4-Holes in point sets

Authors:
Oswin Aichholzer;Ruy Fabila-Monroy;Hernán González-Aguilar;Thomas Hackl;Marco A. Heredia;Clemens Huemer;Jorge Urrutia;Birgit Vogtenhuber
Affiliations:
Institute for Software Technology, Graz University of Technology, Graz, Austria;Departamento de Matemáticas, Cinvestav, D.F. México, Mexico;Facultad de Ciencias, Universidad Autónoma de San Luis Potosí, San Luis Potosí, Mexico;Institute for Software Technology, Graz University of Technology, Graz, Austria;Posgrado en Ciencia e Ingeniería de la Computación, Universidad Nacional Autónoma de México, D.F. México, Mexico;Departament de Matemítica Aplicada IV, Universitat Politècnica de Catalunya, Barcelona, Spain;Instituto de Matemáticas, Universidad Nacional Autónoma de México, D.F. México, Mexico;Institute for Software Technology, Graz University of Technology, Graz, Austria
Venue:
Computational Geometry: Theory and Applications
Year:
2014

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Abstract

We consider a variant of a question of Erdos on the number of empty k-gons (k-holes) in a set of n points in the plane, where we allow the k-gons to be non-convex. We show bounds and structural results on maximizing and minimizing the number of general 4-holes, and maximizing the number of non-convex 4-holes. In particular, we show that for n=9, the maximum number of general 4-holes is (n4); the minimum number of general 4-holes is at least 52n^2-@Q(n); and the maximum number of non-convex 4-holes is at least 12n^3-@Q(n^2logn) and at most 12n^3-@Q(n^2).