Variance asymptotics for random polytopes in smooth convex bodies

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 $K$