Nonexpansive bijections between unit balls of Banach 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 nonexpansive bijection $F:B_M\to B_M$ of its unit ball $B_M$ is an isometry. We extend these results to nonexpansive 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 nonexpansive bijection $F:B_E\to B_M$ is an isometry.