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Isometric copies of ℓ∞n in Lipschitz-free spaces over finite metric spaces
Abstract
For every n∈ℕ, we construct a finite metric subspace Mn of ℓ∞n such that the Lipschitz-free space ℱ(Mn) contains a linear isometric copy of ℓ∞n. This answers a question posed by Khan, Mim, and Ostrovskii, who obtained examples in dimensions three and four. As a consequence, the finite-dimensional real Banach spaces that admit a linear isometric embedding into a Lipschitz-free space over a finite metric space are precisely the polyhedral spaces.
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BibTeX
@misc{haller2026isometric,
title = {{Isometric copies of $\ell_\infty^n$ in Lipschitz-free spaces over finite metric spaces}},
author = {Rainis Haller},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.16230},
eprint = {2609.16230},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.16230},
}
RIS
TY - UNPB
TI - Isometric copies of ℓ_∞^n in Lipschitz-free spaces over finite metric spaces
AU - Rainis Haller
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.16230
UR - https://arxiv.org/abs/2609.16230
AB - For every n∈ℕ, we construct a finite metric subspace M_n of ℓ_∞^n such that the Lipschitz-free space ℱ(M_n) contains a linear isometric copy of ℓ_∞^n. This answers a question posed by Khan, Mim, and Ostrovskii, who obtained examples in dimensions three and four. As a consequence, the finite-dimensional real Banach spaces that admit a linear isometric embedding into a Lipschitz-free space over a finite metric space are precisely the polyhedral spaces.
ER -