Minimum number of different distances defined by a finite number of points

Abstract

We study the minimum number of different distances defined by a finite number of points in the following cases: a) we consider metrics different from the euclidean distance in the plane, b) we consider the euclidean distance but restricted to subsets of the plane of special interest, c) we consider other topological surfaces: the cylinder and the flat torus. All these results extend those obtained by Erdös and other mathematicians for the euclidean distance in the plane.