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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 \(K\).