Gaussian polytopes: variances and limit theorems

The convex hull of n independent random points in Rd, chosen according to the normal distribution, is called a Gaussian polytope. Estimates for the variance of the number of i-faces and for the variance of the ith 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 of i-faces and for the ith intrinsic volume of a Gaussian polytope as n -> co.