥ + ∥y∗ +i ∥. Consider on the one side +˜Si := +� +s ∈ BX : Re x∗ +i (s) > ∥x∗ +i ∥ − δi +2 +� +, +and pick on the other side any ti ∈ BY such that Re y∗ +i (ti) > ∥y∗ +i ∥ − δi +2 . Since ˜C := �n +i=1 λi ˜Si is a ccs +of BX, we can find for every ε > 0 an element s := �n +i=1 λisi in ˜C such that ∥x − s∥ > 2 − ε. Then, +if we let t := �n +i=1 λiti, we get (si, ti) ∈ BE and +Re fi(si, ti) = Re x∗ +i (si) + y∗ +i (ti) > ∥x∗ +i ∥ + ∥y∗ +i ∥ − δi = 1 − δi +for every i, so that (si, ti) ∈ Si, and (s, t) = �n +i=1 λi(si, ti) ∈ C. Finally, +∥(x, y) − (s, t)∥ ⩾ ∥x − s∥ > 2 − ε, +so (x, y) is a ccs Daugavet point as stated. +□ + +DIAMETRAL NOTIONS FOR ELEMENTS OF THE UNIT BALL OF A BANACH SPACE +19 +4. Examples and counterexamples of diametral elements +In this section we aim to include a number of examples and counterexamples of diametral elements +on the unit sphere of Banach spaces. We first characterize the notion in some spaces which have +natural relations with the Daugavet property, such as L1-preduals spaces, M¨untz spaces, and L1- +spaces. +Next, we will remark on some examples which have previously appear in the literature, +including some improvements in some cases (as for Lipschitz free spaces). Finally, we will include +some complicated examples which will be needed to see that no implication in Figure 1 in page 8 +reverses and also to negate some other possible implications between the notions. A summary of all +the relations between properties will be included in Subsection 4.7. +4.1. Characterization in C(K)-spaces, L1-preduals, and M¨untz spaces. It was shown in +[3, Theorems 3.4 and 3.7] that the notions of ∆-point and Daugavet point coincide for L1-preduals. +The authors first characterize the ∆-points in C(K) spaces and then get the result for L1-preduals by +using the principle of local reflexivity. Later on, a characterization of ∆-points (equivalently, Daugavet +points) of L1-preduals was provided in [42, Theorem 3.2] which implicitly prove that actually ∆-points, +and super Daugavet points coincides in this setting. Let us state this result here for further reference. +Let us observe that the authors of [3] works with real Banach spaces, but it is immediate that the +proof of [3, Theorems 3.4] works in the complex case as well; the paper [42] works in both the real +and the complex case. +Proposition 4.1 ([3, Theorems 3.4 and 3.7], [42, Theorem 3.2]). Let X be an L1-predual and let x ∈ +SX. The following assertions are equivalent. +(1) x is a Daugavet point. +(2) x is a ∆-point +(3) For every δ > 0, the weak∗ slice S(JX(x), δ; BX∗) contain infinitely many pairwise linearly +independent extreme points of BX∗. +(4) For every element y ∈ BX, there exists a sequence (x∗∗ +n ) in BX∗∗ such that ∥x − x∗∗ +n ∥ −→ 2 +and +����� +� +n⩾1 +an(y − x∗∗ +n ) +����� ⩽ 2 ∥a∥∞ +for every a := (an) ∈ c00. +(5) For every element y ∈ BX, there exists a sequence (x∗∗ +n ) in BX∗∗ which converges weak∗ to y +and such that ∥x − x∗∗ +n ∥ −→ 2. +In the case that X = C(K) for a Hausdorff topological space K, the above is also equivalent to: +(6) x attains its norm at an accumulation point of K. +We will show that, in fact, ∆-points also coincide with the ccs versions for L1 preduals. +Our +approach will be analogous to the one used in [3] for ∆-points and Daugavet points: we first prove the +result for C(K) spaces and then deduce it for all L1-preduals using that the bidual of an L1-predual +is a C(K)-space. In the case of C(K) spaces, we first prove a sufficient condition for ccs Daugavet +points which, for the same price, can be proved for vector-valued spaces. Recall that given a compact +Hausdorff topological space K and a Banach space X, C(K, X) denotes the Banach space of those +continuous functions from K to X endowed with the supremum norm. + +20 +MART´IN, PERREAU, AND RUEDA ZOCA +Theorem 4.2. Let K be a compact Hausdorff topological space, X a Banach space, and let t0 be an +accumulation point of K. If a function f ∈ SC(K,X) satisfies ∥f(t0)∥ = 1, then f is a ccs Daugavet +point. +Proof. Pick x∗ ∈ SX∗ such that Re x∗(f(t0)) = ∥f(t0)∥ = 1. +Let C := �L +i=1 λiSi be a convex +combination of slices of BC(K). For every i ∈ {1, . . . , L}, pick a function gi ∈ Si. Since K is compact +and t0 is an accumulation point of K we have the following. +Claim. There exists a sequence (Un)n⩾0 of open neighborhoods of t0 such that: +(1) U0 = K, +(2) Un+1 is a proper subset of Un for every n ⩾ 0, +(3) Re(x∗ ◦ f)|Un ⩾ 1 − 1 +n and +���gi|Un − gi(t0) +��� ⩽ 1 +n for every i ∈ {1, . . . , L} and every n ⩾ 1. +Indeed, we construct the sequence inductively. Let U0 := K and assume that U0, . . . , Un are con- +structed for some n ⩾ 0. +Since K is normal, we can find an open subset U of Un such that +t0 ∈ U ⊂ U ⊂ Un. +Also since t0 is an accumulation point of K and since K is Hausdorff, we +can find an open subset V of U such that V is a proper subset of U (pick any point in U distinct from +t0 and separate the two points with open sets). By continuity of f and of the finitely many g′ +is, we +can then find an open subset W of V such that +Re(x∗ ◦ f)|W > 1 − +1 +n + 1 +and +���gi|W − gi(t0) +��� < +1 +n + 1 +for every i ∈ {1, . . . , L}. The set Un+1 := W does the job. +Now, let us pick (Un)n⩾0 as in the claim and let us define Fn := Un\Un+1 for every n ⩾ 0. By +construction, the F ′ +ns are closed non-empty subsets of K and cover K\ +�� +n⩾0 Un +� +, and each Fn may +only intersects its neighbors Fn−1 and Fn+1. By Urysohn’s lemma, for every n ⩾ 1 we can find a +function pn ∈ C(K) satisfying: +(1) 0 ⩽ pn ⩽ 1, +(2) pn|Fn+1 = 1, +(3) pn|F0∪···∪Fn−1∪Un+3 = 0. +The sequence (pn) is normalized and converges pointwise to 0, so it converges weakly to 0. Moreover, +observe that +∥gi − (1 + gi(t0))pn∥∞ ⩽ 1 + 1 +n +for every i ∈ {1, . . . , L} since +���gi|Un − gi(t0) +��� ⩽ +1 +n and pn|(K\Un) = 0 by construction. So all the +functions +gi,n := +n +n + 1 (gi − (1 + gi(t0))pn) +belong to BC(K) and the sequences (gi,n)n∈N converges weakly to gi for every i ∈ {1, . . . , L}. Since +the finitely many S′ +is are all weakly open, we may thus find some N ⩾ 1 such that gi,n ∈ Si for every +i and every n ⩾ N. In particular, the function +gn := +L +� +i=1 +λigi,n + +DIAMETRAL NOTIONS FOR ELEMENTS OF THE UNIT BALL OF A BANACH SPACE +21 +belongs to C for every n ⩾ N. To conclude, fix t ∈ Fn+1 ⊂ Un+1 ⊂ Un and observe that +Re x∗f(t) ⩾ 1 − 1 +n +and that +Re x∗gi,n(t) = +n +n + 1 Re x∗ (gi(t) − (1 + gi(t0)) +⩽ +n +n + 1 Re x∗ +� +gi(t0) + 1 +n − (1 + gi(t0)) +� += −1 + +2 +n + 1 +for every i ∈ {1, . . . L}. Hence, +∥f − gn∥∞ ⩾ Re x∗ +� +f(t) − +L +� +i=1 +λi Re(gi,n(t)) +� +⩾ 2 − 1 +n − +2 +n + 1. +□ +Combining the previous result with Proposition 4.1, we get the promised characterization of diame- +tral points in C(K) spaces. +Corollary 4.3. Let K be a Hausdorff topological compact space. Then the six concepts of diametral +points are equivalent in C(K). +For vector-valued spaces, the situation is not that easy, but we may provide with some results. +Observe that, clearly, if t0 is an isolated point of a compact Hausdorff topological space K and X is +a Banach space, then C(K, X) = C(K \ {t0}, X) ⊕∞ X. +Remark 4.4. Let K be a Hausdorff topological compact space, let X be a Banach space. and let +f ∈ C(K, X) be a function with ∥f∥ = 1. +(1) If f ∈ C(K, X) with ∥f∥ = 1 attains its norm at an accumulation point of K, then f is a ccs +Daugavet point (by Theorem 4.2) and hence, f satisfies the six diametral notions. +(2) If f ∈ C(K, X) with ∥f∥ = 1 attains its norm at an isolated point t0 and f(t0) is a Dau- +gavet (respectively, super Daugavet, ccs Daugavet) point, then f is a Daugavet (respectively, +super Daugavet, ccs Daugavet) point (by [3, Section 4], Remark 3.28, and Theorem 3.31, +respectively). +(3) Suppose that K contains an isolated point t0, let x0 ∈ SX, and let f ∈ C(K, X) be given by +f(t0) = x0 and f(t) = 0 for every t ∈ K \ {t0}. Then: +(3.1) If x0 is a ∆- (respectively, super ∆-) point of X, then f is a ∆- (respectively, super ∆-) +point of C(K, X) (by [3, Section 4] and Proposition 3.25, respectively). +(3.2) If x0 is a Daugavet (respectively, super Daugavet, ccs Daugavet) point of X, then f +is a Daugavet (respectively, super Daugavet, ccs Daugavet) point of C(K, X) (by [45, +Proposition 3.3.11], Remark 3.28 Theorem 3.31, respectively). +(3.3) If f is a ∆- (respectively, Daugavet) point of C(K, X), then x0 is a ∆- (respectively, +Daugavet) point of X (by [45, Theorem 3.4.4], [45, Theorem 3.3.13], respectively). +(4) It is now easy to show that the six diametral notions do not coincide in C(K, X) spaces. +Indeed, let K be a compact Hausdorff topological space containing an isolated point t0, let X +a Banach space containing a ∆-point x0 which is not a Daugavet point (e.g. any x0 in the unit +sphere of X = C[0, 1] ⊕2 C[0, 1]), see Propositions 3.25 and 3.27), and consider the function +f ∈ C(K, X) given by f(t0) = x0 and f(t) = 0 for every t ∈ K \ {t0}. Then, f is a ∆-point by +(3.1) but it is not a Daugavet point by (3.3). +We are now ready to extend Corollary 4.3 to general L1-predual spaces. + +22 +MART´IN, PERREAU, AND RUEDA ZOCA +Corollary 4.5. Let X be an L1-predual and let x ∈ SX be a ∆-point. Then, x is a ccs Daugavet +point. Hence the six diametral notions are equivalent for L1-preduals. +Proof. If x is a ∆-point in X, then as mentioned in item (3) of Remark 2.6, we have that JX(x) is +a ∆-point in X∗∗. Now, X∗∗ is isometric to a C(K) space so Theorem 4.2 gives that JX(x) is a ccs +Daugavet point in X∗∗. Then, using now item (4) of Remark 2.6 (or using a