Tilings and coverings by balls in ${\ell }_1$

Abstract

A famous result of Klee from 1981 is that the Banach space ${\ell }_1(\kappa )$ admits a disjoint tiling by balls of radius $1$, for all cardinals $\kappa$ with ${\kappa }^{\omega }=\kappa$. Klee also observed that the smallest cardinal in which such a tiling might exist is $\kappa =2^{{\aleph }_0}$, leaving open the question whether, for $\kappa <2^{{\aleph }_0}$, ${\ell }_1(\kappa )$ might admit a tiling by balls at all. Our main result answers this question in the negative, proving in particular that ${\ell }_1$ does not admit any tiling by balls. We also give a companion result about star-$n$-finite coverings by balls of ${\ell }_1(\kappa )$ and we give a construction of a star-finite tiling of $\mathcal{X}{\oplus }_{\infty }{c}_{00}$, for each space $\mathcal{X}$ whose dimension is at most countable.