Quermassintegrals of a Random Polytope in a Convex Body

with volume jKj = 1. We choose N n + 1 points x1 ; : : : ; xN independently and uniformly from K, and write C(x1 ; : : : ; xN ) for their convex hull. Let f : R be a continuous strictly increasing function and 0 i n 1. Then, the quantity f [W i (C(x1 ; : : : ; xN ))]dxN : : : dx1 is minimal if K is a ball (W i is the i-th quermassintegral of a compact convex set). If f is convex and strictly increasing and 1 i n 1, then the ball is the only extremal body. These two facts generalize a result of H. Groemer on moments of the volume of C(x1 ; : : : ; xN )