On $k$-Gons and $k$-Holes in Point Sets
Discrete Mathematics
2014-09-02 v1 Computational Geometry
Abstract
We consider a variation of the classical Erd\H{o}s-Szekeres problems on the existence and number of convex -gons and -holes (empty -gons) in a set of points in the plane. Allowing the -gons to be non-convex, we show bounds and structural results on maximizing and minimizing their numbers. Most noteworthy, for any and sufficiently large , we give a quadratic lower bound for the number of -holes, and show that this number is maximized by sets in convex position.
Keywords
Cite
@article{arxiv.1409.0081,
title = {On $k$-Gons and $k$-Holes in Point Sets},
author = {Oswin Aichholzer and Ruy Fabila-Monroy and Hernán González-Aguilar and Thomas Hackl and Marco A. Heredia and Clemens Huemer and Jorge Urrutia and Pavel Valtr and Birgit Vogtenhuber},
journal= {arXiv preprint arXiv:1409.0081},
year = {2014}
}