Intrinsic volumes of random polytopes with vertices on the boundary of a convex body

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Let K be a convex body in ᵈ, let j∈{1,..., d-1}, and let be a positive and continuous probability density function with respect to the (d-1)-dimensional Hausdorff measure on the boundary ∂ K of K. Denote by Kₙ the convex hull of n points chosen randomly and independently from ∂ K according to the probability distribution determined by. For the case when ∂ K is a C² submanifold of 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 Kₙ, as n→∞. In this article, we extend this result to the case when the only condition on K is that a ball rolls freely in K.