The combinatorial structure of random polytopes

Choose n random points in , let P_n be their convex hull, and denote by f_i(P_n) the number of i-dimensional faces of P_n. A general method for computing the expectation of f_i(P_n), i=0,…,d−1, is presented. This generalizes classical results of Efron (in the case i=0) and Rényi and Sulanke (in the case i=d−1) to arbitrary i. For random points chosen in a smooth convex body a limit law for f_i(P_n) is proved as n→∞. For random points chosen in a polytope the expectation of f_i(P_n) is determined as n→∞. This implies an extremal property for random points chosen in a simplex.