Isometries of the unit sphere
type: Item
instance of: scholarly article
title: Isometries of the unit sphere
zbMATH Open document ID: 0615.51005
DOI: 10.1007/BF00147942
author: Daryl Tingley
published in: Geometriae Dedicata
publication date: 1987-01-01T00:00:00Z
review text: The following main theorem is shown: Suppose that S and S' are the unit spheres of finite dimensional Banach spaces. If \(f: S\to S'\) is a surjective isometry, then \(f(-x)=-f(x)\) for all \(x\in S\). This is a first step to generalize the theorem of \textit{S. Mazur} and \textit{S. Ulam} [C.R. Acad. Sci., Paris 194, 946-948 (1932; Zbl 0004.02103)] on isometries between Banach spaces. One main tool in the proof is a result on the extendability of isometries from subsets to the ambient spaces shown by \textit{P. Mankiewicz} [Bull. Acad. Pol. Sci., Ser. Sci. Math. Astron. Phys. 20, 367-371 (1972; Zbl 0234.46019)]. The other considerations use standard techniques from the theory of convex bodies. The assumption of finite dimension is essential for the proof.
Mathematics Subject Classification ID: 51F99
Mathematics Subject Classification ID: 46B20
Mathematics Subject Classification ID: 52A20
zbMATH DE Number: 3996509
zbMATH Keywords: antipodal points
zbMATH Keywords: unit spheres of finite dimensional Banach spaces
zbMATH Keywords: isometry
reviewed by: Bernd Wegner
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/bf00147942
OpenAlex ID: W2061273287
description: scientific article; zbMATH DE number 3996509