A characterization of isometries on an open convex set

Abstract

Let X be a real Hilbert space with dim X >= 2 and let Y be a real normed space which is strictly convex. In this paper, we generalize a theorem of Benz by proving that if a mapping f, from an open convex subset of X into Y, has a contractive distance rho and an extensive one N rho (where N >= 2 is a fixed integer), then f is an isometry.