On Weighted Sums of Numbers of Convex Polygons in Point Sets

Abstract

Let S be a set of n points in general position in the plane, and let X_{k,ell }(S) be the number of convex k-gons with vertices in S that have exactly ell points of S in their interior. We prove several equalities for the numbers X_{k,ell }(S). This problem is related to the Erdős–Szekeres theorem. Some of the obtained equations also extend known equations for the numbers of empty convex polygons to polygons with interior points. Analogous results for higher dimension are shown as well.