Discrete & Computational Geometry

, Volume 49, Issue 2, pp 200–220 | Cite as

Asymptotics for Some Combinatorial Characteristics of the Convex Hull of a Poisson Point Process in the Clifford Torus

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Abstract

Let \(\mathcal P _\lambda \) be a homogeneous Poisson point process of rate \(\lambda \) in the Clifford torus \(T^2\subset \mathbb E ^4\). Let \((f_0, f_1, f_2, f_3)\) be the \(f\)-vector of conv\(\,\mathcal P _\lambda \) and let \(\bar{v}\) be the mean valence of a vertex of the convex hull. Asymptotic expressions for \(\mathsf E \, f_1\), \(\mathsf E \, f_2\), \(\mathsf E \, f_3\) and \(\mathsf E \, \bar{v}\) as \(\lambda \rightarrow \infty \) are proved in this paper.

Keywords

Clifford torus Poisson point process Random polytope Poisson–voronoi tessellation 

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Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

  1. 1.Steklov Mathematical Institute of Russian Academy of SciencesMoscowRussian Federation

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