On disjoint tilings of ℓ1(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 ℝn for n∈ℕ∖{0}) by less than 2ω closed sets in the context of covers of infinite dimensional Banach spaces ℓ1(κ) for infinite κ under geometric conditions arising in tiling theory and in approximation theory. Specifically, for κ=ω1, ω2, 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 ℓ1(κ), built from norm-compact pieces separated by a common positive distance, and the existence of a normal disjoint tiling of ℓ1(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 ℓ1(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 κω=κ, recent results of De Bernardi, Russo, Sezgek and Somaglia, and a ZFC a machine-discovered result (included in the appendix) that there are no nontrivial discrete Chebyshev sets in ℓ1(κ) when κ<2ω.
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BibTeX
@misc{koszmider2026disjoint,
title = {{On disjoint tilings of $\ell_1(κ)$ by star-shaped bodies}},
author = {Piotr Koszmider and Piotr Szewczak},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.35180},
eprint = {2609.35180},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.35180},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On disjoint tilings of ℓ_1(κ) by star-shaped bodies
AU - Piotr Koszmider
AU - Piotr Szewczak
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2609.35180
UR - https://arxiv.org/abs/2609.35180
AB - We employ the set-theoretic methods yielding the consistency of the existence of nontrivial pairwise disjoint covers of the unit interval (or equivalently ℝ^n for n∈ℕ∖{0}) by less than 2^ω closed sets in the context of covers of infinite dimensional Banach spaces ℓ_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 ℓ_1(κ), built from norm-compact pieces separated by a common positive distance, and the existence of a normal disjoint tiling of ℓ_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 ℓ_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 ZFC a machine-discovered result (included in the appendix) that there are no nontrivial discrete Chebyshev sets in ℓ_1(κ) when κ<2^ω.
ER -