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

Expected Length of the Voronoi Path in a High Dimensional Poisson–Delaunay Triangulation Let X be a d dimensional Poisson point process. We prove that the expected length of the Voronoi path between two points at distance 1 in the Delaunay triangulation associated with X is $$\sqrt{\frac{2d}{\pi }}+O(d^{-\frac{1}{2}})$$ 2 d π + O ( d - 1 2 ) when $$d\rightarrow \infty$$ d → ∞ . In any dimension, we also provide a precise interval containing the actual value; in 3D the expected length is between 1.4977 and 1.50007. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Discrete & Computational Geometry Springer Journals

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

, Volume 60 (1) – Feb 23, 2017
20 pages

/lp/springer_journal/expected-length-of-the-voronoi-path-in-a-high-dimensional-poisson-nLxu4PeO3L
Publisher
Springer US
Subject
Mathematics; Combinatorics; Computational Mathematics and Numerical Analysis
ISSN
0179-5376
eISSN
1432-0444
D.O.I.
10.1007/s00454-017-9866-y
Publisher site
See Article on Publisher Site

### Abstract

Let X be a d dimensional Poisson point process. We prove that the expected length of the Voronoi path between two points at distance 1 in the Delaunay triangulation associated with X is $$\sqrt{\frac{2d}{\pi }}+O(d^{-\frac{1}{2}})$$ 2 d π + O ( d - 1 2 ) when $$d\rightarrow \infty$$ d → ∞ . In any dimension, we also provide a precise interval containing the actual value; in 3D the expected length is between 1.4977 and 1.50007.

### Journal

Discrete & Computational GeometrySpringer Journals

Published: Feb 23, 2017

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