On distance-preserving mappings

Abstract

We generalize a theorem of W. Benz by proving the following result: Let H(theta) be a half space of a real Hilbert space with dimension greater than or equal to 3 and let Y be a real normed space which is strictly convex. If a distance p > 0 is contractive and another distance Nrho (N greater than or equal to 2) is extensive by a mapping f : H(theta) --> Y, then the restriction f\H(theta+rho/2) is an isometry, where H(theta+rho/2) is also a half space which is a proper subset of H(theta). Applying the above result, we also generalize a. classical theorem of Beckman and Quarles.