Mean width of random polytopes in a reasonably smooth convex body

AbstractLet K be a convex body in Rd and let Xn=(x1,…,xn) be a random sample of n independent points in K chosen according to the uniform distribution. The convex hull Kn of Xn is a random polytope in K, and we consider its mean width W(Kn). In this article, we assume that K has a rolling ball of radius ϱ>0. First, we extend the asymptotic formula for the expectation of W(K)−W(Kn) which was earlier known only in the case when ∂K has positive Gaussian curvature. In addition, we determine the order of magnitude of the variance of W(Kn), and prove the strong law of large numbers for W(Kn). We note that the strong law of large numbers for any quermassintegral of K was only known earlier for the case when ∂K has positive Gaussian curvature