Isometrien in normierten Räumen. (Isometries in normed spaces) (Q1073297)

Let \(\rho >0\) and \(N>1\) be a fixed real number and integer respectively. Let X, Y be real normed linear spaces with the following properties: (*) If \(a\in X\) and \(\| a\| <1\), then there exists \(b\in X\) such that \(\| a-b\| =1=\| a+b\|.\) (**) If a,b\(\in Y\) and \(\| a\| =1=\| b\|\) and \(\| a+b\| =2\), then \(a=b.\) The author shows that if \(f: X\to Y\) is a mapping such that \(\| x- y\| =\rho \Rightarrow \| f(x)-f(y)\| \leq \rho\) and \(\| x- y\| =N\rho \Rightarrow \| f(x)-f(y)\| \geq N\rho\) for all x,y\(\in X\), then f is an isometry from X to Y. This generalizes a result of \textit{E. M. Schröder} [Aequationes Math. 19, 89-92 (1979; Zbl 0413.51002)]. Since (**) is equivalent to Y being strictly convex, it now follows from a result of \textit{J. A. Baker} [Am. Math. Mon. 78, 655-658 (1971; Zbl 0214.127)] that \(x\to f(x)-f(0)\) is linear. Using a result of \textit{F. S. Beckman} and \textit{D. A. Quarles jun.} [Proc. Am. Math. Soc. 4, 810-815 (1953; Zbl 0052.182)] the author shows that when \(X=Y={\mathbb{R}}^ n\) (n\(\geq 2)\) his theorem is true for \(n=1\) also.