A note on point-finite coverings by balls

TL;DR

This paper offers an elementary proof that certain infinite-dimensional Hilbert and Banach spaces cannot be covered by finitely many open or closed balls of positive radius, extending previous results with new techniques.

Contribution

It provides a simplified proof of non-coverability by point-finite collections of balls in specific infinite-dimensional spaces, including Hilbert and uniformly rotund and smooth Banach spaces.

Findings

Infinite-dimensional Hilbert spaces with density less than continuum cannot be point-finitely covered by balls.

The non-covering result extends to uniformly rotund and smooth Banach spaces.

The proof uses a variation of Lindenstrauss and Phelps' argument related to extreme points.

Abstract

We provide an elementary proof of a result by V.P.~Fonf and C.~Zanco on point-finite coverings of separable Hilbert spaces. Indeed, by using a variation of the famous argument introduced by J.~Lindenstrauss and R.R.~Phelps \cite{LP} to prove that the unit ball of a reflexive infinite-dimensional Banach space has uncountably many extreme points, we prove the following result: Let $X$ be an infinite-dimensional Hilbert space satisfying $\mathrm{dens}(X)<2^{\aleph_0}$, then $X$ does not admit point-finite coverings by open or closed balls, each of positive radius. In the second part of the paper, we follow the argument introduced by V.P. Fonf, M. Levin, and C. Zanco in \cite{FonfLevZan14} to prove that the previous result holds also in infinite-dimensional Banach spaces that are both uniformly rotund and uniformly smooth.