Covering $L^p$ spaces by balls

TL;DR

This paper proves that in certain infinite-dimensional Banach spaces, any covering by closed balls inevitably overlaps infinitely often at some point, highlighting a fundamental property of such coverings.

Contribution

It establishes a new infinite-dimensional covering property for separable uniformly rotund and smooth Banach spaces, showing unavoidable infinite overlaps.

Findings

Existence of points covered infinitely often in such spaces

Universal property for coverings by closed balls

Implications for geometric structure of Banach spaces

Abstract

We prove that, given any covering of any separable infinite-dimensional uniformly rotund and uniformly smooth Banach space $X$ by closed balls each of positive radius, some point exists in $X$ which belongs to infinitely many balls.