Limit theorems for random polytopes with vertices on convex surfaces

Abstract

The random polytope $K_n$, defined as the convex hull of $n$ points chosen uniformly at random on the boundary of a smooth convex body, is considered. Proofs for lower and upper variance bounds, strong laws of large numbers and central limit theorems for the intrinsic volumes of $K_n$ are presented. A normal approximation bound from Stein's method and estimates for surface bodies are among the involved tools.