Extremal problems on triangle areas in two and three dimensions

The study of extremal problems on triangle areas was initiated in a series of papers by Erdos 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(n44/19 ) = O(n2.3158 ) upper bound on the number of unit-area triangles spanned by n points, which is the first breakthrough improving the classical bound of O(n7/3 ) 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 (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 (n2 - n); there exist n-element point sets (for arbitrarily large n) that span (6/2 - o(1))n2 minimum-area triangles. (iii) The number of acute...

• Type: Conference
• Year: 2008
• Where: COMPGEOM
• Authors: Adrian Dumitrescu, Micha Sharir, Csaba D. Tóth
• Keywords: COMPGEOM 2008, Discrete Geometry, N-element Point Sets, Point Sets, Triangles