On uniform continuity of convex bodies with respect to measures in Banach spaces

TL;DR

This paper investigates the conditions under which convex bodies in Banach spaces are continuous with respect to probability measures, focusing on $ ext{mu}$-continuity of sets like balls and half-spaces, with implications for the Glivenko-Cantelli theorem.

Contribution

It provides new insights into $ ext{mu}$-continuity of convex bodies in Banach spaces and addresses a question posed by F. Tops{ exto}e.

Findings

Characterization of $ ext{mu}$-continuity for convex bodies

Results on $ ext{mu}$-continuity of balls and half-spaces

Answer to F. Tops{ exto}e's question

Abstract

Let $\mu$ be a probability measure on a separable Banach space $X$. A subset $U\subset X$ is $\mu$-continuous if $\mu(\partial U)=0$. In the paper the $\mu$-continuity and uniform $\mu$-continuity of convex bodies in $X$, especially of balls and half-spaces, is considered. The $\mu$-continuity is interesting for study of the Glivenko-Cantelli theorem in Banach spaces. Answer to a question of F. Tops{\o}e is given.