## 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.

### MSC:

 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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### References:

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