Quermassintegrals of a random polytope in a convex body

Abstract. Let K be a convex body in $ \mathbb{R}^n $ with volume $ |K| = 1 $. We choose $ N \geq n+1 $ points $ x_1,\ldots, x_N $ independently and uniformly from K, and write $ C(x_1,\ldots, x_N) $ for their convex hull. Let $ f : \mathbb{R^+} \rightarrow \mathbb{R^+} $ be a continuous strictly increasing function and $ 0 \leq i \leq n-1 $. Then, the quantity¶¶$ E (K, N, f \circ W_{i}) = \int\limits_{K} \ldots \int\limits_{K} f[W_{i}(C(x_1, \ldots, x_N))]dx_{N} \ldots dx_1 $ ¶¶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 \leq i \leq 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(x_1,\ldots, x_N) $.