A DISCRETE FORM OF THE BECKMAN-QUARLES THEOREM FOR TWO-DIMENSIONAL STRICTLY CONVEX NORMED SPACES

Abstract

Let rho> 0 be a fixxed real number. Let X be a real normed vector space, dim X geq 2. We prove that if x, y in X and |x-y| / rho is a rational number then there exists a finite set. It implies that each map from X to Y that preserves the distancer rho is an affine isometry.