Strong Diameter Two Property and Convex Combinations of Slices Reaching the Unit Sphere

Abstract

We characterise the class of those Banach spaces in which every convex combination of slices of the unit ball intersects the unit sphere as the class of those spaces in which every convex combination of slices of the unit ball contains two points at distance exactly two. Also, we study when the convex combinations of slices of the unit ball are relatively open or has non-empty relative interior for different topologies, studying the relationship between them and studying these properties for L∞-spaces and preduals of L 1 -spaces.

Attaining strong diameter two property for infinite cardinals

“…It is known that the projective tensor product preserves both the SD2P [8, Corollary 3.6] and the ASD2P [29,Proposition 3.6]. We begin this section by extending this result to the 1-ASD2P κ .…”

“…In [2, Section 3] it is wondered which Banach spaces satisfy this weaker condition. Motivated by this question, the property was studied under the name (P3) in [20], (CS) in [29] and attaining strong diameter two property (ASD2P for short) in the recent preprint [30]. From now on we will use the name ASD2P.…”

“…In [29,Theorem 3.4] it is proved that a Banach space X has the ASD2P if and only if for every finite convex combination of slices C of B X there are x, y ∈ C such that x + y = 2. Therefore, the ASD2P implies that every finite convex combination of slices has diameter two.…”

“…This property is called the strong diameter two property (SD2P for short) and it was first introduced in [3]. Note that there are spaces with the SD2P, but failing to have the ASD2P [29,Example 3.3].…”

Bidual Octahedral Renormings and Strong Regularity in Banach Spaces

“…Observe that the above result gives, in particular, that SD2P and w * -SD2P in X * are equivalent under renorming, whenever X is separable. However, these two properties are not equivalent (see [1], see also [14])).…”