On disjoint tilings of $\ell_1(κ)$ by star-shaped bodies

Abstract

We employ the set-theoretic methods yielding the consistency of the existence of nontrivial pairwise disjoint covers of the unit interval (or equivalently $\mathbb{R}^n$ for $n\in\mathbb{N}\setminus\{0\}$) by less than $2^ω$ closed sets in the context of covers of infinite dimensional Banach spaces $\ell_1(κ)$ for infinite $κ$ under geometric conditions arising in tiling theory and in approximation theory. Specifically, for $κ=ω_1, ω_2$, using side-by-side Sacks forcing we prove the consistency of an arbitrarily large continuum above $κ$ with the existence of a highly disconnected proximinal set in $\ell_1(κ)$, built from norm-compact pieces separated by a common positive distance, and the existence of a normal disjoint tiling of $\ell_1(κ)$ by star-shaped bodies of the form $X+B$, where $X$ is norm compact and $B$ is the unit ball. We also prove that if $κ$ is an uncountable cardinal of countable cofinality, then $\ell_1(κ)$ does not admit any normal disjoint tiling by bodies of the form $X+B$, where $X$ is closed and norm-separable and $B$ is the unit ball. These results complement a result of Klee of 1981 obtained for $κ$ satisfying $κ^ω=κ$, recent results of De Bernardi, Russo, Sezgek and Somaglia, and a $\mathsf{ZFC}$ a machine-discovered result (included in the appendix) that there are no nontrivial discrete Chebyshev sets in $\ell_1(κ)$ when $κ<2^ω$.