Random polytopes with vertices on the boundary of a convex body

Let K be a convex body in ℝⁿ and let f : ∂K → ℝ+ be a continuous, positive function with ∫_∂K f (x) dμ∂K (X) = 1, where μ∂K is the surface measure on ∂K. Let ℙ_f be the probability measure on ∂K given by dℙ _f (x) = f(x) dμ∂K (x). Let K be the (generalized) Gauß-Kronecker curvature and E(f, N) the expected volume of the convex hull of N points chosen randomly on ∂K with respect to ℙ _f. Then, under some regularity conditions on the boundary of K, (formula presented) where c_n is a constant depending on the dimension n only. The minimum at the right-hand side is attained for the normalized affine surface area measure with density (formula presented)