Tight bounds on the expected number of holes in random point sets

Martin Balko, Manfred Scheucher, Pavel Valtr

Research output: Contribution to journalArticlepeer-review

Abstract

For integers (Formula presented.) and (Formula presented.), a (Formula presented.) -hole in a set (Formula presented.) of points in general position in (Formula presented.) is a (Formula presented.) -tuple of points from (Formula presented.) in convex position such that the interior of their convex hull does not contain any point from (Formula presented.). For a convex body (Formula presented.) of unit (Formula presented.) -dimensional volume, we study the expected number (Formula presented.) of (Formula presented.) -holes in a set of (Formula presented.) points drawn uniformly and independently at random from (Formula presented.). We prove an asymptotically tight lower bound on (Formula presented.) by showing that, for all fixed integers (Formula presented.) and (Formula presented.), the number (Formula presented.) is at least (Formula presented.). For some small holes, we even determine the leading constant (Formula presented.) exactly. We improve the currently best-known lower bound on (Formula presented.) by Reitzner and Temesvari (2019). In the plane, we show that the constant (Formula presented.) is independent of (Formula presented.) for every fixed (Formula presented.) and we compute it exactly for (Formula presented.), improving earlier estimates by Fabila-Monroy, Huemer, and Mitsche and by the authors.

Original languageEnglish
JournalRandom Structures and Algorithms
DOIs
StateAccepted/In press - 1 Jan 2022
Externally publishedYes

Keywords

  • convex position
  • k-hole
  • random point set
  • stochastic geometry

ASJC Scopus subject areas

  • Software
  • Mathematics (all)
  • Computer Graphics and Computer-Aided Design
  • Applied Mathematics

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