straightforward argument +based on the principle of local reflexivity as in [3, Theorem 3.7]), we get that x is a ccs Daugavet point +in X. +□ +Let us observe that the proof of Theorem 4.2 also works for M¨untz spaces (by using [3, Lemma 3.10] +to provide suitable replacements for the functions pn). We recall that given an an increasing sequence +Λ = (λn)∞ +n=0 of non-negative real numbers with λ0 = 0 such that �∞ +i=1 +1 +λi < ∞, then the real Banach +space +M(Λ) := span{tλn : n ⩾ 0} ⊆ C[0, 1] +is called the M¨untz space associated with Λ. Excluding the constant functions form M(Λ), we have +the subspace M0(Λ) := span{tλn : n ⩾ 1} of M(Λ). +So, adapting the proof of Theorem 4.2 to M¨untz spaces (for real scalar-valued functions attaining +its norm at 1 ∈ [0, 1]) and also using [3, Proposition 3.12], we get the following result analogous to +Corollaries 4.3 and 4.5. +Corollary 4.6. Let X = M(Λ) or X = M0(Λ) for an increasing sequence Λ of non-negative real +numbers with λ0 = 0 such that �∞ +i=1 +1 +λi < ∞. Then, every ∆-point of X is a ccs Daugavet point (and +hence the six diametral notions are equivalent). +4.2. Characterization in L1-spaces. In [3, Theorem 3.1] the equivalence between the notions of +Daugavet point and ∆-point was obtained for elements of σ-finite L1-spaces in the real case. Actually, +it is not complicated to extend the results to arbitrary measures and also to the complex case. +Proposition 4.7 ([3, Theorem 3.1] for the σ-finite real case). Let (Ω, Σ, µ) be a measure space, and +let f be a norm one element in L1(µ). Then, the following assertions are equivalent. +(1) f is a Daugavet point. +(2) f is a ∆-point. +(3) The support of the function f contains no atom. +Observe that (1) implies (2) is immediate. For (2) implies (3), suppose that f is a ∆-point and let +A be an atom of finite measure (the only ones that can be contained in the support of an integrable +function). Then, we clearly have that L1(µ) = L1(µ|Ω\A) ⊕1 K (as integrable functions are constant +on atoms), and we may write f = (f1, c) for suitable f1 ∈ L1(µ|Ω\A) and c = f(A) ∈ K. If c ̸= 0, then +∥f1∥ ̸= 1 and it follows from [45, Theorem 3.4.4] that 1 ∈ K is a ∆-point, a contradiction. This shows +that the support of f does not contain any atom. +To get that (3) implies (1), we actually prove the following more general result. Recall that given a +measured space (Ω, Σ, µ) and a Banach space X, L1(µ, X) denotes the Banach space of all B¨ochner- +integrable functions from Ω to X. +Theorem 4.8. Let (Ω, Σ, µ) be a measured space, let X be a Banach space, and let f be a norm one +element in L1(µ, X). If the support of the function f contains no atom, then f is a super Daugavet +point. + +DIAMETRAL NOTIONS FOR ELEMENTS OF THE UNIT BALL OF A BANACH SPACE +23 +Proof. Let us write S := supp f which contains no atom by hypothesis. Let us first prove that f is +a super Daugavet point. Since S contains no atoms, we have that L1(µ|S, X) satisfies the Daugavet +property (see e.g. [54, Example in p. 81]). In particular, f is a super Daugavet point in this space. +Since L1(µ, X) = L1(µ|S, X) ⊕1 L1(µ|Σ\S, X), we get that f is a super Daugavet point in L1(µ, X) by +the transfer results from Subsection 3.2 (see Remark 3.28). +□ +Our next goal is to discuss the relationship with the ccs diametral notions. For real L1(µ)-spaces, +and using a result from [1], we may actually get that real-valued integrable functions with atomless +support are ccs ∆-points. +Proposition 4.9. Let (Ω, Σ, µ) be a measured space and let f be a norm one element in the real space +L1(µ). If the support of the function f contains no atom, then f is a ccs ∆-point. +Proof. Take ε > 0 and D := �n +i=1 λiSi a ccs of BL1(µ) containing f. Write f := �n +i=1 λigi with gi ∈ Si +for every i. Consider the measurable subset ˜S := supp f ∪ �n +i=1 supp gi of Ω and let ˜µ be the σ-finite +measure ˜µ := µ| ˜S on ( ˜S, Σ| ˜S). Then D induces a ccs ˜D of BL1(˜µ) by restriction of the support which +contains the function ˜f which is just f viewed as an element of L1(˜µ) and hence, the support of ˜f +does not contains atoms. Since ˜f belongs to the unit sphere of the real space L1(˜µ), we have by [1, +Theorem 5.5] that ˜f is an interior point of ˜D for the relative weak topology of BL1(˜µ). As we have +already shown that ˜f is a super Daugavet point in Theorem 4.8 (and hence a super ∆-point), we can +find ˜g ∈ ˜D such that +�� ˜f − ˜g +�� > 2 − ε. By just considering the extension g of ˜g to the whole Ω by 0, +we get that g ∈ D and that ∥f − g∥ = +�� ˜f − ˜g +�� > 2 − ε. +□ +Let us comment that it is not clear whether ccs ∆-points transfer through absolute sums, but we +have used specific geometric properties of L1-spaces in the previous proof. +Remark 4.10. Observe that since [1, Theorem 5.5] is also valid for convex combination of relative +weakly open subsets of BL1(µ), we in fact have that every ∆-point in a real L1(µ) space is actually a +ccw ∆-point. +Putting together Proposition 4.7, Theorem 4.8, and Proposition 4.9, we get the following corollary. +Corollary 4.11. Let (Ω, Σ, µ) be a measured space and let f be a norm one element in L1(µ). Then, +the following notions are equivalent for f: ∆-points, Daugavet point, super ∆-point, and super Dau- +gavet point. Moreover, in the real case, the previous four notions are also equivalent to being ccs +∆-point. +We now deal with ccs Daugavet points in L1(µ)-spaces. Observe that if Ω admits an atom A of +finite measure, then we have L1(µ) ≡ L1(µ|Ω\A) ⊕1 K. In particular, in this case L1(µ) fails to have +ccs Daugavet points by Proposition 3.29. We then have the following characterization of the presence +of a ccs Daugavet point in an L1-space. +Proposition 4.12. Let (Ω, Σ, µ) be a measure space. Then, the following assertions are equivalent. +(1) L1(µ) has the Daugavet property. +(2) L1(µ) contains a ccs Daugavet point. +(3) L1(µ) has the SD2P. +(4) µ admits no atom of finite measure. + +24 +MART´IN, PERREAU, AND RUEDA ZOCA +Proof. (1)⇔(4) is well known (see [54, Section 2, Example (b)]); (1)⇒(2) is also known; (2)⇒(3) is +contained in Proposition 3.12. Finally, (3)⇒(4) follows from Proposition 3.29 and the comment before +the statement of this proposition. +□ +4.3. Remarks on some examples from the literature. +4.3.1. Two examples in Lipschitz-free spaces. In [51], Veeorg constructed a surprising example of a +space satisfying the Radon-Nikod´ym property and containing a Daugavet point. We slightly improve +this result by showing that this point is also a ccs ∆-point by proving a general fact about extreme +∆-molecules in Lipschitz-free spaces. For the necessary definitions we refer to the cited paper [51] and +to [9, 10, 32]; for further background on Lipschitz-free spaces, we refer to the book [53]. +For this purpose, we start by recalling the following characterization of molecules which are ∆-points +on Lipschitz-free spaces from [32]. +Proposition 4.13 ([32, Theorem 4.7]). Let M be a pointed metric space and let x ̸= y ∈ M. The +molecule mx,y is a ∆-point if and only if every slice S of BF(M) containing mx,y also contains for +every ε > 0 a molecule mu,v with u ̸= v ∈ M satisfying d(u, v) < ε. +In the case in which the molecule is an extreme point, we have the following improved result. +Theorem 4.14. Let M be a pointed metric space, and let x ̸= y ∈ M. If the molecule mx,y is an +extreme point and a ∆-point, then mx,y is a ccs ∆-point. +Observe that this result cannot be obtained from Proposition 3.13: molecules of Lipschitz-free +spaces which are preserved extreme points are denting points, hence very far from being ∆-points. +To give the proof of the theorem, we need a result which is just an equivalent reformulation of a +result in [32]. +Lemma 4.15 ([32, Theorem 2.6]). Let M be a pointed metric space, and let µ ∈ SF(M). For every +ε > 0, there exists δ > 0 such that given u ̸= v ∈ M with d(u, v) < δ we have ∥µ ± mu,v∥ > 2 − ε. +Using this result and a homogeneity argument similar to the one from [15, Lemma 2.3], we can +provide the pending proof. +Proof of Theorem 4.14. Let C := �n +i=1 λiSi be a ccs of BF(M) containing mx,y and let ε > 0. Since +mx,y is extreme, we have that mx,y ∈ �n +i=1 Si, and by Proposition 4.13 every Si contains molecules +of F(M) supported at arbitrarily close points. Using Lemma 4.15, we construct inductively for every +η > 0 a finite sequence (mui,vi)n +i=1 of molecules in F(M) such that +(1) mui,vi ∈ Si for every i. +(2) +���mx,y − �k +i=1 λimui,vi +��� > 1 + �k +i=1 λi − kε +n for every k ⩽ n. +Indeed, since S1 contains molecules of F(M) supported at arbitrarily close points, we can find by +Lemma 4.15 u1 ̸= v1 ∈ M such that mu1,v1 ∈ S1 and ∥mx,y − mu1,v1∥ > 2 − ε +n. +It follows that +∥mx,y − λ1mu1,v1∥ ⩾ ∥mx,y − mu1,v1∥−(1−λ1) > 1+λ1− ε +n. Let us assume that mu1,v1, . . . , muk,vk are +constructed as desired for a given k ∈ {1, . . . , n−1}. Since Sk+1 contains molecules of F(M) supported +at arbitrarily close points, we can find by Lemma 4.15 uk+1 ̸= vk+1 ∈ M such that muk+1,vk+1 ∈ Sk+1 + +DIAMETRAL NOTIONS FOR ELEMENTS OF THE UNIT BALL OF A BANACH SPACE +25 +and +������ +mx,y − �k +i=1 λimui,vi +���mx,y − �k +i=1 λimui,vi +��� +− muk+1,vk+1 +������ +> 2 − +ε +n +���mx,y − �k +i=1 λimui,vi +��� +. +Then, +������ +mx,y − �k+1 +i=1 λimui,vi +���mx,y − �k +i=1 λimui,vi +��� +������ +⩾ +������ +mx,y − �k +i=1 λimui,vi +���mx,y − �k +i=1 λimui,vi +��� +− muk+1,vk+1 +������ +− +� +�1 − +λk+1 +���mx,y − �k +i=1 λimui,vi +��� +� +� +> 1 + +λk+1 +���mx,y − �k +i=1 λimui,vi +��� +− +ε +n +���mx,y − �k +i=1 λimui,vi +��� +. +By the