A characterization of isometries on an open convex set

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 ρ and an extensive one Nρ (where N ≥ 2 is a fixed integer), then f is an isometry.