Covering $B_X$ by finitely many convex sets

TL;DR

This paper investigates the structure of finite coverings of the unit ball in infinite-dimensional Banach spaces by convex sets, focusing on the existence of large-dimensional or infinite-dimensional subsets within the covering sets.

Contribution

It establishes that such coverings always contain a set with arbitrarily high finite dimension, but only under restrictive conditions can they contain infinite-dimensional sets with large radius.

Findings

Existence of high finite-dimensional subsets within coverings

Limitations on infinite-dimensional subsets with large radius

Conditions under which large-radius infinite-dimensional sets exist

Abstract

Given a finite covering by closed convex sets of $B_X$, the unit ball of an infinite-dimensional Banach space, we investigate whether there is a set of the covering that contains balls of radius close to $1$ and (a) arbitrarily high finite dimension or (b) infinite dimension. In case (a) the answer is affirmative, but for the case (b) we just get radius close to $1/2$ and finite codimension under much more restrictive hypotheses.