, Volume 102, Issue 2, pp 91-102

On a conjecture of R. E. Miles about the convex hull of random points

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Denoty byp d+i (B d ,d+m) the probability that the convex hull ofd+m points chosen independently and uniformly from ad-dimensional ballB d possessesd+i(i=1,...,m) vertices. We prove Mile's conjecture that, given any integerm, p d+m (B d ,d+m)»1 asd»∞. This is obvious form=1 and was shown by Kingman form=2 and by Miles form=3. Further, a related result by Miles is generalized, and several consequences are deduced.

Dedicated to Professor E. Halwaka on the occasion of his seventieth