Some extremal problems in geometry

Abstract

The question of how often the same distance can occur between k distinct points in n -dimensional Euclidean space E n 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 E n 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 ck 3− ϵ triangles with the same non-zero area chosen from among k points in E 5 , where ϵ is a positive constant. Since there can be ck 3 such triangles in E 6 , 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.