Big slices versus big relatively weakly open subsets in Banach spaces

TL;DR

This paper investigates the size differences between slices and relatively weakly open subsets of the unit ball in Banach spaces, demonstrating that spaces containing $c_0$ can be renormed to exhibit maximal slice diameter while having small weakly open subsets.

Contribution

It proves that Banach spaces with $c_0$ can be renormed to show maximal slice diameter alongside arbitrarily small weakly open subsets, resolving an open problem.

Findings

Spaces with $c_0$ can be renormed to have slices of diameter 2.

Such spaces can contain weakly open subsets with arbitrarily small diameter.

The results highlight fundamental differences between slices and weakly open subsets.

Abstract

We study the unknown differences between the size of slices and relatively weakly open subsets of the unit ball in Banach spaces. We show that every Banach space containing isomorphic copies of $c_0$ can be equivalently renormed so that every slice of its unit ball has diameter 2 and satisfying that its unit ball contains nonempty relatively weakly open subsets with diameter arbitrarily small, which answers an open problem and stresses the differences of diameter between slices and relatively weakly open subsets of the unit ball in Banach spaces.