Mazur-Ulam Theorem With Gromov-Hausdorff Distance
Metric Geometry
2026-02-18 v1 Functional Analysis
Abstract
It is shown that two Banach spaces are linearly isometric if and only if the Gromov--Hausdorff distance between them is finite, in particular, zero. The proof is compilative and relies on results obtained by many researchers on the approximability of almost-surjective almost-isometries by linear surjective isometries. In the finite-dimensional case, previously obtained by I.~Mikhailov, a simpler proof under weaker assumptions is given. In the finite-dimensional case, a criterion for isometry in terms of finite (compact) subsets is also given.
Cite
@article{arxiv.2602.15525,
title = {Mazur-Ulam Theorem With Gromov-Hausdorff Distance},
author = {S. A. Bogaty and A. A. Tuzhilin},
journal= {arXiv preprint arXiv:2602.15525},
year = {2026}
}
Comments
14 pages