On weighted sums of numbers of convex polygons in point sets

TL;DR

This paper derives new equalities relating to the counts of convex polygons with specific interior points in point sets, extending classical results and exploring higher-dimensional analogs.

Contribution

It introduces novel equalities for convex polygons with interior points, extending existing equations and applying to higher dimensions.

Findings

Derived new equalities for convex polygons with interior points

Extended known equations for empty convex polygons

Presented results for higher-dimensional 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 Erd\H{o}s-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.