Nonexpansive bijections between unit balls of Banach spaces. (English) Zbl 1400.46013

Summary: 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.


46B20 Geometry and structure of normed linear spaces
46B04 Isometric theory of Banach spaces
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
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