Intrinsic Volumes of Inscribed Random Polytopes in Smooth Convex Bodies

LetK be a d-dimensional convex bodywith a twice continuously differentiable boundary and everywhere positive Gauss–Kronecker curvature. Denote byKn the convex hull of n points chosen randomly and independently fromK according to the uniform distribution. Matching lower andupper bounds are obtained for the orders ofmagnitudeof the variances of the sth intrinsic volumes Vs(Kn) of Kn for s ∈ {1, . . . , d}. Furthermore, strong laws of large numbers are proved for the intrinsic volumes of Kn. The essential tools are the economic cap covering theorem of Bárány and Larman, and the Efron–Stein jackknife inequality.