Volume thresholds for Gaussian and spherical random polytopes and their duals

Abstract

Let g be a Gaussian random vector in R n . Let N = N(n) be a positive integer and let KN be the convex hull of N independent copies of g. Fix R > 0 and consider the ratio of volumes VN := Evol(KN \RB n 2 )/vol(RB n 2 ). For a large range of R = R(n), we establish a sharp threshold for N, above which VN ! 1 as n ! 1, and below which VN ! 0 as n ! 1. We also consider the case when KN is generated by independent random vectors distributed uniformly on the Euclidean sphere. In this case, similar threshold results are proved for both R 2 (0,1) and R = 1. Lastly, we prove complementary results for polytopes generated by random facets.