Expected Length of the Voronoi Path in a High Dimensional Poisson-Delaunay Triangulation

Pedro Machado Manhães de Castro 1 Olivier Devillers 2
2 VEGAS - Effective Geometric Algorithms for Surfaces and Visibility
Inria Nancy - Grand Est, LORIA - ALGO - Department of Algorithms, Computation, Image and Geometry
Abstract : Let $X_n$ be a $d$ dimensional Poisson point process of intensity $n$. We prove that the expected length of the Voronoi path between two points at distance 1 in the Delaunay triangulation associated with $X_n$ is $\sqrt{\frac{2d}{\pi}}+O(d^{-\frac{1}{2}})$ for all $n\in\mathbb{N}$ and $d\rightarrow\infty$. In any dimension, we provide a precise interval containing the exact value, in 3D the expected length is between 1.4977 and 1.50007.
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Pedro Machado Manhães de Castro, Olivier Devillers. Expected Length of the Voronoi Path in a High Dimensional Poisson-Delaunay Triangulation. [Research Report] RR-8947, Inria. 2016. ⟨hal-01353735⟩

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