Mean width of random polytopes in a reasonably smooth convex body

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