Mazur-Ulam Theorem With Gromov-Hausdorff Distance

TL;DR

This paper establishes a characterization of Banach spaces being linearly isometric based on the finiteness of their Gromov-Hausdorff distance, providing new proofs and criteria especially in finite dimensions.

Contribution

It proves that Banach spaces are linearly isometric iff their Gromov-Hausdorff distance is finite, with simplified proofs and new finite-dimensional criteria.

Findings

Banach spaces are linearly isometric iff Gromov-Hausdorff distance is finite

Finite-dimensional case allows simpler proofs and weaker assumptions

Finite subsets can determine isometry in finite-dimensional spaces

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.