On the shape of the convex hull of random points
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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-ɛ)→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.
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