A continuous analogue of the Upper Bound Theorem

For an absolutely continuous probability measure mu on R-d and a nonnegativ e integer k, let (S) over tilde (k) (mu, 0) denote the probability that the convex hull of k + d + 1 random points which are i.i.d. according to mu co ntains the origin 0. For d and k given, we determine a tight upper bound on (S) over tilde (k) (mu, 0), and we characterize the measures in R-d which attain this bound. As we will see, this result can be considered a continuo us analogue of the Upper Bound Theorem for the maximal number of faces of c onvex polytopes with a given number of vertices. For our proof we introduce so-called h-functions, continuous counterparts of h-vectors of simplicial convex polytopes.