Diametral strong diameter two property of Banach spaces is stable under direct sums with 1-norm

TL;DR

This paper demonstrates that the diametral strong diameter two property of Banach spaces remains intact when taking their direct sum with the 1-norm, ensuring stability of this geometric property under such operations.

Contribution

It establishes the stability of the diametral strong diameter 2 property under 1-sums of Banach spaces, a previously unproven geometric invariance.

Findings

The diametral strong diameter 2 property is preserved under 1-sums.

The property is stable when combining Banach spaces with the 1-norm.

The result extends understanding of geometric properties in Banach space theory.

Abstract

We prove that the diametral strong diameter 2 property of a Banach space (meaning that, in convex combinations of relatively weakly open subsets of its unit ball, every point has an "almost diametral" point) is stable under 1-sums, i.e., the direct sum of two spaces with the diametral strong diameter 2 property equipped with the 1-norm has again this property.