Monotonicity of facet numbers of random convex hulls

TL;DR

This paper investigates conditions under which the expected number of facets of random convex hulls increases monotonically as more random points are added, using tools from integral geometry.

Contribution

It characterizes classes of probability distributions for which the mean facet number of convex hulls increases with the number of points.

Findings

Mean facet number is strictly increasing for certain distribution classes.

Uses Blaschke-Petkantschin formulae from integral geometry.

Provides conditions for monotonicity in convex hull facet counts.

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 $n$.