assumption, +�����mx,y − +k+1 +� +i=1 +λimui,vi +����� > +�����mx,y − +k +� +i=1 +λimui,vi +����� + λk+1 − ε +n > 1 + +k+1 +� +i=1 +λi − (k + 1)ε +n +. +As a consequence, µ := �n +i=1 λimui,vi belongs to C and satisfies ∥mx,y − µ∥ > 2 − ε. +□ +In particular, we have, as announced, that the molecule mx,y in the example from [51] is a ccs +∆-point. Note that it cannot be a ccs Daugavet point by Proposition 3.12 since the space has the +RNP, but we do not know whether it is a super ∆-point or even a super Daugavet point. Let us state +the result for further reference. +Example 4.16. Let M be the metric space constructed in [51, Example 3.1] and let x, y be the points +described there. Then, F(M) has the RNP, the molecule mx,y is an extreme point of the unit ball of +F(M) which is a Daugavet point. Hence, by our Theorem 4.14, mx,y is a ccs-∆-point. +Another interesting example in the Lipschitz-free space setting is the following one which uses a +metric space constructed by Aliaga, Noˆus, Petitjean, and Proch´azka [10]. +Example 4.17. Let M be the metric space from [10, Examples 4.2]. Then one can check that the +molecule m0,q is an extreme point of BF(M) and, since the points 0 and q are discretely connectable, it +follows from an easy adjustment of [32, Proposition 4.2] that this molecule is a ∆-point. In particular, +it follows from Theorem 4.14 that this molecule is also a ccs ∆-point. However, it is not difficult to +show that there exists denting points in BF(M) that are at distance strictly less than 2 to m0,q (take +any among the molecules mxn +i ,xn +i+1), so this molecule is not a Daugavet point. Also observe that this +space has the RNP since the metric space M is countable and complete [9, Theorem 4.6]. +Let us finally remark that the spaces F(M) of Examples 4.16 and 4.17 have the RNP, so they +are strongly regular and hence strongly regular points are norm dense, but both examples have ccs +∆-points. They cannot contain ccs Daugavet points by Proposition 3.12. +Let us also comment that the use of Theorem 4.14 above cannot be omitted, as the molecule m0,q +is not a preserved extreme point, hence Proposition 3.13 is again not applicable. + +26 +MART´IN, PERREAU, AND RUEDA ZOCA +4.3.2. An example of a Banach space with the DD2P, the restricted DSD2P, but containing ccs of +arbitrarily small diameter. In [2, Theorem 2.12], Abrahamsen, H´ajek, Nygaard, Talponen, and Troy- +anski constructed a space X which has the DLD2P, which is midpoint locally uniformly rotund (in +particular, satisfying that pre-ext (BX) = SX), and such that BX contains convex combinations of +slices of arbitrarily small diameter. It then follows from Proposition 3.13 that every element of SX is +actually a super ∆-point and a ccs ∆-point (that is, X has the DD2P and the restricted DSD2P). But +containing ccs of arbitrarily small diameter, X fails the SD2P. The obvious explanation for the failure +of the SD2P and the fact that every element in the unit sphere is a ccs ∆-point is that none of the +convex combinations of slices of diameter strictly smaller than 2 intersects the unit sphere. On the +other hand, the space X is constructed as the ℓ2-sum of spaces, and so X does not contain Daugavet +points by [3, Proposition 4.6] (see Proposition 3.27). +Observe further that X has the restricted DSD2P and the DD2P, but fails the DSD2P (which is +equivalent to the Daugavet property by [34]). +4.3.3. An example in a space with one-unconditional basis. Abrahamsen, Lima, Martiny, and Troy- +anski constructed in [6, Section 4] a Banach space XM with one-unconditional basis which contains a +subset DB ⊆ SXM satisfying: +• Every element in DB is both a Daugavet point and a point of continuity; +• BXM = co(DB); +• DB is weakly dense in the unit ball. +Observe that no element of DB is a super ∆-point (it is exactly the opposite!). By Theorem 3.19, no +element of DB is a ccs ∆-point. +4.4. A super ∆-point which fails to be a Daugavet point in an extreme way. In order to +put into a context the following result, let us recall that Daugavet points are at distance 2 from any +denting point (see [32, Proposition 3.1]). With this in mind, the following result can be interpreted as +the existence of super ∆-points which fail to be Daugavet points in an extreme way. +Theorem 4.18. Let X be a Banach space with the Daugavet property. Then, for every ε > 0, there +exists an equivalent norm | · | and two points x, y ∈ B(X,|·|) such that +(1) y is a super ∆-point. +(2) x is strongly exposed. +(3) |x − y| < ε. +Proof. Take a subspace Y ⊆ X with dim(X/Y ) = 1. Observe that Y has the Daugavet property (see +e.g. [50, Theorem 6 (a)]). Take x ∈ SX with 0 < d(x, Y ) < ε (this can be settled taking a non-zero +element v ∈ X/Y with quotient norm smaller than ε). Now, we can find an element y ∈ SY such that +∥x − y∥ < ε. By the Hahn-Banach theorem, we can take f ∈ SX∗ with Re f(x) > 0 and f = 0 on Y . +This means that x belongs to the slice T := {z ∈ BX : Re f(z) > α} for some α > 0. Take δ > 0 such +that ∥x−y∥ +1−δ +< ε. By Lemma 2.1 we can find x∗ ∈ SX∗ such that x ∈ S(x∗, δ; BX) ⊆ T. By the above +inclusion we conclude that S(x∗, δ; BX) ∩ BY = ∅ or, in other words, that Re x∗(z) ⩽ 1 − δ for every +z ∈ BY . Set +B := co(BY ∪ (1 − δ)BX ∪ {±x}). + +DIAMETRAL NOTIONS FOR ELEMENTS OF THE UNIT BALL OF A BANACH SPACE +27 +B is the unit ball of an equivalent norm |·| which satisfies, in view of the inclusions (1−δ)BX ⊆ B ⊆ BX, +that +∥x∥ ⩽ |x| ⩽ +1 +1 − δ∥x∥ +for every x ∈ X. Let us prove that | · |, x and y satisfies our requirements. First, observe that +|x − y| ⩽ ∥x − y∥ +1 − δ +< ε. +Next, we claim that y is a super ∆ point. Indeed, since Y has the Daugavet property we can find a net +{ys} ⊆ BY with {ys} −→ y weakly and ∥y − ys∥ −→ 2. Notice that the weak convergence {ys} −→ y +is still guaranteed on X because i: (Y, ∥ · ∥) −→ (X, | · |) is weak to weak continuous as ∥ · ∥ and | · | +are equivalent. Moreover, notice that ys ∈ BY ⊆ B for every s, so |ys| ⩽ 1 for every s. Finally, +|ys − y| ⩾ ∥ys − y∥ −→ 2, +and since y ∈ BY ⊆ B, we conclude |ys − y| −→ 2. From there, y is clearly a super ∆-point for the +norm | · |. +It remains to prove that x is strongly exposed. Indeed, we will prove that Re x∗ strongly exposes +B at x, for which it is enough to prove that Re x∗ strongly exposes co(BY ∪ (1 − δ)BX ∪ {±x}) at +x. Take z := αu + β(1 − δ)v + (γ − ω)x ∈ co(BY ∪ (1 − δ)BX ∪ {±x}) with α + β + γ + ω = 1. +Observe that 1 − δ < Re x∗(x) ⩽ |x∗| ⩽ ∥x∗∥ due to the inclusion B ⊆ BX. Taking into account that +Re x∗(u) ⩽ 1 − δ since u ∈ BY as BY ∩ S(x∗, δ; BX) = ∅, we conclude +Re x∗(z) ⩽ (1 − δ)(α + β) + (γ − ω) Re x∗(x). +Since Re x∗(x) > 1 − δ, we get that sup +� +Re x∗(z): z ∈ co(BY ∪ (1 − δ)BX ∪ {±x}) +� += Re x∗(x). If we +take a sequence +zn := αnun + βn(1 − δ)vn + (γn − ωn)x ∈ co(BY ∪ (1 − δ)BX ∪ {±x}) +with αn + βn + γn + ωn = 1 such that Re x∗(zn) −→ Re x∗(x), it follows from the previous argument +that αn → 0, βn → 0, ωn → 0 and γn → 1, which means zn → x in norm. +□ +Remark 4.19. Using the previous theorem and Proposition 3.25 it is easy to construct (considering +ℓ2-sums, for instance) a Banach space X containing a sequence of super ∆-points (yn) such that the +distance from yn to the set of strongly exposed points is going to zero. +4.5. A super ∆-point which is a strongly regular point. In the present subsection, as well +as in the next, we aim to distinguish the super and ccs notions of ∆- and Daugavet points. The +following result shows that there are plenty of examples of spaces containing super ∆-points which are +strongly regular points (hence far from being ccs ∆-points). We do the construction in for real spaces +for simplicity. +Theorem 4.20. Every real Banach space with the Daugavet property can be equivalently renormed +so that the new unit ball has a point which is simultaneously super-∆ and a point of strong regularity +(hence, far away of being ccs ∆-point). +We will use the following immediate result which follows from the fact that a convex combination +of ccs is again a ccs. +Lemma 4.21. Let X be a Banach space and let C be a closed, convex, bounded subset of X. Then +the set of strongly regular points of C is a convex set. + +28 +MART´IN, PERREAU, AND RUEDA ZOCA +Proof of Theorem 4.20. Let X be a Banach space with the Daugavet property. Take a 1-codimensional +subspace Y of X. Since Y is complemented in X then X = Y ⊕ R, so we will see X in such way. Take +r > 0, y0 ∈ SY and f ∈ SX∗ such that f(y0) = 1, and consider on X = Y ⊕ R the equivalent norm +| · | whose unit ball is B := co +� +BY × {0} ∪ {±(y0, r)} ∪ {±(y0, −r)} +� +. It readily follows that | · | agrees +with the original norm ∥ · ∥ on the elements of the form (y, 0). +We claim that (y0, 0) satisfies our requirements. First of all, let us prove that (y0, 0) is a super-∆ +point. Since Y is one-codimensional, it has the Daugavet property (see e.g. [50, Theorem 6 (a)]). +Consequently, there exists a net (ys) −→ y0 weakly in BY such that ∥y0 −ys∥ −→ 2. Then, (ys, 0) −→ +(y0, 0) weakly in (X, | · |). Moreover, it is clear that (ys, 0) ∈ B for every s. Finally, +|(ys, 0) − (y0, 0)| = |(ys − y0, 0)| = ∥ys − y0∥ −→ 2. +Let us now prove that (y0, 0) is a point of strong regularity. +To do so, it is enough, in view of +Lemma 4.21, to show that (y0, ±r) is a strongly exposed point (we will prove that for (y0, r), being +the other case completely analogous). Let us prove that Re(f, 1) strongly exposes (y0, r) in the set +BY × {0} ∪ {±(y0, r)} ∪ {±(y0, −r)}. On the one hand, we have +Re(f, 1)(y0, r) = Re f(y0) + r = 1 + r. +On the other