Diameter two properties, convexity and smoothness

TL;DR

This paper investigates how diameter 2 properties in Banach spaces influence the smoothness and strict convexity of their biduals, revealing that strong diameter 2 properties prevent these geometric features.

Contribution

It establishes that the strong diameter 2 property inhibits the bidual from being strictly convex or smooth and introduces a new property related to slices and points close to diameter 2.

Findings

Strong diameter 2 property prevents bidual from being smooth or strictly convex.

Characterization of a property involving slices and points close to diameter 2.

Spaces with this property have non-smooth biduals.

Abstract

We study smoothness and strict convexity of (the bidual) of Banach spaces in the presence of diameter 2 properties. We prove that the strong diameter 2 property prevents the bidual from being strictly convex and being smooth, and we initiate the investigation whether the same is true for the (local) diameter 2 property. We also give characterizations of the following property for a Banach space $X$: "For every slice $S$ of $B_X$ and every norm-one element $x$ in $S$, there is a point $y\in S$ in distance as close to 2 as we want." Spaces with this property are shown to have non-smooth bidual.