On weighted sums of numbers of convex polygons in point sets

Abstract

The version of record of this article, first published in Discrete & Computational Geometry, is available online at Publisher’s website: https://doi.org/10.1007/s00454-022-00395-8

Let S be a set of n points in general position in the plane, and let Xk,(S) be the number of convex k-gons with vertices in S that have exactly points of S in their interior. We prove several equalities for the numbers Xk,(S). This problem is related to the Erd¿os–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.

Research of C.H. was partially supported by project MTM2015-63791-R (MINECO/ FEDER), PID-2019-104129GB-I00/MCIN/AEI/10.13039/501100011033, and by project Gen. Cat. DGR 2017SGR1336. D.O. was partially supported by project PAPIIT IG100721 and CONACyT 282280. P. P-L. was partially supported by project DICYT 041933PL Vicerrectoría de Investigación, Desarrollo e Innovación USACH (Chile), and Programa Regional STICAMSUD 19-STIC-02. Research of B.V. was partially supported by the Austrian Science Fund (FWF) within the collaborative DACH project Arrangements and Drawings as FWF Project I 3340-N35. We thank an anonymous referee for helpful comments. This project has been supported by the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 734922

Postprint (author's final draft)