A Continuous Analogue of the Upper Bound Theorem

For an absolutely continuous probability measure on Rd and a nonnegative integer k, let ~ sk( ;0) denote the probability that the convex hull of k + d + 1 random points which are i.i.d. according to contains the origin 0. For d and k given, we determine a tight upper bound on ~ sk( ;0), and we characterize the measures in Rd which attain this bound. As we will see, this result can be considered a continuous analogue of the Upper Bound Theorem for the maximal number of faces of convex 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.