Mean width of random polytopes in a reasonably smooth convex body

Abstract

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 ϱ > 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 ∂ 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 ∂ K has positive Gaussian curvature.