hand, given (y, 0) ∈ BY × 0 we have Re(f, 1)(y, 0) = f(y) ⩽ 1 < 1 + r. Moreover, +Re(f, 1)(y0, −r) = 1 − r and Re(f, 1)(−y0, ±r) = −1 ± r < 1 + r. Consequently, +sup{Re(f, 1)(a, b): (a, b) ∈ BY × {0} ∪ {±(y0, r)} ∪ {±(y0, −r)}, (a, b) ̸= (y0, r)} +⩽ 1 < 1 + r = Re(f, 1)(y0, r). +This is enough to guarantee that Re(f, 1) strongly exposes (y0, r) in B, so we are done. +□ +4.6. A super Daugavet point which is not ccs ∆-point. The previous example shows that we +can distinguish the notion of super ∆-point and the one of ccs ∆-point. It seems natural then that +we should be able to distinguish the notions of super Daugavet point and the one of ccs ∆-point. In +order to do so, we need to consider an involved construction but, as a consequence, we will prove +that there are super Daugavet points which are contained in convex combinations of slices of small +diameter. The construction will be very similar to that of [14, Theorem 2.4], with a slight variation +which makes the resulting norm with a stronger Daugavet flavour. As in the previous subsection, we +will only work with real spaces here. +In order to do so, let us recall a construction from Argyros, Odell, and Rosenthal [11]. Pick a +nonincreasing null sequence {εn} in R+. We construct an increasing sequence of closed, bounded and +convex subsets {Kn} in the real space c0 and a sequence {gn} in c0 as follows: First define K1 = {e1}, +g1 = e1 and K2 = co(e1, e1 + e2). Choose l2 > 1 and g2, . . . , gl2 ∈ K2 an ε2-net in K2. Assume that +n ⩾ 2 and that mn, ln, Kn, and {g1, . . . , gln} have been constructed, with Kn ⊆ Bspan{e1,...,emn} and +gi ∈ Kn for every 1 ⩽ i ⩽ ln. Define Kn+1 as +Kn+1 = co(Kn ∪ {gi + emn+i : 1 ⩽ i ⩽ ln}). +Consider mn+1 = mn + ln and choose {gln+1, . . . , gln+1} ∈ Kn+1 so that {g1, . . . , gln+1} is an εn+1-net +in Kn+1. Finally, we define K0 = ∪nKn. Then it follows that K0 is a non-empty closed, bounded +and convex subset of c0 such that x(n) ⩾ 0 for every n ∈ N and ∥x∥∞ ⩽ 1 for every x ∈ K0 and so +diam (K0) ⩽ 1. +Now, for a fixed i, we have from the construction that {gi + emn+i}n is a sequence in K0 (for n +large enough) which is weakly convergent to gi, and ∥(gi − emn+i) − gi∥ = ∥emn+i∥ = 1 holds for every + +DIAMETRAL NOTIONS FOR ELEMENTS OF THE UNIT BALL OF A BANACH SPACE +29 +n. Then diam (K0) = 1. We will freely use the set K0 and the above construction throughout the +subsection. Observe that, from the above construction, it follows that +K0 = {gi : i ∈ N} +w = {gi : i ∈ N}. +Observe finally that, by the inductive construction, gi has finite support for every i ∈ N. +By [11, Theorem 1.2] we have that K0 contains convex combinations of slices of arbitrarily small +diameter. However, all the points in K0 are “super Daugavet points” in the following sense. +Proposition 4.22. For every x0 ∈ K0, every ε > 0, and every non-empty weakly open subset W of +K0, there exists y ∈ W satisfying that ∥x0 − y∥ > 1 − ε = diam (K0) − ε. +Proof. Take ε > 0 and a non-empty relatively weakly open subset of K0. By a density argument, we +can find i ∈ N satisfying that ∥x0 − gi∥ < ε. Again by a density argument there exists gk ∈ W for +certain k ∈ N. +As we explained above, by the definition of K0 we have that the sequence gk +emn+k ∈ K0 for every +n ∈ N. Since +� +gk + emn+k +� +n∈N −→ gk weakly, we can find n ∈ N large enough so that gk + emn+k ∈ W +and mn + k /∈ supp(gi) ∪ supp(gk) (this is possible because the previous set is finite). +So taking +y = gk + emn+k, we get y(mm + k) = 1 and so +∥gi − y∥ ⩾ y(mn + k) − gi(mn + k) = 1 − 0 = 1. +As a consequence, ∥x0 − y∥ ⩾ ∥gi − y∥ − ∥gi − x0∥ > 1 − ε, and the proof is finished. +□ +It is time to construct the announced renorming of C[0, 1]. Take a sequence of non-empty pairwise +disjoint open subsets Vn of [0, 1] satisfying that 0 /∈ � +n∈N +Vn. By Urysohn lemma, we can find, for +every n ∈ N, a function hn ∈ SC[0,1] with 0 ⩽ hn ⩽ 1 and such that supp(hn) ⊆ Vn. If we consider +Z := span{hn : n ∈ N}, we get that Z is lattice isometrically isomorphic to c0 (indeed, the mapping +en �−→ hn is an isometric Banach lattice isomorphism). Consequently, we can consider the set K0 +constructed in Z, obtaining that K0 ⊆ BC[0,1] is a set of positive functions (because the latter linear +isometry preserves the lattice structure) which contains convex combination of slices of arbitrarily small +diameter but enjoying the property exhibited in Proposition 4.22. Moreover, by the construction of +the functions hn, f(0) = 0 for every f ∈ Z so, in particular, f(0) = 0 for every f ∈ K0. +Now, take 0 < ε < 1 and write +Bε := co +� +2 +� +K0 − 1 +2 +� +∪ 2 +� +−K0 + 1 +2 +� +∪ ((1 − ε)BC[0,1] + εBker(δ0)) +� +, +where 1 stands for the constant function 1 in C[0, 1]. +Consider ∥·∥ε the norm on (the real version of) C[0, 1] whose unit ball is Bε. As we have indicated, +the renorming technique follows the scheme of the renorming given in [14, Theorem 2.4] with the +difference that we use Bker(δ0) instead of Bc0 in the last term because ker(δ0) is a Banach space with +the Daugavet property. +We have the following result. +Theorem 4.23. The space (X, ∥ · ∥ε) satisfies that: +(1) Every element of 2(K0 − 1 +2 ) is a super Daugavet point. +(2) For every η > 0 there exists a convex combination of slices D of Bε with D ∩ 2(K0 − 1 +2 ) ̸= ∅ +and such that diam (D) < η. + +30 +MART´IN, PERREAU, AND RUEDA ZOCA +In particular, there are super Daugavet points which are not ccs−∆ points. +Proof. (1). Take a ∈ K0, and let us prove that 2a − 1 is a super Daugavet point. In order to do so, +pick a non-empty relatively weakly open subset W of Bε. Write +A := 2(K0 − 1 +2 ) and B := (1 − ε)BC[0,1] + εBker(δ0). +Since Bε = co(A ∪ −A ∪ B) we have that W has non-empty intersection with co(A ∪ −A ∪ B). Now +observe that A−A +2 += K0 −K0 ⊆ Bker(δ0) ⊆ B so that co(A∪−A∪B) = co(A∪B)∪co(−A∪B) by [14, +Lemma 2.4]. Consequently, either W ∩ co(A ∪ B) or W ∩ co(−A ∪ B) is non-empty. Let us distinguish +by cases. +Assume first that W ∩ co(A ∪ B) is non-empty, so find a′ ∈ K0, f ∈ BC[0,1], g ∈ Bker(δ0), and +α, β ∈ [0, 1] with α + β = 1 satisfying that +α(2a′ − 1) + β((1 − ε)f + εg) ∈ W. +Take η > 0. By Proposition 4.22, there exists a net (as) −→ a′ weakly with as ∈ K0 for every s and +satisfying that ∥a − as∥ −→ 1. Since (2as − 1) −→ 2a′ − 1 weakly, we can find s large enough so that +α(2as − 1) + β((1 − ε)f + εg) ∈ W +and +∥(2a − 1) − (2as − 1)∥ = 2∥a − as∥ > 2 − η. +Observe that 2a − 1 and 2as − 1 are functions in BC[0,1] since a, as are positive functions of norm at +most one. Since ∥(2a − 1) − (2as − 1)∥ > 2 − η, there exists t0 ∈ [0, 1] and θ ∈ {−1, 1} such that +θ(2a − 1)(t0) > 1 − η and θ(2as − 1)(t0) < −1 + η (observe that t0 ̸= 0 since a(t0) = as(t0) = 0 by +construction). Consequently, the set +U := {t ∈ [0, 1]: θ(2a − 1)(t) > 1 − η and θ(2as − 1)(t) < −1 + η} +is a non-empty open subset of [0, 1], and we can construct a sequence of non-empty pairwise disjoint +open sets Wn ⊆ U. Observe that 0 /∈ � +n∈N Wn since 0 /∈ U. Take pn ∈ Wn for every n ∈ N. We +can construct, for every n ∈ N, two functions fn and gn in the unit ball of C[0, 1] satisfying fn = f +and gn = g in [0, 1] \ Wn and fn(pn) = gn(pn) = −θ. Observe that the sequence of functions (f − fn) +have pairwise disjoint supports, so (f − fn) −→ 0 weakly or, in other words, (fn) −→ f weakly. A +similar argument shows that (gn) −→ g weakly. Notice also that, given n ∈ N, since 0 /∈ Wn then +gn(0) = g(0) = 0, so (gn) ⊆ ker(δ0). Henceforth α(2as − 1) + β((1 − ε)fn + εgn) is a sequence in Bε +which converges in n weakly to α(2as − 1) + β((1 − ε)f + εg) ∈ W. Consequently, we can find n large +enough such that α(2as −1)+β((1−ε)fn +εgn) ∈ W. Finally, observe that the inclusion Bε ⊆ BC[0,1] +implies that ∥z∥ ⩽ ∥z∥ε, so +��(2a − 1) − α(2as − 1) − β((1 − ε)fn + εgn) +�� +ε ⩾ +��(2a − 1) − α(2as − 1) − β((1 − ε)fn + εgn) +�� +⩾ θ((2a − 1) − α(2as − 1) − β((1 − ε)fn)(pn) += θ(2a − 1)(pn) − θα(2as − 1)(pn) +− θβ((1 − ε)fn(pn) + θεgn(pn)) +> 1 − η − α(−1 + η) − β(−1) += 1 + α + β − (1 + α)η = 2 − 2η. +Since η > 0 was arbitrary this finishes the case W ∩ co(A ∪ B) ̸= ∅. + +DIAMETRAL NOTIONS FOR ELEMENTS OF THE UNIT BALL OF A BANACH SPACE +31 +For the case W ∩ co(−A ∪ B) ̸= ∅, find a′ ∈ K0, f ∈ BC[0,1], g ∈ Bker(δ0), and α, β ∈ [0, 1] with +α + β = 1 satisfying that +α(−2a′ + 1) + β((1 − ε)f + εg) ∈ W. +This case is simpler because ∥(2a − 1) − (−2a′ + 1)∥ ⩾ (2a − 1) − (−2a′ + 1)(0) = 2. Now, an +approximation argument for fn and gn similar to that of the above case (working on a non-empty +open subset of (0, 1) in order to get gn(0) = 0) finishes this case and, consequently, the proof of (1). +(2). The first part of the proof will be a repetition of the argument of [14, Theorem 2.4]. Fix γ > 0. +From [11, Theorem 1.2] there exist slices S1, · · · , Sn of K0 such that +diam +� +1 +n +n +� +i=1 +Si +� +< 1 +4(1 − ε)γ. +We can assume that Si = {x ∈ K0 : x∗ +i (x) > 1 − �δ} where 0 < �δ < 1, x∗ +i ∈ C[0, 1]∗ and sup x∗ +i (K0) = 1 +holds for every i = 1, . . . , n. It is clear that +sup x∗ +i +� +2(K0 − 1 +2 ) +� += 2(1 − x∗ +i (1 +2 )), +for all i = 1, · · · , n. We put ρ, δ > 0 such that 1 +2ρ∥x∗ +i ∥ + δ < �δ, 2ρ < ε, ρ∥x∗ +i ∥ < 4δ, and (7−2ε)ρ +(1−ε) +< γ, +for all i = 1, . . . , n. We consider the relatively weakly open set of Bε given by +Ui := +� +x ∈ Bε : x∗ +i (x) > 2 +� +1 − δ − x∗ +i +�1 +2 +�� ++ 1 +2ρ∥x∗ +i ∥, x(0) = δ0(x) < −1 + ρ2 +� +for every i = 1, . . . , n. It is clear that ∥x∗ +i ∥ε ⩽ ∥x∗ +i ∥ for every i = 1, . . . , n and ∥δ0∥ε = ∥δ0∥ = 1. +Since ρ∥x∗ +i ∥ < 4δ, we have that 2(1 − x∗ +i ( 1 +2 )) > 2(1 − δ − x∗ +i ( 1 +2)) + 1 +2ρ∥x∗ +i ∥. Now, we have that +sup x∗ +i (2(K0 − 1 +2 )) = 2(1 − x∗ +i ( 1 +2)), then there exists x ∈ K0 such that +x∗ +i (2(x − 1 +2 )) > 2(1 − δ − x∗ +i (1 +2)) + 1 +2ρ∥x∗ +i ∥ and δ0(2(x − 1 +2)) = −1 < −1 + ρ2. +This implies that Ui ̸= ∅ for every i = 1, . . . , n. In order to estimate the diameter of 1 +n +�n +i=1 Ui, it is +enough to compute the diameter of +1 +n +n +� +i=1 +Ui ∩ co +� +2 +� +K0 − 1 +2 +� +∪ −2 +� +K0 − 1 +2 +� +∪ [(1 − ε)BX + εBker(δ0)] +� +. +Since 2(K0 − 1 +2 ) and (1 − ε)BC[0,1] + εBker(δ0) are convex subsets of Bε, given x ∈ Bε, we can assume +that x = λ12(a − 1 +2 ) + λ22(−b + 1 +2 ) + λ3[(1 − ε)x0 + εy0], where λi ∈ [0, 1] with �3 +i=1 λi = 1 and +a, b ∈ K0, x0 ∈ BC[0,1], and y0 ∈ Bker(δ0). +So given x, y ∈ 1 +n +�n +i=1 Ui, for i = 1, · · · , n, there exist ai, a′ +i, bi, b′ +i ∈ K0, λ(i,j), λ′ +(i,j) ∈ [0, 1] with +j = 1, 2, 3 and, xi, x′ +i ∈ BC[0,1], and yi, y′ +i ∈ BKer(δ0), such that +ui := 2λ(i,1) +� +ai − 1 +2 +� ++ 2λ(i,2) +� +−bi + 1 +2 +� ++ λ(i,3)[(1 − ε)xi + εyi] +u′ +i := 2λ′ +(i,1) +� +a′ +i − 1 +2 +� ++ 2λ′ +(i,2) +� +−b′ +i + 1 +2 +� ++ λ′ +(i,3)[(1 − ε)x′ +i + εy′ +i] +belong to Ui for every i ∈ {1, . . . , n}, and such that +x = 1 +n +n +� +i=1 +ui and y = 1 +n +n +� +i=1 +u′ +i. + +32 +MART´IN, PERREAU, AND RUEDA ZOCA +For i ∈ {1, . . . , n} we have that ui ∈ Ui so +δ0(ui) = δ0 +� +2λ(i,1) +� +ai − 1 +2 +� ++ 2λ(i,2) +� +−bi + 1 +2 +� ++ λ(i,3)[(1 − ε)xi + εyi] +� +< −1 + ρ2. +Observe that, by construction, +δ0 +� +ai − 1 +2 +� += −1 +2, δ0 +� +−bi + 1 +2 +� += 1 +2 and δ0((1 − ε)xi + εyi) = δ0((1 − ε)xi) ⩾ −(1 − ε). +This implies that +2λ(i,2) + λ(i,3)ε − 1 = −λ(i,1) + λ(i,2) − λ(i,3)(1 − ε) < −1 + ρ2. +Since 2ρ < ε, we deduce that λ(i,2) + λ(i,3) < 1 +2ρ. As a consequence we get that +(4.1) +λ(i,1) > 1 − 1 +2ρ, +and, similarly, we get that +(4.2) +λ′ +(i,1) > 1 − 1 +2ρ, +for every i = 1, . . . , n. Now, the previous inequalities imply that +∥x − y∥ε ⩽ 1 +n +����� +n +� +i=1 +2λ(i,1) +� +ai − 1 +2 +� +− 2λ′ +(i,1) +� +a′ +i − 1 +2 +������ +ε ++ 1 +n +n +� +i=1 +����2λ(i,2) +� +−bi + 1 +2 +����� +ε ++ 1 +n +n +� +i=1 +����2λ′ +(i,2) +� +−b′ +i + 1 +2 +����� +ε ++ 1 +n +n +� +i=1 +∥λ(i,3)[(1 − ε)xi + εyi]∥ε + 1 +n +n +� +i=1 +∥λ′ +(i,3)[(1 − ε)x′ +i + εy′ +i]∥ε +⩽ 1 +n +����� +n +� +i=1 +2λ(i,1) +� +ai − 1 +2 +� +− 2λ′ +(i,1) +� +a′ +i − 1 +2 +������ +ε ++ 1 +n +n +� +i=1 +� +λ(i,2) + λ(i,3) +� ++ 1 +n +n +� +i=1 +� +λ′ +(i,2) + λ′ +(i,3) +� +and, by using (4.1),(4.2), +⩽ 1 +n +����� +n +� +i=1 +2λ(i,1) +� +ai − 1 +2 +� +− 2λ′ +(i,1) +� +a′ +i − 1 +2 +������ +ε ++ ρ +⩽ 2 +n +����� +n +� +i=1 +λ(i,1)ai − λ′ +(i,1)a′ +i +����� +ε ++ 1 +n +n +� +i=1 +|λ(i,1) − λ′ +(i,1)|∥1∥ε + ρ +⩽ 2 +n +����� +n +� +i=1 +λ(i,1)ai − λ′ +(i,1)a′ +i +����� +ε ++ (3 − 2ε) +2(1 − ε)ρ. + +DIAMETRAL NOTIONS FOR ELEMENTS OF THE UNIT BALL OF A BANACH SPACE +33 +Now, +����� +n +� +i=1 +λ(i,1)ai − λ′ +(i,1)a′ +i +����� +ε +⩽ +����� +n +� +i=1 +(λ(i,1) − 1)ai +����� +ε ++ +����� +n +� +i=1 +ai − a′ +i +����� +ε ++ +����� +n +� +i=1 +(λ′ +(i,1) − 1)a′ +i +����� +ε +⩽ +1 +1 − ε +����� +n +� +i=1 +ai − a′ +i +����� + +n +� +i=1 +1 +1 − ε|λ(i,1) − 1|∥ai∥ + +n +� +i=1 +1 +1 − ε|λ′ +(i,1) − 1|∥a′ +i∥ +⩽ +1 +1 − ε +����� +n +� +i=1 +ai − a′ +i +����� + +1 +1 − εnρ. +(In the previous estimate observe that ∥ai∥ ⩽ 1 and ∥a′ +i∥ ⩽ 1 since ai, a′ +i ∈ K0 ⊆ Bker(δ0) ⊆ Bε). +Hence, +(4.3) +∥x − y∥ε ⩽ +2 +1 − ε +����� +1 +n +n +� +i=1 +ai − a′ +i +����� + (7 − 2ε) +2(1 − ε)ρ. +Now, in order to prove that the previous norm is small we will prove that both elements 1 +n +�n +i=1 ai, 1 +n +�n +i=1 a′ +i +are elements of 1 +n +�n +i=1 Si, which has small diameter. To this end, note that +x∗ +i +� +2λ(i,1) +� +ai − 1 +2 +� ++ 2λ(i,2) +� +−bi + 1 +2 +� ++ λ(i,3)[(1 − ε)xi + εyi] +� +> 2 +� +1 − δ − x∗ +i +�1 +2 +�� ++ ρ +2∥x∗ +i ∥, +for every i ∈ {1, . . . , n}. Then, +x∗ +i +� +2λ(i,1) +� +ai − 1 +2 +�� ++ 1 +2ρ∥x∗ +i ∥ ⩾ x∗ +i +� +2λ(i,1) +� +ai − 1 +2 +�� ++ λ(i,2)∥x∗ +i ∥ + λ(i,3)∥x∗ +i ∥ +⩾ x∗ +i +� +2λ(i,1) +� +ai − 1 +2 +�� ++ λ(i,2)∥x∗ +i ∥ε + λ(i,3)∥x∗ +i ∥ε +⩾ x∗ +i +� +2λ(i,1) +� +ai − 1 +2 +� ++ 2λ(i,2) +� +−bi + 1 +2 +� ++ λ(i,3)[(1 − ε)xi + εyi] +� +. +We have that +x∗ +i +� +2λ(i,1) +� +ai − 1 +2 +�� +> 2 +� +1 − δ − x∗ +i +�1 +2 +�� +, +and hence +x∗ +i (λ(i,1)ai) > 1 − δ − (1 − λ(i,1))x∗ +i +�1 +2 +� +⩾ 1 − δ − 1 +2ρ∥x∗ +i ∥. +We recall that δ+ 1 +2ρ∥x∗ +i ∥ < �δ, so x∗ +i (λ(i,1)ai) > 1−�δ. It follows that x∗ +i (ai) > 1−�δ. Then, ai ∈ K0∩Si +and, similarly, we get that a′ +i ∈ K0 ∩ Si, for every i = 1, . . . , n. Therefore, +1 +n +n +� +i=1 +ai, 1 +n +n +� +i=1 +a′ +i ∈ 1 +n +n +� +i=1 +Si. + +34 +MART´IN, PERREAU, AND RUEDA ZOCA +Since the diameter of 1 +n +�n +i=1 Si is less than 1 +4(1 − ε)γ, we deduce that 1 +n∥ �n +i=1 ai − a′ +i∥ < 1 +4(1 − ε)γ. +Finally, we conclude from (4.3) and the above estimate that ∥x − y∥ε ⩽ γ. Hence, the set C := +1 +n +�n +i=1 Ui has diameter at most γ for the norm ∥ · ∥ε. +Now, Bourgain’s lemma (see Lemma 2.2) ensures the existence of a convex combination of slices +�pi +j=1 αijTij ⊆ Ui for every 1 ⩽ i ⩽ n. Using this fact, we will find a convex combination of slices of +B of diameter smaller than γ + +4ρ2 +(1− ρ2 +ε )ε and such that every slice contains points of 2(K0 − 1 +2 ). Since +ρ and γ can be taken as small as we wish, we will be done. In order to do so, fix 1 ⩽ i ⩽ n and define +Ai := +� +j ∈ {1, . . . , pi}: Tij ∩ +� +2K0 − 1 +2 +� += ∅ +� +; +Bi := {1, . . . , pi} \ Ai. +Given xij ∈ Tij we have that, for j ∈ Ai, that δ0(xij) ⩾ −1 + ε by the definition of the unit ball Bε. +Since �pi +j=1 αijxij ∈ �pi +j=1 αijTij ⊆ Ui we derive −1 + ρ2 > δ0 +��pi +j=1 αijxij +� +. Hence +−1 + ρ2 > +� +j∈Ai +αijδ0(xij) + +� +i∈Bi +αijδ0(xij) ⩾ (−1 + ε) +� +j∈Ai +αij − +� +j∈Bi +αij = −1 + ε +� +j∈Ai +αij. +From the above inequality we infer that � +j∈Ai αij < ρ2 +ε holds for every 1 ⩽ i ⩽ n. Now, we set +Λi := � +j∈Bi λij, which belongs to the interval [1 − ρ2 +ε , 1] for 1 ⩽ i ⩽ n and set +D := 1 +n +n +� +i=1 +� +j∈Bi +αij +ΛI +Tij. +Observe that D is a convex combination of slices of Bε since every Tij is a slice of Bε and since +1 +n +n +� +i=1 +� +j∈BI +αij +Λi +αij = 1. +We claim that D ⊆ C + +2 +1− ρ2 +ε +ρ2 +ε Bε. This is enough to finish the proof because the above condition +implies that +diam (D) ⩽ diam (C) + +4 +1 − ρ2 +ε +ρ2 +ε ⩽ γ + +4 +1 − ρ2 +ε +ρ2 +ε . +So let us prove the above inclusion. Take z := 1 +n +�n +i=1 +� +j∈Bi +αij +Λi xij ∈ D for certain xij ∈ Tij. Write +z′ := 1 +n +�n +i=1 +� +j∈Bi αijxij. Then +|z − z′| ⩽ 1 +n +n +� +i=1 +� +j∈Bij +����1 − 1 +Λi +���� αij|xij| < +1 +1 − ρ2 +ε +ρ2 +ε . +On the other hand, for 1 ⩽ i ⩽ n and j ∈ Ai take xij ∈ Tij. Define +z′′ := 1 +n +n +� +i=1 +pi +� +j=1 +αijxij ∈ 1 +n +n +� +i=1 +pi +� +j=1 +αijTij ⊆ 1 +n +n +� +i=1 +Ui = C. +Moreover, we have +|z′ − z′′| ⩽ 1 +n +n +� +i=1 +� +j∈Ai +αij < ρ2 +ε . + +DIAMETRAL NOTIONS FOR ELEMENTS OF THE UNIT BALL OF A BANACH SPACE +35 +Consequently z = z′′ + (z − z′′) ∈ C + +2 +1− ρ2 +ε +ρ2 +ε Bε since +|z − z′′| ⩽ |z − z′| + |z′ − z′′| < +1 +1 − ρ2 +ε +ρ2 +ε + ρ2 +ε < +2 +1 − ρ2 +ε +ρ2 +ε . +□ +4.7. A summary of relations between the properties. Figure 2 below is an scheme which +complements Figure 1 with the counterexamples following from known results and from the results in +this section. +ccs Daugavet +ccs ∆ +super Daugavet +super ∆ +∆ +Daugavet +? +/ +(d) +/ +(a) +/ +(b) +/ +(e) +/ +(c) +/ +(f) +Figure 2. Scheme of all relations between the diametral notions +Let us list the corresponding counterexamples. +(a) The example in Subsection 4.6. +(b) The example in Subsection 4.5 negates this implication in the strongest possible way. +(c) Any of the elements in DB in Paragraph 4.3.3. They also show directly that ∆-points are not +necessarily ccs ∆-points. +(d) Every element of the unit sphere of the space X given in Paragraph 4.3.2 is ccs ∆-point but +not Daugavet point. Another example is the molecule m0,q of Example 4.17. +(e) In X = C[0, 1]⊕2 C[0, 1], every element in the unit sphere is super ∆-point (Proposition 3.25); +but X contains no Daugavet point (Proposition 3.27). Also, every element of the unit sphere +of the space X given in Paragraph 4.3.2 is super ∆-point but not Daugavet point. +(f) Any of the elements in DB in Paragraph 4.3.3. +5. Diametral-properties for elements of the open unit ball +As mentioned in Section 2, the DSD2P is equivalent to the Daugavet property by [34], but the +ccs ∆-points on the unit sphere of a Banach space do not characterize the DSD2P, but the restricted +DSD2P, which is not equivalent to the Daugavet property (see Paragraph 4.3.2). Actually, the elements +in the open unit ball play a decisive role in the proof in [34] of the equivalence between the DSD2P +and the Daugavet property. Our objective here is to introduce and study the diametral notions for +interior points, providing interesting applications, and to investigate the behavior of Daugavet- and +∆-elements on rays in the unit ball of a given Banach space. +The definition of the Daugavet notions for elements in the open unit ball is the natural extension +of the definitions for elements of norm one given in Definition 2.5. + +36 +MART´IN, PERREAU, AND RUEDA ZOCA +Definition 5.1. Let X be a Banach space and let x ∈ BX. We say that +(1) x is a Daugavet point if supy∈S ∥x − y∥ = ∥x∥ + 1 for every slice S of BX, +(2) x is a super Daugavet point if supy∈V ∥x − y∥ = ∥x∥ + 1 for every non-empty relatively weakly +open subset V of BX, +(3) x is a ccs Daugavet point if supy∈C ∥x − y∥ = ∥x∥ + 1 for every ccs C of BX. +It turns out that the existence of a non-zero Daugavet kind element