Compactivorous Sets in Banach Spaces

TL;DR

This paper introduces the concept of compactivorous sets in Banach spaces, characterizes their properties in various contexts, and extends key results to products of locally compact Polish groups.

Contribution

It provides new characterizations of compactivorous sets in both separable and nonseparable Banach spaces and extends the main theorem to countable products of locally compact Polish groups.

Findings

Compactivorous sets guarantee Haar nonnegligibility in separable Banach spaces.

Characterizations of compactivorous sets in different Banach space contexts.

Extension of main theorem to products of locally compact Polish groups.

Abstract

A set $E$ in a Banach space $X$ is compactivorous if for every compact set $K$ in $X$ there is a nonempty, (relatively) open subset of $K$ which can be translated into $E$. In a separable Banach space, this is a sufficient condition which guarantees the Haar nonnegligibility of Borel subsets. We give some characterisations of this property in both separable and nonseparable Banach spaces and prove an extension of the main theorem to countable products of locally compact Polish groups.