Relatively weakly open convex combinations of slices

TL;DR

This paper demonstrates that for certain Banach spaces like $c_0$ and $C(K)$ with scattered compact $K$, finite convex combinations of slices of the unit ball are relatively weakly open, revealing geometric properties of these spaces.

Contribution

It establishes that specific Banach spaces have the property that convex combinations of slices are relatively weakly open, extending understanding of their geometric structure.

Findings

Finite convex combinations of slices are relatively weakly open in $c_0$.

This property holds for $C(K)$ spaces with scattered compact $K$.

The result contributes to the geometric theory of Banach spaces.

Abstract

We show that $c_0$, and in fact $C(K)$ for any scattered compact Hausdorff space $K$, have the property that finite convex combinations of slices of the unit ball are relatively weakly open.