Probability Theory and Related Fields

, Volume 77, Issue 2, pp 231–240

On the shape of the convex hull of random points

  • Imre Bárány
  • Zoltán Füredi
Article

DOI: 10.1007/BF00334039

Cite this article as:
Bárány, I. & Füredi, Z. Probab. Th. Rel. Fields (1988) 77: 231. doi:10.1007/BF00334039

Summary

Denote by En the convex hull of n points chosen uniformly and independently from the d-dimensional ball. Let Prob(d, n) denote the probability that En has exactly n vertices. It is proved here that Prob(d, 2d/2d)→1 and Prob(d, 2d/2d(3/4)+ɛ)→0 for every fixed ɛ>0 when d→∞. The question whether En is a k-neighbourly polytope is also investigated.

Copyright information

© Springer-Verlag 1988

Authors and Affiliations

  • Imre Bárány
    • 1
  • Zoltán Füredi
    • 2
  1. 1.School of OR & IECornell UniversityIthacaUSA
  2. 2.Department of MathematicsMassachusetts Institute of TechnologyCambridgeUSA