On duality of diameter 2 properties

TL;DR

This paper explores duality relationships between diameter 2 properties in Banach spaces and introduces new forms of octahedrality, providing simplified proofs and deeper understanding of the geometric structure of Banach spaces.

Contribution

It introduces two new versions of octahedrality that are dual to diameter 2 properties, enhancing the theoretical framework and stability analysis of these geometric properties.

Findings

Established dual relationships between diameter 2 properties and octahedrality.

Introduced new forms of octahedrality related to diameter 2 properties.

Provided simplified proofs for known results using stability properties.

Abstract

It is known that a Banach space has the strong diameter 2 property (i.e. every convex combination of slices of the unit ball has diameter 2) if and only if the norm on its dual space is octahedral (a notion introduced by Godefroy and Maurey). We introduce two more versions of octahedrality, which turn out to be dual properties to the diameter 2 property and its local version (i.e., respectively, every relatively weakly open subset and every slice of the unit ball has diameter 2). We study stability properties of different types of octahedrality, which, by duality, provide easier proofs of many known results on diameter 2 properties.