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On 5-gons and 5-holes. (English) Zbl 1374.52020
Márquez, Alberto (ed.) et al., Computational geometry. XIV Spanish meeting on computational geometry, EGC 2011, dedicated to Ferran Hurtado on the occasion of his 60th birthday, Alcalá de Henares, Spain, June 27–30, 2011. Revised selected papers. Berlin: Springer (ISBN 978-3-642-34190-8/pbk). Lecture Notes in Computer Science 7579, 1-13 (2012).
Summary: We consider an extension of a question of Erdős on the number of \(k\)-gons in a set of \(n\) points in the plane. Relaxing the convexity restriction we obtain results on 5-gons and 5-holes (empty 5-gons). In particular, we show a direct relation between the number of non-convex 5-gons and the rectilinear crossing number, provide an improved lower bound for the number of convex 5-holes any point set must contain, and prove that the number of general 5-holes is asymptotically maximized for point sets in convex position.
For the entire collection see [Zbl 1253.68016].

MSC:
52C10 Erdős problems and related topics of discrete geometry
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