Central limit theorems for random boundary polytopes

Abstract

The number of faces of the convex hull of $n$ independent and identically distributed random points chosen on the boundary of a smooth convex body in $\mathbb{R}^d$ is investigated. In dimensions two and three the number of $k$-faces is known to be constant almost surely and in dimension four and higher the variance is known to be non-zero if $k\ge 1$. We show that it is of order $n$. This is complemented by a central limit theorem with a Berry-Esseen bound which is of optimal order $n^{-1/2}$. We derive similar results for the Poissonized model, where additionally the number of random points is Poisson distributed. As a main tool, we develop a representation of the number of faces as a sum of exponentially stabilizing score functions.