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 ℓ1, then every nonexpansive bijection F:BM→BM of its unit ball BM is an isometry. We extend these results to nonexpansive bijections F:BE→BM 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 ℓ1, then every nonexpansive bijection F:BE→BM is an isometry.