Variance asymptotics for random polytopes in smooth convex bodies |
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Authors: | Pierre Calka J E Yukich |
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Institution: | 1. Laboratoire de Mathématiques Rapha?l Salem, Université de Rouen, Avenue de l’Université, BP.12, Technop?le du Madrillet, 76801, Saint-Etienne-du-Rouvray, France 2. Department of Mathematics, Lehigh University, Bethlehem, PA, 18015, USA
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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 $K$ . |
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