Monotonicity of facet numbers of random convex hulls

Abstract

Let $X_1,\ldots,X_n$ be independent random points that are distributed according to a probability measure on $\mathbb{R}^d$ and let $P_n$ be the random convex hull generated by $X_1,\ldots,X_n$ ($n\geq d+1$). Natural classes of probability distributions are characterized for which, by means of Blaschke-Petkantschin formulae from integral geometry, one can show that the mean facet number of $P_n$ is strictly monotonically increasing in $n$.