An inequality for distances among n points and distance preserving mappings

Abstract

Using familiar properties of norm and inner product, we will prove a new inequality concerning distances between each pair of n points in an inner product space, where n is an integer larger than 3. Moreover, we investigate the Aleksandrov-Rassias problem by proving that if the distance 1 is contractive and the golden ratio is extensive by a mapping f , then f is a linear isometry up to translation.