Rigidity of unit spheres in finite-dimensional real normed spaces
Abstract
We prove that finite-dimensional real normed spaces with isometric unit spheres are linearly isometric, where each sphere carries the distance induced by the ambient norm. This answers the object-level question of Kadets and Martín in finite dimensions, without asserting linear extension of a prescribed sphere isometry. We associate with each norm a family of Plücker contraction bodies generated by volume-normalized maximal minors of linear contractions into finite-dimensional ℓ∞ spaces. Null-Lagrangian identities show that the sphere metric determines these bodies. A finite exposed-face construction recovers almost-norming contractions from the convexified data, and compactness together with an exact volume identity yields a linear isometry.
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BibTeX
@misc{nakamura2026rigidity,
title = {{Rigidity of unit spheres in finite-dimensional real normed spaces}},
author = {Jumpei Nakamura and Ryotaro Tanaka},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.31096},
eprint = {2609.31096},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.31096},
}
RIS
TY - UNPB
TI - Rigidity of unit spheres in finite-dimensional real normed spaces
AU - Jumpei Nakamura
AU - Ryotaro Tanaka
PY - 2026
DA - 2026/09/25
PB - arXiv
DO - 10.48550/arxiv.2609.31096
UR - https://arxiv.org/abs/2609.31096
AB - We prove that finite-dimensional real normed spaces with isometric unit spheres are linearly isometric, where each sphere carries the distance induced by the ambient norm. This answers the object-level question of Kadets and Martín in finite dimensions, without asserting linear extension of a prescribed sphere isometry. We associate with each norm a family of Plücker contraction bodies generated by volume-normalized maximal minors of linear contractions into finite-dimensional ℓ_∞ spaces. Null-Lagrangian identities show that the sphere metric determines these bodies. A finite exposed-face construction recovers almost-norming contractions from the convexified data, and compactness together with an exact volume identity yields a linear isometry.
ER -