Facets of spherical random polytopes

Abstract

Facets of the convex hull of 𝑛 independent random vectors chosen uniformly at random from the unit sphere in ℝ 𝑑 are studied. A particular focus is given on the height of the facets as well as the expected number of facets as the dimension increases. Regimes for 𝑛 and 𝑑 with different asymptotic behavior of these quantities are identified and asymptotic formulas in each case are established. Extensions of several known results in fixed dimension to the case where dimension tends to infinity are described.

Improved Bounds for the Expected Number of k-Sets

“…In [5], Bonnet and O'Reilly consider the convex hull of random points from the unit sphere in R d . They call such polytopes spherical random polytopes and they provide asymptotic expressions for the expected number of facets as n and d grow at different rates.…”

“…Our result shows that this correspondence continues for the case when n is proportional to d: for any α > 1, Theorem 8 says that the expected number of facets of a Gaussian random polytope with n ∼ αd vertices is equal to C(α) d+o(d) for some constant C(α). For spherical random polytopes, the case when the number of vertices is equal to n ∼ αd for some α > 1 is dealt with in [5,Theorem 4.2]. The asymptotic formula given there is also of the form C(α) d+o(d) for some constant C(α).…”