The expected number of facets and the typical facet height of the convex hull of n uniform points on S^{d-1} are determined asymptotically in every regime where n and d tend to infinity.
Facets of high-dimensional Gaussian polytopes
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abstract
We study the number of facets of the convex hull of n independent standard Gaussian points in d-dimensional Euclidean space. In particular, we are interested in the expected number of facets when the dimension is allowed to grow with the sample size. We establish an explicit asymptotic formula that is valid whenever d/n tends to zero. We also obtain the asymptotic value when d is close to n.
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Facets of spherical random polytopes
The expected number of facets and the typical facet height of the convex hull of n uniform points on S^{d-1} are determined asymptotically in every regime where n and d tend to infinity.