actually forces the whole ray +to which it belongs to be composed of similar elements. +Proposition 5.2. Let X be a Banach space, and let x ∈ SX. The following assertions are equivalent. +(1) x is a Daugavet- (resp. super Daugavet-, resp. ccs Daugavet-) point. +(2) rx is a Daugavet- (resp. super Daugavet-, resp. ccs Daugavet-) point for every r ∈ [0, 1]. +(3) rx is a Daugavet- (resp. super Daugavet- , resp. ccs Daugavet-) point for some r ∈ (0, 1). +Let us recall the following elementary but very useful result from [34] due to Kadets. +Lemma 5.3 ([34, Lemma 2.2]). Let X be a normed space. If x, y ∈ X and ε > 0 satisfies that +∥x + y∥ > ∥x∥ + ∥y∥ − ε, +then for every a, b > 0, it is satisfied that +∥ax + by∥ > a ∥x∥ + b ∥y∥ − max{a, b}ε. +Proof of Proposition 5.2. We will only do the proof for Daugavet points, being the other cases com- +pletely analogous. So let us first assume that x is a Daugavet point. Take r ∈ [0, 1], ε > 0, and S +a slice of BX. Then, there exists y ∈ S such that ∥x − y∥ > 2 − ε. In particular, ∥y∥ > 1 − ε. As +∥y∥ ⩽ 1, +∥x − y∥ > 2 − ε ⩾ ∥x∥ + ∥y∥ − ε. +It follows from Lemma 5.3 that +∥rx − y∥ > ∥rx∥ + ∥y∥ − ε > ∥rx∥ + 1 − 2ε. +Hence, rx is also a Daugavet point. +Now, let us assume that rx is a Daugavet point for some r ∈ (0, 1). Again take ε > 0 and S slice +of BX, and pick y ∈ S such that ∥rx − y∥ > ∥rx∥ + 1 − rε. In particular, ∥y∥ > 1 − rε. As ∥y∥ ⩽ 1, +we have +∥rx − y∥ > ∥rx∥ + ∥y∥ − rε. +Hence, by Lemma 5.3, we get that +∥x − y∥ > ∥x∥ + ∥y∥ − ε > 2 − (1 + r)ε +and so x is a Daugavet point. +□ +As mentioned in the discussion preceding Proposition 3.12, the presence of a ccs Daugavet point in +a given Banach space forces the space to satisfy the SD2P. This can now be viewed as a consequence +of the previous proposition and the following immediate reformulation of [41, Theorem 3.1]. +Proposition 5.4 ([41, Theorem 3.1]). Let X be a Banach space. Then, X has the SD2P if and only +if 0 is a ccs Daugavet point. + +DIAMETRAL NOTIONS FOR ELEMENTS OF THE UNIT BALL OF A BANACH SPACE +37 +Note that c0 has the SD2P but has no non-zero Daugavet points (use Proposition 4.1, for instance). +Compare Proposition 5.4 with the following obvious remark. +Remark 5.5. A Banach space X is infinite-dimensional if and only if 0 is a super Daugavet point. +We also mention that 0 is always a Daugavet point (in finite or infinite dimension), as every slice +of the unit ball has to intersect the unit sphere. +Let us also point out that [41, Theorem 3.1] admits the following scaled version. +Proposition 5.6. Let X be a Banach space, and let r ∈ (0, 1]. Then, the following assertions are +equivalent. +(1) Every ccs of BX has diameter greater than or equal to 2r. +(2) sup{∥x∥: x ∈ C} ⩾ r for every ccs C of BX. +(3) sup{∥x∥: x ∈ D} ⩾ r for every symmetric ccs D of BX (so containing 0). +Proof. Suppose (1) holds. Then, for any given ccs C of BX, and for any fixed ε > 0, there exists +x, y ∈ C such that ∥x − y∥ > 2r − 2ε. In particular, it follows that ∥x∥ > r − ε or ∥y∥ > r − ε, giving +(2). (2)⇒(3) is immediate. Suppose that (1) fails, that is, that there exists a ccs C of BX and ε > 0 +such that diam (C) ⩽ 2r − 2ε. We consider the ccs D of BX given by D := 1 +2(C − C). Then D is +symmetric, and for every u := x−y +2 +and u′ := x′−y′ +2 +in D we have ∥u − u′∥ = +��� x+y′ +2 +− x′+y +2 +���. Now, +x, x′, y, y′ belong to C, and C is convex, so x+y′ +2 +and x′+y +2 +do also belong to C, so ∥u − u′∥ ⩽ 2r − 2ε. +Being D symmetric, it implies that D ⊂ (r − ε)BX, hence (3) fails. +□ +The following is a nice consequence of the proposition above outside the diametral notions. +Corollary 5.7. Let X be a Banach space. Then, BX contains ccs of arbitrarily small diameter if and +only if 0 is a strongly regular point of BX. +Now let us consider the ∆ notions for points of the open unit ball which are just the adaptation of +the notions given in Definition 2.4. +Definition 5.8. Let X be a Banach space and let x ∈ BX. We say that +(1) x is a ∆-point if supy∈S ∥x − y∥ = ∥x∥ + 1 for every slice S of BX containing x, +(2) x is a super ∆-point if supy∈V ∥x − y∥ = ∥x∥ + 1 for every non-empty relatively weakly open +subset V of BX containing x, +(3) x is a ccs ∆-point if supy∈C ∥x − y∥ = ∥x∥ + 1 for every slice ccs C of BX containing x. +With this definitions in hands, we may get an improvement of Proposition 5.4 from Proposition 5.6. +Corollary 5.9. Let X be a Banach space. Then, X has the SD2P if and only if 0 is ccs ∆-point. +Compare the previous corollary with the following obvious remark which is analogous to Remark 5.5. +Remark 5.10. A Banach space X is infinite-dimensional if and only if 0 is a super ∆-point. +Observe that the definition of ccs ∆-points for elements in BX gives a localization of the DSD2P, +that is, X has the DSD2P (and hence the Daugavet property [34]) if and only if all the elements of +BX are ccs ∆-points. Recall that the DSD2P is not equivalent to the restricted DSD2P (meaning that +all points in SX are ccs ∆-points), see Paragraph 4.3.2. + +38 +MART´IN, PERREAU, AND RUEDA ZOCA +The following result is a localization of Kadets’ theorem [34] on the equivalence of the DSD2P and +the DPr. +Theorem 5.11. Let X be a Banach space and let x ∈ SX. If rx is a ccs ∆-point for every r ∈ (0, 1), +then x is a ccs Daugavet point. Moreover, it is enough that inf{r ∈ (0, 1): rx is a ccs ∆-point} = 0. +Proof. Fix a ccs C of BX and ε > 0. Since ˜C := 1 +2(C − C) is also a ccs of BX and since 0 ∈ ˜C is a +norm interior point of ˜C by [41, Proposition 2.1], we have that rx belongs to ˜C for every r ∈ (0, δ) +for some δ > 0. By hypothesis, there is r > 0 such that rx is a ccs ∆-point and rx ∈ ˜C. So there +exists y ∈ ˜C such that ∥rx − y∥ > r + 1 − rε. Then if we write y := y1 − y2 with y1, y2 ∈ C, we have +∥rx − y1∥ > r +1−rε or ∥rx − y2∥ > r +1−rε by the triangle inequality; in particular, ∥y1∥ > 1−rε +and ∥y2∥ > 1 − rε. In both cases, we have that there is y ∈ C such that ∥rx − y∥ > r∥x∥ + ∥y∥ − rε +and it follows from Lemma 5.3 that +∥x − y∥ > ∥x∥ + ∥y∥ − ε > 2 − (1 + r)ε > 2 − 2ε. +□ +At this point, it is natural to ask whether an equivalent formulation of Proposition 5.2 is valid for +some of the various ∆-notions. For ccs ∆-points, the answer is negative as follows from Theorem 5.11 +and, for instance, the example in Paragraph 4.3.2. Another, maybe simpler, example showing that is +the following one. +Example 5.12. Let us assume that a positive measure µ admits an atom of finite measure and also +has a non-empty non-atomic part. Then, the real space L1(µ) contains no ccs Daugavet point by +Proposition 4.12. However, it contains elements in the unit sphere which are ccs ∆-points and super +Daugavet points by Theorem 4.8 and Proposition 4.9. In particular, as a consequence of Theorem 5.11 +and of Proposition 5.2, there must exist f in the unit sphere which is a ccs ∆-point and t ∈ (0, 1) such +that tf is not a ccs ∆-point but it is a super ∆-point. +For ∆-points, we have the following result. +Proposition 5.13. Let X be a Banach space, and let x ∈ SX. If x is a ∆-point, then rx is a ∆-point +for every r ∈ (0, 1). +Proof. Let us assume that x is a ∆-point and let us fix r ∈ (0, 1). +Take ε > 0 and a slice S of +BX containing rx. Now, either x belongs to S or −x belongs to S. In the first case, we can find +y ∈ S such that ∥x − y∥ ⩾ 2 − ε, and using Lemma 5.3, we get ∥rx − y∥ ⩾ ∥rx∥ + 1 − 2ε. Else, +∥rx − (−x)∥ = r + 1 = ∥rx∥ + 1 and we are done. +□ +For super ∆-points, it is currently quite obscure whether they behave like ∆-points up on rays. +6. Kuratowski measure and large diameters +Let M be a metric space. +The Kuratowski measure of non-compactness α(A) of a non-empty +bounded subset A of M is defined as the infimum of all real numbers ε > 0 such that A can be covered +by a finite number of subsets of M of diameter smaller than or equal to ε. +From the definition, we clearly have α(A) = 0 if and only if A is totally bounded (a.k.a. precompact). +It follows that every complete subset A of M with α-measure 0 is compact, and in particular, if M is +a complete metric space, that α(A) = 0 if and only if A is compact, where A stands for the closure of +the set A. The α-measure can be thus seen as a way to measure how far a given (non-empty) bounded +and closed subset of M is from being a compact space. It was introduced by C. Kuratowski in [38] in + +DIAMETRAL NOTIONS FOR ELEMENTS OF THE UNIT BALL OF A BANACH SPACE +39 +order to provide a generalization of the famous intersection theorem from Cantor. A general theory +on measures of non-compactness was later developed, and it turned out to provide important results +in metric fixed point theory, and in particular to have applications in functional equations or optimal +control. We refer e.g. to [12] for an introduction to the topic and for more precise applications. +Observe that A ⊆ B implies α(A) ⩽ α(B), and that α(A) = α(A). Also note that α(A ∪ B) = +max{α(A), α(B)} for every non-empty bounded subsets A, B of M. Furthermore, if M = X is a +normed space, then α is known to enjoy additional useful properties: it is symmetric, translation +invariant, positively homogeneous, sub-additive, and satisfies α(co A) = α(A). The α-measure has +proved to be a powerful tool for the study of the geometry of Banach spaces and we refer e.g. to the +works [46], [47] and [44] in connection with property (α), with drop property, and with an isomorphic +characterization of reflexive Banach spaces. +From the definition it is clear that the Kuratowski measure of A is smaller than or equal to its +diameter. Obviously, equality