Volume thresholds for Gaussian and spherical random polytopes and their duals

Abstract

Abstract. Let g be a Gaussian random vector in R n . Let N = N (n) be a positive integer and let K N be the convex hull of N independent copies of g. Fix R > 0 and consider the ratio of volumesFor a large range of R = R(n), we establish a sharp threshold for N , above which V N → 1 as n → ∞, and below which V N → 0 as n → ∞. We also consider the case when K N is generated by independent random vectors distributed uniformly on the Euclidean sphere. In this case, similar threshold results are proved for both R ∈ (0, 1) and R = 1. Lastly, we prove complementary results for polytopes generated by random facets.

Threshold phenomena for high-dimensional random polytopes

“…Moreover, since the uniform distribution on the unit sphere S n−1 arises as the weak limit of the beta distribution, as β → −1 (see for example the proof of Theorem 2.7 in [10]), the result of Theorem 2.4 in [14] can be recovered by Theorem 1.1. Corollary 1.3.…”

“…Moreover, Pivovarov considered the dual setting of polytopes generated as sections of random halfspaces with respect to the same probability measures. We stress that the authors in both [9] and [14] exploit the method of [7], which due to its geometric viewpoint seems to be applicable for a wide variety of probability distributions. Let N and n be natural numbers, N > n, and X 1 , X 2 , .…”

“…We state it here in a slightly more explicit form than in Theorem 2.2.1 from [15]. For a related result where the log concave isotropic measures are replaced by the volume ratios of the intersection of Gaussian polytopes with balls of arbitrary radii, see Theorem 2.1 from [14]. Corollary 1.6.…”

“…The proofs of Theorem 1.1 and Theorem 1.5 follow the method introduced in [7] and exploited in [14]. We thus define, for every x ∈ R n , the functions Proof.…”

Approximate Voronoi cells for lattices, revisited

“…The volume of such random polytopes has been previously studied in e.g. [31,32,35,40,41], and in particular the case of points from the sphere was analyzed in [31]. For the application to approximate Voronoi cells we need bounds for the case when points are drawn uniformly at random from a ball, which to the best of our knowledge has not been explicitly studied before.…”