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Holes and Islands in Random Point Sets

Authors Martin Balko, Manfred Scheucher, Pavel Valtr

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ACM Subject Classification
  • Mathematics of computing → Combinatoric problems
  • Theory of computation → Computational geometry
Keywords
  • stochastic geometry
  • random point set
  • Erdős-Szekeres type problem
  • k-hole
  • k-island
  • empty polytope
  • convex position
  • Horton set

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Abstract

For d ∈ ℕ, let S be a finite set of points in ℝ^d in general position. A set H of k points from S is a k-hole in S if all points from H lie on the boundary of the convex hull conv(H) of H and the interior of conv(H) does not contain any point from S. A set I of k points from S is a k-island in S if conv(I) ∩ S = I. Note that each k-hole in S is a k-island in S.
For fixed positive integers d, k and a convex body K in ℝ^d with d-dimensional Lebesgue measure 1, let S be a set of n points chosen uniformly and independently at random from K. We show that the expected number of k-islands in S is in O(n^d). In the case k=d+1, we prove that the expected number of empty simplices (that is, (d+1)-holes) in S is at most 2^(d-1) ⋅ d! ⋅ binom(n,d). Our results improve and generalize previous bounds by Bárány and Füredi [I. Bárány and Z. Füredi, 1987], Valtr [P. Valtr, 1995], Fabila-Monroy and Huemer [Fabila-Monroy and Huemer, 2012], and Fabila-Monroy, Huemer, and Mitsche [Fabila-Monroy et al., 2015].

Cite As Get BibTex

Martin Balko, Manfred Scheucher, and Pavel Valtr. Holes and Islands in Random Point Sets. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 14:1-14:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.SoCG.2020.14

Author Details

Martin Balko
  • Department of Applied Mathematics, Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic
Manfred Scheucher
  • Institut für Mathematik, Technische Universität Berlin, Germany
Pavel Valtr
  • Department of Applied Mathematics, Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic
  • Department of Computer Science, ETH Zürich, Switzerland

Funding

  • Balko, Martin: was supported by the grant no. 18-19158S of the Czech Science Foundation (GAČR), by the Center for Foundations of Modern Computer Science (Charles University project UNCE/SCI/004), and by the PRIMUS/17/SCI/3 project of Charles University. This article is part of a project that has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 810115).
  • Scheucher, Manfred: was supported by DFG Grant FE 340/12-1.
  • Valtr, Pavel: was supported by the grant no. 18-19158S of the Czech Science Foundation (GAČR) and by the PRIMUS/17/SCI/3 project of Charles University.

References

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