Most likely balls in Banach spaces: existence and non-existence

TL;DR

This paper investigates the conditions under which convex sets, like balls of fixed radius, exist with maximal probability in Banach spaces, providing both criteria and counterexamples.

Contribution

It introduces a general criterion for the existence of fixed-shape convex sets with maximal probability in Banach spaces and presents counterexamples where such sets do not exist.

Findings

Established a criterion for existence of convex sets of fixed shape in Banach spaces.

Provided counterexamples demonstrating non-existence in certain cases.

Highlighted the limitations of existing assumptions in the theory.

Abstract

We establish a general criterion for the existence of convex sets of fixed shape as, e.g., balls of a given radius, of maximal probability on Banach spaces. We also provide counterexamples showing that their existence my fail even in some common situations.