On k-gons and k-holes in point sets
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We consider a variation of the classical Erdos-Szekeres problems on the existence and number of convex k-gons and k-holes (empty k-gons) in a set of n points in the plane. Allowing the k-gons to be non-convex, we show bounds and structural results on maximizing and minimizing their numbers. Most noteworthy, for any k and sufficiently large n, we give a quadratic lower bound for the number of k-holes, and show that this number is maximized by sets in convex position. (C) 2014 Elsevier B.V. All rights reserved.
CitationAichholzer, O., Fabila, R., Gonzalez, H., Hackl, T., Heredia, M., Huemer, C., URRUTIA, J., Valtr, P., Vogtenhuber, B. On k-gons and k-holes in point sets. "Computational geometry: theory and applications", 01 Agost 2015, vol. 48, núm. 7, p. 528-537.