Gaussian polytopes: variances and limit theorems

Abstract

The convex hull ofnindependent random points in ℝd, chosen according to the normal distribution, is called a Gaussian polytope. Estimates for the variance of the number ofi-faces and for the variance of theith intrinsic volume of a Gaussian polytope in ℝd,d∈ℕ, are established by means of the Efron-Stein jackknife inequality and a new formula of Blaschke-Petkantschin type. These estimates imply laws of large numbers for the number ofi-faces and for theith intrinsic volume of a Gaussian polytope asn→∞.