Monotonicity of facet numbers of random convex hulls

Let X1,…,Xn be independent random points that are distributed according to a probability measure on Rd and let Pn be the random convex hull generated by X1,…,Xn (n≥d+1). For natural classes of probability distributions and by means of Blaschke-Petkantschin formulae from integral geometry it is shown that the mean facet number of Pn is strictly monotonically increasing in n.