# Non-expansive bijections between unit balls of Banach spaces (A technical version with some boring proofs included)

## TL;DR

This paper extends known results about non-expansive bijections being isometries from specific Banach spaces to more general cases involving mappings between different Banach spaces, including finite-dimensional, strictly convex, or  spaces.

## Contribution

It generalizes the class of Banach spaces for which non-expansive bijections are necessarily isometries, covering mappings between different spaces.

## Key findings

- Non-expansive bijections between certain Banach spaces are isometries.
- Extension of isometry results to mappings between different Banach spaces.
- Applicable to finite-dimensional, strictly convex, and  spaces.

## Abstract

It is known that if $M$ is a finite-dimensional Banach space, or a strictly convex space, or the space $\ell_1$, then every non-expansive bijection $F: B_M \to B_M$ is an isometry. We extend these results to non-expansive bijections $F: B_E \to B_M$ between unit balls of two different Banach spaces. Namely, if $E$ is an arbitrary Banach space and $M$ is finite-dimensional or strictly convex, or the space $\ell_1$ then every non-expansive bijection $F: B_E \to B_M$ is an isometry.