A contribution to a theorem of Ulam and Mazur (Q1106426)

The authors show the following: Let X be an at least 2-dimensional real Banach space and let Y be another one that is strictly convex. Suppose f:X\(\to Y\) is a mapping such that for some \(\rho >0\) and some integer \(N>1\); \[ \| x-y\| =\rho \Rightarrow \| f(x)-f(y)\| \leq \rho \quad and\quad \| x-y\| =N\rho \Rightarrow \| f(x)-f(y)\| \geq N\rho \] then, f is an affine isometry. This extends an old result of Mazur and Ulam.