# Variance asymptotics for random polytopes in smooth convex bodies

Research paper by Pierre Calka, J. E. Yukich

Indexed on: 09 Feb '13Published on: 09 Feb '13Published in: Probability Theory and Related Fields

#### Abstract

Let $$K \subset \mathbb R ^d$$ be a smooth convex set and let $$\mathcal{P }_{\lambda }$$ be a Poisson point process on $$\mathbb R ^d$$ of intensity $${\lambda }$$. The convex hull of $$\mathcal{P }_{\lambda }\cap K$$ is a random convex polytope $$K_{\lambda }$$. As $${\lambda }\rightarrow \infty$$, we show that the variance of the number of $$k$$-dimensional faces of $$K_{\lambda }$$, 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 $$K$$.