INTRINSIC VOLUMES OF RANDOM POLYTOPES WITH VERTICES ON THE BOUNDARY OF A CONVEX BODY

Let K be a convex body in R-d, let j is an element of {1,..., d - 1}, and let rho be a positive and continuous probability density function with respect to the (d - 1)-dimensional Hausdorff measure on the boundary partial derivative K of K. Denote by K-n the convex hull of n points chosen randomly and independently from partial derivative K according to the probability distribution determined by rho. For the case when partial derivative K is a C-2 submanifold of R-d with everywhere positive Gauss curvature, M. Reitzner proved an asymptotic formula for the expectation of the difference of the jth intrinsic volumes of K and K-n, as n -> infinity. In this article, we extend this result to the case when the only condition on K is that a ball rolls freely in K.