Variance asymptotics for random polytopes in smooth convex bodies

TL;DR

This paper investigates the asymptotic behavior of the variance of face counts in random polytopes formed by Poisson and binomial processes within smooth convex bodies, revealing a connection to the body's affine surface area.

Contribution

It establishes variance asymptotics for face counts of random polytopes in smooth convex bodies, linking these to geometric properties like affine surface area.

Findings

Variance of face counts converges to a multiple of affine surface area.

Results apply to both Poisson and binomial process models.

Asymptotic formulas hold as the intensity parameter tends to infinity.

Abstract

Let $K \subset \R^d$ be a smooth convex set and let $\P_\la$ be a Poisson point process on $\R^d$ of intensity $\la$. The convex hull of $\P_\la \cap K$ is a random convex polytope $K_\la$. As $\la \to \infty$, we show that the variance of the number of $k$-dimensional faces of $K_\la$, when properly scaled, converges to a scalar multiple of the affine surface area of $K$. Similar asymptotics hold for the variance of the number of $k$-dimensional faces for the convex hull of a binomial process in $K$.