Random polytopes with vertices on the boundary of a convex body

Let K be a convex body in R-n and let f : partial derivativeK --> R+ be a c ontinuous, positive function with integral (partial derivativeK) f(x) d mu (partial derivativeK)(x) = 1, where mu (partial derivativeK) is the surface measure on partial derivativeK. Let P-f be the probability measure on part ial derivativeK given by dP(f)(x) = f(x) d mu (partial derivativeK)(x). Let kappa be the (generalized) Gauss-Kronecker curvature and E(f, N) the expec ted volume of the convex hull of N points chosen randomly on partial deriva tiveK with respect to P-f. Then, under some regularity conditions on the bo undary of K, [GRAPHICS] 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 meas ure with density [GRAPHICS] (C) 2000 Academie des sciences/Editions scientifiques et medicales Elsevier SAS.