Empty non-convex and convex four-gons in random point sets

Ruy Fabila-Monroy; Clemens Huemer; Dieter Mitsche

doi:10.1556/sscmath.52.2015.1.1301

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Studia Scientiarum Mathematicarum Hungarica

Volume/Issue:: Volume 52: Issue 1

Empty non-convex and convex four-gons in random point sets

Authors:

Ruy Fabila-Monroy

Ruy Fabila-Monroy CINVESTAV-IPN Departamento de Matemáticas Ciudad de México México

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Clemens Huemer

Clemens Huemer Universitat Politècnica de Catalunya, BarcelonaTech Barcelona España

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, and

Dieter Mitsche

Dieter Mitsche Université de Nice Sophia-Antipolis Laboratoire J.A. Dieudonné Alpes France

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Pages:: 52–64

Online Publication Date:: 17 Mar 2015

Publication Date:: 01 Mar 2015

Article Category:: Journal Article

DOI:: https://doi.org/10.1556/sscmath.52.2015.1.1301

Keywords:: Primary 60D05; 52A22; random point set; empty four-gon; polygon; geometric probability

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Let S be a set of n points distributed uniformly and independently in a convex, bounded set in the plane. A four-gon is called empty if it contains no points of S in its interior. We show that the expected number of empty non-convex four-gons with vertices from S is 12n²logn + o(n²logn) and the expected number of empty convex four-gons with vertices from S is Θ(n²).

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Studia Scientiarum Mathematicarum Hungarica

Print ISSN:: 0081-6906

Online ISSN:: 1588-2896

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Gábor SIMONYI (Rényi Institute of Mathematics)
András STIPSICZ (Rényi Institute of Mathematics)
Géza TÓTH (Rényi Institute of Mathematics)

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Gábor SÁGI (Rényi Institute of Mathematics)

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Imre BÁRÁNY (Rényi Institute of Mathematics)
Károly BÖRÖCZKY (Rényi Institute of Mathematics and Central European University)
Péter CSIKVÁRI (ELTE, Budapest)
Joshua GREENE (Boston College)
Penny HAXELL (University of Waterloo)
Andreas HOLMSEN (Korea Advanced Institute of Science and Technology)
Ron HOLZMAN (Technion, Haifa)
Satoru IWATA (University of Tokyo)
Tibor JORDÁN (ELTE, Budapest)
Roy MESHULAM (Technion, Haifa)
Frédéric MEUNIER (École des Ponts ParisTech)
Márton NASZÓDI (ELTE, Budapest)
Eran NEVO (Hebrew University of Jerusalem)
János PACH (Rényi Institute of Mathematics)
Péter Pál PACH (BME, Budapest)
Andrew SUK (University of California, San Diego)
Zoltán SZABÓ (Princeton University)
Martin TANCER (Charles University, Prague)
Gábor TARDOS (Rényi Institute of Mathematics)
Paul WOLLAN (University of Rome "La Sapienza")

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Studia Scientiarum Mathematicarum Hungarica
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