# Strong diameter two property and convex combination of slices reaching the unit sphere

## TL;DR

This paper characterizes Banach spaces where convex combinations of slices of the unit ball intersect the sphere, linking geometric properties to the diameter two property and analyzing convex combinations' topological features.

## Contribution

It provides a characterization of Banach spaces with the strong diameter two property through convex combinations of slices and explores their topological and geometric properties.

## Key findings

- Convex combinations of slices intersect the sphere iff they contain points at distance two.
- Characterization of spaces where convex combinations of slices are relatively open.
- Analysis of these properties in $L_{}$-spaces and preduals of $L_1$-spaces.

## Abstract

We characterise the class of those Banach spaces in which every convex combination of slices of the unit ball intersects the unit sphere as the class of those spaces in which every convex combination of slices of the unit ball contains two points at distance exactly two. Also, we study when the convex combinations of slices of the unit ball are relatively open or has non-empty relative interior for different topologies, studying the relationship between them and studying these properties for $L_{\infty}$-spaces and preduals of $L_1$-spaces.