Isometries of the unit sphere (Q1820427)

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.