Probability Theory and Related Fields

, Volume 77, Issue 2, pp 231-240

First online:

On the shape of the convex hull of random points

  • Imre BárányAffiliated withSchool of OR & IE, Cornell University
  • , Zoltán FürediAffiliated withDepartment of Mathematics, Massachusetts Institute of Technology

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