Distinct triangle areas in a planar point set

Abstract Erd""os, Purdy, and Straus conjectured that the number of distinct (nonzero) areas of the trianglesdetermined by n noncollinear points in the plane is at least b n-12 c, which is attained for dn/2e andrespectively b n/2c equally spaced points lying on two parallel lines. We show that this number isat least 17 38 n- O(1) ss 0.4473n. The best previous bound, (p2- 1)n- O(1) ss 0.4142n, whichdates back to 1982, follows from the combination of a result of Burton and Purdy [5] and Ungar's theorem [23] on the number of distinct directions determined by n noncollinear points in the plane. 1 Introduction Let S be a finite set of points in the plane. Consider the (nondegenerate) triangles determined by triplesof points of S. There are at most \Gamma n3\Delta triangles, some of which may have the same area. Denote by g(S)the number of distinct (nonzero) areas of the triangles determined b