Isometries of the unit sphere
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.
unit spheres of finite dimensional Banach spaces
scientific article; zbMATH DE number 3996509