Tight Bounds on the Expected Number of Holes in Random Point Sets

Martin Balko, Manfred Scheucher, Pavel Valtr

Research output: Chapter in Book/Report/Conference proceedingChapterpeer-review


For integers d≥ 2 and k≥ d+ 1, a k-hole in a set S of points in general position in Rd is a k-tuple of points from S in convex position such that the interior of their convex hull does not contain any point from S. For a convex body K⊆ Rd of unit volume, we study the expected number EHd,kK(n) of k-holes in a set of n points drawn uniformly and independently at random from K. We prove an asymptotically tight lower bound on EHd,kK(n)≥Ω(nd) for all fixed d≥ 2 and k≥ d+ 1. For small holes, we even determine the leading constant limn→∞n-dEHd,kK(n) exactly. We improve the best known lower bound on limn→∞n-dEHd,d+1K(n) and we show that our bound is tight for d≤ 3. We show that limn→∞n-2EH2,kK(n) is independent of K for every fixed k≥ 3 and we compute it exactly for k= 4, improving several earlier estimates.

Original languageEnglish
Title of host publicationTrends in Mathematics
PublisherSpringer Science and Business Media Deutschland GmbH
Number of pages6
StatePublished - 1 Jan 2021
Externally publishedYes

Publication series

NameTrends in Mathematics
ISSN (Print)2297-0215
ISSN (Electronic)2297-024X


  • Convex position
  • Holes
  • Random point set
  • Stochastic geometry

ASJC Scopus subject areas

  • Mathematics (all)


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