Daugavet- and delta-points in Banach spaces with unconditional bases

Abstract

We study the existence of Daugavet-and delta-points in the unit sphere of Banach spaces with a 1-unconditional basis. A norm one element x in a Banach space is a Daugavet-point (resp. delta-point) if every element in the unit ball (resp. x itself) is in the closed convex hull of unit ball elements that are almost at distance 2 from x. A Banach space has the Daugavet property (resp. diametral local diameter two property) if and only if every norm one element is a Daugavet-point (resp. delta-point). It is wellknown that a Banach space with the Daugavet property does not have an unconditional basis. Similarly spaces with the diametral local diameter two property do not have an unconditional basis with suppression unconditional constant strictly less than 2.We show that no Banach space with a subsymmetric basis can have delta-points. In contrast we construct a Banach space with a 1-unconditional basis with delta-points, but with no Daugavetpoints, and a Banach space with a 1-unconditional basis with a unit ball in which the Daugavet-points are weakly dense.

Daugavet points and $Δ$-points in Lipschitz-free spaces

“…The first example of a ∆-point which is not a Daugavet point [2, Example 4.7] required a study of absolute normalised norms (which was pushed quite further in [15]). See also [3] for more technical examples of Banach spaces containing ∆-points which are not Daugavet points.…”

Delta-points in Banach spaces generated by adequate families

“…From [ALMT20] we need the notion of minimal norming subsets for vectors in a Banach space with 1-unconditional basis (see also the notion of 1-sets in [BDHQ19]).…”