On a problem of Mazur and Sternbach

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.