Summary
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, 2d/2 d -ɛ)→1 and Prob(d, 2d/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.
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Bárány, I., Füredi, Z. On the shape of the convex hull of random points. Probab. Th. Rel. Fields 77, 231–240 (1988). https://doi.org/10.1007/BF00334039
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DOI: https://doi.org/10.1007/BF00334039