Extremal problems on triangle areas in two and three dimensions

Abstract

The study of extremal problems on triangle areas was initiated in a series of papers by Erdős and Purdy in the early 1970s. In this paper we present new results on such problems, concerning the number of triangles of the same area that are spanned by finite point sets in the plane and in 3-space, and the number of distinct areas determined by the triangles. In the plane, our main result is an O ( n 44 / 19 ) = O ( n 2.3158 ) upper bound on the number of unit-area triangles spanned by n points, which is the first breakthrough improving the classical bound of O ( n 7 / 3 ) from 1992. We also make progress in a number of important special cases. We show that: (i) For points in convex position, there exist n-element point sets that span Ω ( n log n ) triangles of unit area. (ii) The number of triangles of minimum (nonzero) area determined by n points is at most 2 3 ( n 2 − n ) ; there exist n-element point sets (for arbitrarily large n) that span ( 6 / π 2 − o ( 1 ) ) n 2 minimum-area triangles. (iii) The number of acute triangles of minimum area determined by n points is O ( n ) ; this is asymptotically tight. (iv) For n points in convex position, the number of triangles of minimum area is O ( n ) ; this is asymptotically tight. (v) If no three points are allowed to be collinear, there are n-element point sets that span Ω ( n log n ) minimum-area triangles (in contrast to (ii), where collinearities are allowed and a quadratic lower bound holds). In 3-space we prove an O ( n 17 / 7 β ( n ) ) = O ( n 2.4286 ) upper bound on the number of unit-area triangles spanned by n points, where β ( n ) is an extremely slowly growing function related to the inverse Ackermann function. The best previous bound, O ( n 8 / 3 ) , is an old result of Erdős and Purdy from 1971. We further show, for point sets in 3-space: (i) The number of minimum nonzero area triangles is at most n 2 + O ( n ) , and this is worst-case optimal, up to a constant factor. (ii) There are n-element point sets that span Ω ( n 4 / 3 ) triangles of maximum area, all incident to a common point. In any n-element point set, the maximum number of maximum-area triangles incident to a common point is O ( n 4 / 3 + ε ) , for any ε > 0 . (iii) Every set of n points, not all on a line, determines at least Ω ( n 2 / 3 / β ( n ) ) triangles of distinct areas, which share a common side.