Let K be a d-dimensional convex body with a twice continuously differentiable boundary and everywhere positive Gauss-Kronecker curvature. Denote by K n the convex hull of n points chosen randomly and independently from K according to the uniform distribution. Matching lower and upper bounds are obtained for the orders of magnitude of the variances of the sth intrinsic volumes V s (K n ) of K n for s ∈ {1,…,d}. Furthermore, strong laws of large numbers are proved for the intrinsic volumes of K n . The essential tools are the economic cap covering theorem of Bárány and Larman, and the Efron-Stein jackknife inequality.
