Mean width of random polytopes in a reasonably smooth convex body

Let K be a convex body in R-d and let X-n = (x(1),..., x(n)) be a random sample of n independent points in K chosen according to the uniform distribution. The convex hull K-n of X-n is a random polytope in K, and we consider its mean width W (K-n). In this article, we assume that K has a rolling ball of radius Q > 0. First, we extend the asymptotic formula for the expectation of W (K) - W (K-n) which was earlier known only in the case when partial derivative K has positive Gaussian curvature. In addition, we determine the order of magnitude of the variance of W(K-n), and prove the strong law of large numbers for W(K-n). We note that the strong law of large numbers for any quermassintegral of K was only known earlier for the case when partial derivative K has positive Gaussian curvature. (C) 2009 Elsevier Inc. All rights reserved.