does not always hold, but a fruitful relationship between the notion +of ∆-points and the Kuratowski measure of slices was discovered in [5] and completed in [52]. In +particular, the following result was obtained (see [52, Corollary 2.2]). +Theorem 6.1. Let X be a Banach space and let x ∈ SX. If x is a ∆-point, then α(S) = 2 for every +slice S of BX containing x. Besides, α(S(x, δ; BX∗)) = 2 in BX∗ for every δ > 0. +Observe that the converse does not hold in general, as the following example shows. +Example 6.2. Consider X := L1([0, 1]) ⊕∞ ℓ1. It follows that both X and X∗ enjoy the SD2P [15, +Remark 2.6], so Theorem 6.3 below implies that given any slice S = S(x∗, δ; BX) we have α(S) = 2 +and the same holds for the slices in the dual. However, there are points which are not ∆-points because +X fails the DLD2P [30, Theorem 3.2] since ℓ1 fails it, so it remains to take any point x ∈ SX which +is not a ∆-point to get the desired counterexample. +Observe that the connection between having big slices in diameter and having big slices in Kura- +towski index goes beyond Theorem 6.1. The following result was first pointed out in [20]. +Theorem 6.3 ([20, Proposition 3.1]). Let X be a Banach space and let β ∈ (0, 2]. The following +assertions are equivalent. +(1) Every slice of BX has diameter greater than or equal to β. +(2) Every slice of BX has Kuratowski measure greater than or equal to β. +In this section, we aim to prove analogues to this result for relative weakly open subsets, as well +as for convex combinations of slices or of weakly open sets, and to extend Veeorg’s result to super +∆-points and ccw ∆-points. +6.1. Kuratowski measure and diameter two properties. The analogue to Theorem 6.3 for +non-empty weakly open subsets is the following. +Theorem 6.4. Let X be a Banach space and let β ∈ (0, 2]. The following assertions are equivalent. +(1) Every non-empty relatively weakly open subset of BX has diameter greater than or equal to β. +(2) Every non-empty relatively weakly open subset of BX has Kuratowski measure greater than or +equal to β. +Proof. (2)⇒(1) is immediate, so let us prove (1)⇒(2). To this end, fix β ∈ (0, 2] and assume that +every non-empty relatively weakly open subset of BX has diameter greater than or equal to β. Then + +40 +MART´IN, PERREAU, AND RUEDA ZOCA +pick ε > 0, and let us prove by induction on n that for every non-empty relatively weakly open subset +W of BX and for every finite collection C1, . . . , Cn of subsets of X with diam (Ci) ⩽ β − ε for every +i, we have that W ̸⊂ +n� +i=1 +Ci. +For n = 1, it is clear since by assumption diam (W) ⩾ β > β − ε for every non-empty relatively +weakly open subset W of BX. +So assume that the result is true for every non-empty relatively weakly open subset W of BX +and for every collection of n sets, and let us prove the result for collections of n + 1 sets. To this +end, consider C1, . . . , Cn, Cn+1 be subsets of X with diam (Ci) ⩽ β − ε for every i. Observe that +diam (Ci) = diam (Ci +w) ⩽ β − ε by w-lower semicontinuity of the norm of X, so that we may and do +assume that Ci is weakly closed for every i. +Observe that by the case n = 1 we have that W ̸⊂ Cn+1, which means that W \ Cn+1 is non- +empty. Moreover, it is a weakly open subset of BX since Cn+1 is assumed to be weakly closed, and by +induction hypothesis we conclude that W\Cn+1 ̸⊂ +n� +i=1 +Ci. In particular W ̸⊂ +n+1 +� +i=1 +Ci and the theorem +is proved. +□ +Next, let us establish the analogue Theorem 6.3 for convex combinations of slices. To this end, +observe that by Bourgain lemma (see Lemma 2.2) every convex combination of non-empty relatively +weakly open subsets of BX contains a convex combination of slices of BX. This assertion makes valid +the following lemma which allows us to focus our attention in convex combination of weakly open +subsets. +Lemma 6.5. Let X be a Banach space and r > 0. +(1) The following are equivalent: +(a) Every convex combination of slices of BX has diameter greater than or equal to r. +(b) Every convex combination of non-empty relatively weakly open subsets of BX has diameter +greater than or equal to r. +(2) The following are equivalent: +(a) α(C) ⩾ r holds for every convex combination C of slices of BX. +(b) α(D) ⩾ r holds for every convex combination D of non-empty relatively weakly open +subsets of BX. +Now we are able to give the following result. +Theorem 6.6. Let X be a Banach space and let β ∈ (0, 2]. The following assertions are equivalent. +(1) Every convex combination of slices of BX has diameter greater than or equal to β. +(2) Every convex combination of slices of BX has Kuratowski measure greater than or equal to β. +Proof. (2)⇒(1) is immediate, so let us prove (1)⇒(2). To this end, fix β ∈ (0, 2] and assume that every +convex combination of non-empty relatively weakly open subsets of BX has diameter greater than or +equal to β. Then pick ε > 0, and let us prove by induction on n that for every D convex combination +of non-empty relatively weakly open subsets of BX and for every finite collection C1, . . . , Cn of subsets +of X with diam (Ci) ⩽ β − ε for every i, we have that D ̸⊂ +n� +i=1 +Ci. +For n = 1 it is clear since by assumption diam (D) ⩾ β > β − ε for every convex combination of +non-empty relatively weakly open subsets of D of BX. + +DIAMETRAL NOTIONS FOR ELEMENTS OF THE UNIT BALL OF A BANACH SPACE +41 +Assume by inductive step that the result stands for n. +Now pick D convex combination of non-empty relatively weakly open subsets of BX and a finite +collection C1, . . . , Cn+1 of subsets of X with diam (Ci) ⩽ β − ε for every i. We can assume as in the +proof of Theorem 6.4 that every Ci is weakly closed. Write D = �k +i=1 λiWi. Observe that by the case +n = 1 we have that D ̸⊆ Cn+1, so there exists z ∈ D \ Cn+1. Since z ∈ D we can write z = �k +i=1 λixi +where xi ∈ Wi holds for every 1 ⩽ i ⩽ k. Moreover, since z = �k +i=1 λixi /∈ Cn+1, this means that +z = �k +i=1 λixi ∈ X \ Cn+1, and the latter is a weakly open set. By a weak-continuity argument of +the sum we can find weakly open subsets Vi of BX, with xi ∈ Vi for every 1 ⩽ i ⩽ k, satisfying that +z = �k +i=1 λixi ∈ �n +i=1 λiVi ⊆ X \ Cn+1. Up to taking smaller Vi, we can assume Vi ⊆ Wi for every +i. Now call ˜D := �k +i=1 λiVi, which is a convex combination of weakly open subsets of BX. By the +inductive step we get that ˜D ̸⊆ +n� +j=1 +Cj, so there exists y ∈ ˜D with y /∈ Cj for 1 ⩽ j ⩽ n. Observe that +the condition Vi ⊆ Wi implies ˜D ⊆ D, so y ∈ D indeed. Moreover, ˜D ⊆ X \Cn+1 implies in particular +y /∈ Cn+1. This implies that y ∈ D \ +n+1 +� +i=1 +Ci, which is precisely what we wanted to prove. +□ +6.2. Kuratowski measure and ∆-notions. We now prove an analogue to Theorem 6.1 for super +∆-points. +Theorem 6.7. Let X be a Banach space and let x ∈ SX be a super ∆-point. Then every non-empty +relatively weakly open subset W of BX containing x satisfies that α(W) = 2. +The proof will be an obvious consequence of the following result. +Proposition 6.8. Let X be a Banach space, x ∈ SX be a super ∆ point, and W be a weakly open +subset of BX such that x ∈ W. Then, for every ε > 0, there exists a sequence {xn} ⊆ W such that +∥xi − xj∥ > 2 − ε holds for every i ̸= j. +Proof. Set ε > 0. Let us construct by induction a sequence {xn} satisfying that ∥x − xi∥ > 2 − ε +2 and +such that ∥xi − xj∥ > 2 − ε for i ̸= j. +Using that x is a super ∆ point select, by the definition of super ∆, a point x1 ∈ W with ∥x−x1∥ > +2 − ε +2. +Now, assume that x1, . . . , xn have been constructed and let us construct xn+1. By the properties +defining the sequence observe that, given 1 ⩽ i ⩽ n, we have ∥x−xi∥ > 2− ε +2, so we can find gi ∈ SX∗ +with Re gi(x − xi) > 2 − ε +2, which implies Re gi(x) > 1 − ε +2 and Re gi(xi) < −1 + ε +2. Consequently +x ∈ V := W ∩ +n� +i=1 +S +� +gi, ε +2 +� +, +which is a weakly open set. Since x is super ∆ we can find xn+1 ∈ V such that ∥x−xn+1∥ > 2− ε +2. In +order to finish the construction we only must to prove that ∥xi−xn+1∥ > 2−ε holds for every 1 ⩽ i ⩽ n. +But this is clear because, given 1 ⩽ i ⩽ n, the condition xn+1 ∈ V implies that Re gi(xn+1) > 1 − ε +2, +so +∥xn+1 − xi∥ ⩾ Re gi(xn+1 − xi) > 1 − ε +2 + 1 − ε +2 = 2 − ε, +and the proof is finished. +□ +Note that a similar statement than Theorem 6.7 can be established for ccw ∆ points. + +42 +MART´IN, PERREAU, AND RUEDA ZOCA +Theorem 6.9. Let X be a Banach space and let x ∈ SX be a ccw ∆-point. Then every non-empty +convex combination D of relatively weakly open subsets of BX containing x satisfies that α(D) = 2. +As in the previous case, the proof follows directly from the next result. +Proposition 6.10. Let X be a Banach space, x ∈ SX be a ccw ∆ point, and D a ccw of BX such +that x ∈ D. Then, for every ε > 0, there exists a sequence {xn} ⊆ D such that ∥xi − xj∥ > 2 − ε holds +for every i ̸= j. +Proof. Set ε > 0. Write D := �k +i=1 λiWi with λi ̸= 0 for every i. Set δ := +ε +2 min1⩽i⩽k λi . Let us construct +by induction a sequence {xn} ⊆ D satisfying that ∥x − xi∥ > 2 − δ and such that ∥xi − xj∥ > 2 − ε +for i ̸= j. Using that x is a ccw ∆ point select, by the definition of ccw ∆, a point x1 ∈ D with +∥x − x1∥ > 2 − δ. +Now assume that x1, . . . , xn have been constructed and let us construct xn+1. We can write x = +�k +j=1 λjxj and xi := �k +j=1 λjxi +j as being elements of D. +By the properties defining the sequence, observe that, given 1 ⩽ i ⩽ n we have ∥x − xi∥ > 2 − δ, so +we can find gi ∈ SX∗ with +Re gi(x − xi) = +k +� +j=1 +λj Re gi(xj − xi +j) > 2 − δ = 2 − +ε +2 min1⩽j⩽n λj +. +A convexity argument implies