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

Discrete & Computational Geometry - Tập 49 Số 2 - Trang 200-220 - 2013
Magazinov, Alexander1
1Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russian Federation

Tóm tắt

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.

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Discrete & Computational Geometry

Tập 49 Số 2

200-220

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