Quermassintegrals of a random polytope in a convex body
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Let K be a convex body in n with volume |K| = 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: + + be a continuous strictly increasing function and 0 i n - 1. Then, the quantity double-struck E sign (K, N, f Wi) = K ... K f[Wi(C(x1, ..., xN))]dxN ... dx1 is minimal if K is a ball (Wi 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).