that Re gi(xj − xi +j) > 2 − ε +2 holds for every 1 ⩽ j ⩽ k, which implies +that +Re gi(xj) > 1 − ε +2 and +Re gi(xi +j) < −1 + ε +2. +Observe that +xj ∈ Vi := Wi ∩ +n� +i=1 +S +� +gi, ε +2 +� +, +which is a weakly open set. +Since x is a ccw ∆-point and x ∈ �k +j=1 λjVj, we can find a point +xn+1 = �k +j=1 λjzj ∈ �k +j=1 λjVj ⊆ D such that ∥x − xn+1∥ > 2 − δ. In order to finish the construction +we only must to prove that ∥xi − xn+1∥ > 2 − ε holds for every 1 ⩽ i ⩽ n. Given 1 ⩽ j ⩽ k, the +condition zj ∈ Vj implies Re gi(zj) > 1 − ε +2. On the other hand, Re gi(xi +j) < −1 + ε +2, so +∥xn+1 − xi∥ ⩾ Re gi(x − xi) = +k +� +j=1 +λj Re gi(zj − xi +j) > (2 − ε) +k +� +j=1 +λj = 2 − ε, +and the proof is finished. +□ +7. Commented open questions +The only implications between properties which is not known to hold or not is the following one +(see Figure 2 in page 35). +Question 7.1. Let X be a Banach space and let x ∈ SX be a ccs ∆-point. Is x a super ∆-point? +Let us give some comments on this question. On the one hand, it may look that the answer is +positive by Bourgain’s lemma (Lemma 2.2), but this lemma does not say that, in general, given an +element x of a relative weak open subset W of BX, there is a convex combination of slices of BX + +DIAMETRAL NOTIONS FOR ELEMENTS OF THE UNIT BALL OF A BANACH SPACE +43 +contained in W and containing x. The later happens when x ∈ co(pre-ext (BX)) (see Remark 2.3) so, +the answer to Question 7.1 is positive in this case. On the other hand, a possible counterexample to +this problem could be the molecules in Examples 4.16 or 4.17, which are known to be ccs ∆-points +and are extreme points but not preserved extreme points (hence they do not belong to the convex hull +of the set of preserved extreme points). A way to show that these molecules are not super ∆-points +would be to investigate whether RNP spaces may contains super ∆-points. +Theorem 3.19 states that real Banach spaces with a one-unconditional basis do neither contain +super ∆-points nor ccs ∆-points. It is likely that such result also holds true in the complex setting +since we do believe that the preliminary results from [6] are also valid for complex scalars, provided +that one works with the suitable notion of one-uncondional bases (for which [6, Proposition 2.3] holds). +Also, we also expect that the results there can be easily extended to one-uncondional FDDs. Yet, +since a sharper version of this result was obtained in Proposition 3.20 for super ∆-points in a very +general setting, it is natural to ask whether improved results could be simultaneously obtained in both +directions for ccs ∆-points by proving an analogue to Proposition 3.20. So let us ask the following. +Question 7.2. Let X be a Banach space, and let us assume that there exists a subset A ⊆ F(X, X) +satisfying that sup +� +∥Id − T∥ : T ∈ A +� +< 2 and that for every ε > 0 and every x ∈ X, there exists +T ∈ A such that ∥x − Tx∥ < ε. Can X contain a ccs ∆-point? +A negative answer to this question would be interesting, since it would provide an example of a ccs +∆-point that is not a super ∆-point, hence a negative answer to Question 7.1. +Another interesting question could be if a point of continuity could be a ccs ∆-point. +Let us +formalize the questions. +Question 7.3. Let X be a Banach space. +(1) Does X fail the RNP (or even the CPCP) if contains a super ∆-point or a super Daugavet +point? +(2) Is it possible for a point of continuity being a ccs ∆-point? +The surprising examples given in Section 4 shows that the mere existence of some diametral notions +(but ccs Daugavet points) on a Banach space does not imply that the whole space has any diameter +two property nor the Daugavet property. Our question here is how many diametral points has to +contain a Banach space to have any diameter two property or the Daugavet property or fails to have +the RNP or one-unconditional basis. +Question 7.4. How big can be the set of Daugavet points, super Daugavet points, ∆-points, super +∆-points, or ccs ∆-points in a Banach space with the Radon-Nikod´ym property, or with the CPCP, +or being strongly regular, or having one-unconditional basis? +Concerning isometric consequences of the existence of diametral points, there are some recent results +showing that a Banach space containing a ∆-point cannot be uniformly non-square [5] or even locally +uniformly non-square [37], or asymptotic uniformly smooth [5, 52]. +Also, a Banach space having +an unconditional basis with suppression-unconditional constant less that 2 cannot contains super ∆- +points and a Banach space containing a ccs Daugavet point has the SD2P. Taking into account that +it is not known if there exists an stictly convex Banach space with the Daugavet property (see [33, +Section 5]), the following question makes sense. Recall that Paragraph 4.3.2 shows an example of an +strictly convex Banach space in which every norm-one element is a ccs ∆-point and a super ∆-point, +but it does not contain any Daugavet point by the way in which it is constructed. + +44 +MART´IN, PERREAU, AND RUEDA ZOCA +Question 7.5. Is there an strictly convex Banach space containing a Daugavet point? +In view of Proposition 3.13 and of Theorem 4.14, the following question makes sense. +Question 7.6. Let X be a Banach space. Suppose that x ∈ ext (BX) is a ∆-point, does this imply +that x is a ccs ∆-point or a super ∆-point? +By now, the only isomorphic restriction which is known for a Banach space to contain ∆-points +or even Daugavet points is that it cannot be finite-dimensional. It would be interesting to find some +more. +Question 7.7. Find isomorphic restrictions for a Banach space to contain ∆-points or any of the +other diametral notions. In particular, is it possible for a reflexive or even super-reflexive Banach +space to contain ∆-, super ∆-, ccs ∆-, Daugavet or super Daugavet points? +The results about absolute sums in Subsection 3.2 are not complete in the case of super Daugavet +points and they are even less clear in the case of ccs notions. Here are two possible questions. +Question 7.8. Let X, Y be Banach spaces and let N be an absolute sum. +(1) If N is A-octahedral, x ∈ SX and y ∈ SY are super Daugavet points, is (ax, by) a super +Daugavet point in X ⊕N Y when a, b satisfy the conditions in the definition of A-octahedrality? +(2) If N is the ℓ∞-sum, x ∈ SX and y ∈ SY are ccs ∆-points, are the elements of the form (ax, by) +ccs ∆-points in X ⊕∞ Y for a, b ∈ [0, 1] with max{a, b} = 1? +It would be also desirable to study the reversed results to those in Subsection 3.2 as it is done in +[45] for ∆-points and Daugavet points (see the tables in pages 86 and 87 of [45]). +Question 7.9. Let X, Y be Banach spaces, let N be an absolute sum, x ∈ SX, y ∈ SY , and a, b ⩾ 0 +such that N(a, b) = 1. Discuss what happens with x and y supposing that (ax, by) satisfies any of the +six diametral notions. +It maybe the case that some of the arguments given in Subsections 4.1 and 4.2 can be adapted to +other classes of Banach spaces. We propose some possibilities. +Question 7.10. Characterize the six diametral notions in uniform algebras, in Lorentz spaces and +their isometric preduals, and in some vector-valued function spaces as C(K, X) or L∞(µ, X) spaces. +The relations between the weak-star versions of the diametral points (see Remark 2.6) are not yet +clear. For instance, the following questions arises. +Question 7.11. Let X be a Banach space and x ∈ SX. +(1) Is JX(x) a ccs ∆-point in X∗∗ if x is a ccs ∆-point? +(2) Is there any relationship between the DD2P in X and the weak-star super ∆-points in SX∗? +As commented in Remark 4.19, a Banach space X containing a sequence (yn) of super ∆-points +such that the distance of yn to the set of strongly exposed points of BX is going to zero. But the +following question remains open. +Question 7.12. Can a super ∆-point (or even a ∆-point) belong to the closure of the set of denting +points? + +DIAMETRAL NOTIONS FOR ELEMENTS OF THE UNIT BALL OF A BANACH SPACE +45 +The answer to the next question on the behaviour of ∆- and super ∆-points in rays is still unknown, +as we commented in Section 5. +Question 7.13. Let X be a Banach space and let x ∈ SX. +(1) If rx is a ∆-point for some 0 < r < 1, does this imply that x is a ∆-point? +(2) If rx is a super ∆-point for some 0 < r < 1, does this imply that x is a auper ∆-point? +(3) If x is a super ∆-point, does this imply that rx is a super ∆-point for all 0 < r < 1? +As we proved in Section 6, every relative weakly open subset which contains a super ∆-point +(respectively, a ccw ∆-point) has Kuratowski measure 2. Our proofs do not seem to work for convex +combination of slices, so let us ask the following. +Question 7.14. If a ccs of the unit ball contains a ccs ∆-point, does it necessarily have maximal +Kuratowski measure? +Acknowledgments +Part of this work was done during the visit of the second named author at the University of Granada +in September 2022. 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Departamento de An´alisis Matem´atico, 18071 +Granada, Spain +ORCID: 0000-0003-4502-798X +Email address: mmartins@ugr.es +URL: https://www.ugr.es/local/mmartins +(Perreau) University of Tartu, Institute of Mathematics and Statistics, Narva mnt 18, 51009 Tartu +linn, Estonia +ORCID: 0000-0002-2609-5509 +Email address: yoel.perreau@ut.ee +(Rueda Zoca) Universidad de Granada, Facultad de Ciencias. Departamento de An´alisis Matem´atico, +18071 Granada, Spain +ORCID: 0000-0003-0718-1353 +Email address: abrahamrueda@ugr.es +URL: https://arzenglish.wordpress.com +