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The Ball-Covering Property in Lipschitz-Free Spaces
Abstract
We study ball-covering properties in Lipschitz-free spaces. We establish an extension criterion for proving that ℱ(M) fails the ball-covering property and apply it to several classes of nonseparable metric spaces. In contrast, we construct a nonseparable uniformly discrete metric space M such that ℱ(M) has the uniform ball-covering property and is isomorphic to ℓ1(2ω). More precisely, ℱ(M) has the α-ball-covering property for every α∈[−1,1). This example also shows that these quantitative ball-covering properties are not hereditary within the class of Lipschitz-free spaces. Finally, we prove some stability results under sufficiently small bi-Lipschitz perturbations of the metric.
Related
- The renorming-stable ball covering property and C(K)-spaces
- The Dunford-Pettis property for Lipschitz-free spaces over ℝn
- Kwapień's property (K) in Lipschitz-free spaces
- Structural consequences of the Schur p-property for Lipschitz-free p-spaces
- The ball fixed point property for continuous functions on locally compact spaces
BibTeX
@misc{aliaga2026ballcovering,
title = {{The Ball-Covering Property in Lipschitz-Free Spaces}},
author = {Ramón J. Aliaga and Colin Petitjean and Antonín Procházka and Daniele Puglisi},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.27485},
eprint = {2609.27485},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.27485},
}
RIS
TY - UNPB
TI - The Ball-Covering Property in Lipschitz-Free Spaces
AU - Ramón J. Aliaga
AU - Colin Petitjean
AU - Antonín Procházka
AU - Daniele Puglisi
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2609.27485
UR - https://arxiv.org/abs/2609.27485
AB - We study ball-covering properties in Lipschitz-free spaces. We establish an extension criterion for proving that ℱ(M) fails the ball-covering property and apply it to several classes of nonseparable metric spaces. In contrast, we construct a nonseparable uniformly discrete metric space M such that ℱ(M) has the uniform ball-covering property and is isomorphic to ℓ_1(2^ω). More precisely, ℱ(M) has the α-ball-covering property for every α∈[−1,1). This example also shows that these quantitative ball-covering properties are not hereditary within the class of Lipschitz-free spaces. Finally, we prove some stability results under sufficiently small bi-Lipschitz perturbations of the metric.
ER -