Some external problems in geometry

Abstract

The question of how often the same distance can occur between k distinct points in n-dimensional Euclidean space En has been extensively studied by Paul Erdös and others. Sir Alexander Oppenheim posed the somewhat similar problem of investigating how many triangles with vertices chosen from among k points in En can have the same non-zero area. A subsequent article by Erdös and Purdy gave some preliminary results on this problem. Here we carry that work somewhat further and show that there cannot be more than ck3−ϵ triangles with the same non-zero area chosen from among k points in E5, where ϵ is a positive constant. Since there can be ck3 such triangles in E6, the result is in a certain sense best possible. The methods used are mainly combinatorial and geometrical. A main tool is a theorem on generalized graphs due to Paul Erdös.