Symmetric strong diameter two property

TL;DR

This paper investigates a geometric property of Banach spaces called the symmetric strong diameter two property, exploring its preservation under certain sums and its presence in Lipschitz function spaces over various metric spaces.

Contribution

It introduces the symmetric strong diameter two property, analyzes its stability under $ ext{ell}_$-sums, and demonstrates its occurrence in Lipschitz spaces over specific metric spaces.

Findings

The symmetric strong diameter two property is only preserved by $ ext{ell}_$-sums.

Lipschitz spaces $ ext{Lip}_0(M)$ have the weak star version of the property for several classes of metric spaces.

The property is strictly stronger than the strong diameter two property.

Abstract

We study Banach spaces with the property that, given a finite number of slices of the unit ball, there exists a direction such that all these slices contain a line segment of length almost 2 in this direction. This property was recently named the symmetric strong diameter two property by Abrahamsen, Nygaard, and P\~oldvere. The symmetric strong diameter two property is not just formally stronger than the strong diameter two property (finite convex combinations of slices have diameter 2). We show that the symmetric strong diameter two property is only preserved by $\ell_\infty$-sums, and working with weak star slices we show that $\text{Lip}_0(M)$ have the weak star version of the property for several classes of metric spaces $M$.