On the expected number of k-sets

Imre Bárány, William Steiger

Research output: Contribution to journalArticlepeer-review

5 Scopus citations


Given a set S of n points in R d, a subset X of size d is called a k-simplex if the hyperplane aff(X) has exactly k points on one side. We study E d (k,n), the expected number of k-simplices when S is a random sample of n points from a probability distribution P on R d . When P is spherically symmetric we prove that E d (k, n)≤cn d-1 When P is uniform on a convex body K⊂R 2 we prove that E 2 (k, n) is asymptotically linear in the range cn≤k≤n/2 and when k is constant it is asymptotically the expected number of vertices on the convex hull of S. Finally, we construct a distribution P on R 2 for which E 2((n-2)/2, n) is cn log n.

Original languageEnglish (US)
Pages (from-to)243-263
Number of pages21
JournalDiscrete & Computational Geometry
Issue number1
StatePublished - Dec 1994

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • Geometry and Topology
  • Discrete Mathematics and Combinatorics
  • Computational Theory and Mathematics


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