Asymptotics for Some Combinatorial Characteristics of the Convex Hull of Poisson Point Process on the Clifford Torus with Large Value of Rate

Details Asymptotics for Some Combinatorial Characteristics of the Convex Hull of Poisson Point Process on the Clifford Torus with Large Value of Rate Asymptotics for Some Combinatorial Characteristics of the Convex Hull of Poisson Point Process on the Clifford Torus with Large Value of Rate

2010-10-28

arxiv.org/abs/1010.6013v2

Mathematics

Metric Geometry

27 pages

Scientific paper

N. Dolbilin and M. Tanemura studied the convex hulls of finite subsets of the Clifford torus $T$ in $E^4$. They have completely studied the combinatorial structure of the convex hull for a periodic point set. Moreover, there was performed a numerical simulation of the convex hull for the Poisson point process on $T$ that showed that the mean valence of a vertex of the convex hull has asymptotics $O^*(\ln \lambda)$ where $\lambda$ is the rate of the process. N. Dolbilin suggested the author to prove the conjecture on the logarithmic growth of the mean degree of a vertex. In this paper we prove this conjecture and some related theorems.

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