The Annals of Applied Probability

Random inscribed polytopes have similar radius functions as Poisson–Delaunay mosaics

Herbert Edelsbrunner and Anton Nikitenko

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Using the geodesic distance on the $n$-dimensional sphere, we study the expected radius function of the Delaunay mosaic of a random set of points. Specifically, we consider the partition of the mosaic into intervals of the radius function and determine the expected number of intervals whose radii are less than or equal to a given threshold. We find that the expectations are essentially the same as for the Poisson–Delaunay mosaic in $n$-dimensional Euclidean space. Assuming the points are not contained in a hemisphere, the Delaunay mosaic is isomorphic to the boundary complex of the convex hull in $\mathbb{R}^{n+1}$, so we also get the expected number of faces of a random inscribed polytope. As proved in Antonelli et al. [Adv. in Appl. Probab. 9–12 (1977–1980)], an orthant section of the $n$-sphere is isometric to the standard $n$-simplex equipped with the Fisher information metric. It follows that the latter space has similar stochastic properties as the $n$-dimensional Euclidean space. Our results are therefore relevant in information geometry and in population genetics.

Article information

Ann. Appl. Probab., Volume 28, Number 5 (2018), 3215-3238.

Received: May 2017
Revised: February 2018
First available in Project Euclid: 28 August 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]
Secondary: 68U05: Computer graphics; computational geometry [See also 65D18]

Voronoi tessellations Delaunay mosaics inscribed polytopes discrete Morse theory critical simplices intervals stochastic geometry Poisson point process Blaschke–Petkantschin formula Fisher information metric


Edelsbrunner, Herbert; Nikitenko, Anton. Random inscribed polytopes have similar radius functions as Poisson–Delaunay mosaics. Ann. Appl. Probab. 28 (2018), no. 5, 3215--3238. doi:10.1214/18-AAP1389.

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