Expectation of intrinsic volumes of random polytopes

Let K be a convex body in Rd, let j ∈ {1, . . . , d − 1}, and let K(n) be the convex hull of n points chosen randomly, independently and uniformly from K. If ∂K is C2 +, then an asymptotic formula is known due to M. Reitzner (and due to I. Bárány if ∂K is C3 +) for the difference of the jth intrinsic volume of K and the expectation of the jth intrinsic volume of K(n). We extend this formula to the case when the only condition on K is that a ball rolls freely inside K.