On a problem of Mazur and Sternbach

TL;DR

This paper addresses a classical problem in Banach space theory, proving that bijective local isometries are global isometries under certain conditions, using new metric geometry techniques.

Contribution

It provides a novel approach to Problem 155, showing that local isometries are global isometries in Banach spaces with an alternative method.

Findings

Affirmative answer for local isometries in Banach spaces

New metric geometry approach to the problem

Extension of previous results to broader class of maps

Abstract

We investigate Problem 155 form the "Scottish Book" due to S. Mazur and L. Sternbach. In modern terminology they asked if every bijective, locally isometric map between two real Banach spaces is always a global isometry. Recently, an affirmative answer when the source space is separable was obtained by M. Mori, using techniques related to the Mazur-Ulam theorem and its generalizations. The main purpose of this short note is to offer a different approach to Problem 155 motivated by recent advances in metric geometry. We show that it has an affirmative answer under the additional assumption that the map considered is a local isometry.