Asymptotic geometry and delta-points

TL;DR

This paper investigates the geometric properties of Banach spaces related to Daugavet- and delta-points, providing criteria for their absence in certain spaces and constructing examples with such points under specific conditions.

Contribution

It establishes conditions under which Banach spaces do not contain delta-points and constructs examples of superreflexive spaces with delta-points based on weaker assumptions.

Findings

Asymptotically uniformly smooth and reflexive asymptotically uniformly convex spaces lack delta-points.

The James tree space and its predual also do not contain delta-points.

Existence of superreflexive spaces with delta-points depends on weaker conditions.

Abstract

We study Daugavet- and $\Delta$-points in Banach spaces. A norm one element $x$ is a Daugavet-point (respectively a $\Delta$-point) if in every slice of the unit ball (respectively in every slice of the unit ball containing $x$) you can find another element of distance as close to $2$ from $x$ as desired. In this paper we look for criteria and properties ensuring that a norm one element is not a Daugavet- or $\Delta$-point. We show that asymptotically uniformly smooth spaces and reflexive asymptotically uniformly convex spaces do not contain $\Delta$-points. We also show that the same conclusion holds true for the James tree space as well as for its predual. Finally we prove that there exists a superreflexive Banach space with a Daugavet- or $\Delta$-point provided there exists such a space satisfying a weaker condition.