ON DISTANCE-PRESERVING MAPPINGS

Abstract

We generalize a theorem of W. Benz by proving the following result: Let <TEX>$H_{\theta}$</TEX> be a half space of a real Hilbert space with dimension <TEX>$\geq$</TEX> 3 and let Y be a real normed space which is strictly convex. If a distance <TEX>$\rho$</TEX> > 0 is contractive and another distance N<TEX>$\rho$</TEX> (N <TEX>$\geq$</TEX> 2) is extensive by a mapping f : <TEX>$H_{\theta}$</TEX> \longrightarrow Y, then the restriction f│<TEX>$_{\theta}$</TEX> <TEX>$H_{+}$</TEX><TEX>$\rho$</TEX>/2// is an isometry, where <TEX>$H_{\theta}$</TEX>＋<TEX>$\rho$</TEX>/2/ is also a half space which is a proper subset of <TEX>$H_{\theta}$</TEX>. Applying the above result, we also generalize a classical theorem of Beckman and Quarles.