Remarks on diameter 2 properties

TL;DR

This paper investigates the diameter 2 property in Banach spaces, showing that certain spaces and their finite convex combinations have this property, and introduces new examples where it holds.

Contribution

It extends the diameter 2 property to finite convex combinations and provides new examples of spaces with this property beyond previously known cases.

Findings

Finite convex combinations of weakly open subsets have diameter 2.

Forming -sums preserves the diameter 2 property.

New examples of diameter 2 spaces are identified.

Abstract

If $X$ is an infinite-dimensional uniform algebra, if $X$ has the Daugavet property or if $X$ is a proper $M$-embedded space, every relatively weakly open subset of the unit ball of the Banach space $X$ is known to have diameter 2, i.e., $X$ has the diameter 2 property. We prove that in these three cases even every finite convex combination of relatively weakly open subsets of the unit ball have diameter 2. Further, we identify new examples of spaces with the diameter 2 property outside the formerly known cases; in particular we observe that forming $\ell_p$-sums of diameter 2 spaces does not ruin diameter 2 structure.