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

Denoty bypd+i(Bd,d+m) the probability that the convex hull ofd+m points chosen independently and uniformly from ad-dimensional ballBd possessesd+i(i=1,...,m) vertices. We prove Mile's conjecture that, given any integerm, pd+m(Bd,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.

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