Holes and islands in random point sets

Martin Balko, Manfred Scheucher, Pavel Valtr

Research output: Contribution to journalArticlepeer-review

2 Scopus citations


For (Formula presented.), let S be a set of points in (Formula presented.) in general position. A set I of k points from S is a k-island in S if the convex hull (Formula presented.) of I satisfies (Formula presented.). A k-island in S in convex position is a k-hole in S. For (Formula presented.) and a convex body (Formula presented.) of volume 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-holes in S is in (Formula presented.). Our estimate improves and generalizes all previous bounds. In particular, we estimate the expected number of empty simplices in S by (Formula presented.). This is tight in the plane up to a lower-order term. Our method gives an asymptotically tight upper bound (Formula presented.) even in the much more general setting, where we estimate the expected number of k-islands in S.

Original languageEnglish
Pages (from-to)308-326
Number of pages19
JournalRandom Structures and Algorithms
Issue number3
StatePublished - 1 May 2022
Externally publishedYes


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

ASJC Scopus subject areas

  • Software
  • General Mathematics
  • Computer Graphics and Computer-Aided Design
  • Applied Mathematics


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