Combinatorial properties of convex polygons in point sets

Abstract

The Erdős-Szekeres theorem is a famous result in Discrete geometry that inspired a lot of research and motivated new problems. The theorem states that for every integer n ≥ 3 there is another integer N_0 such that any set of N ≥ N_0 points in general position in the plane contains the vertex set of a convex n-gon. Related is the question on the number of empty (without interior points) convex k-gons, X_k, in a set of n points, for k=3,4,5,... . A known result states that the alternating sum of the X_k's only depends on the number n of points, but not on the precise positions of the points. A proof was given by Pinchasi, Radoičić, and Sharir in 2006. In this thesis we extend this result to the numbers of convex k-gons with l interior points, X_{k,l}, and provide several formulas involving these numbers. All these formulas only depend on the number n of points of the set. The proofs are based on a continuous motion argument. We further show that with this proof technique at most n-2 linearly independent equations for the X_{k,l}'s can be obtained and we provide n-2 such equations. We also obtain several other formulas, building upon a work by Huemer, Oliveros, Pérez-Lantero, and Vogtenhuber. The obtained formulas could further be useful to solve some open problems related to the Erdős-Szekeres theorem. This thesis also surveys several known results and questions related to this classical problem for point sets in the plane.