A Note on the Number of Empty Triangles

Abstract

AbstractLet P be a set of n points on the plane, in general position, H of them placed on the boundary of the convex hull of P. In this note we prove that there is a well defined family of empty triangles, the family of empty triangles not generated by an empty convex pentagon, containing exactly n2 − 5n + H + 4 empty triangles. This result immediately implies a slight improvement on the lower bound on the number of empty triangles that every set of n points in the plane must determine.KeywordsEmpty triangleempty convex pentagonconvex hullpoints in general position