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October 2018 Random inscribed polytopes have similar radius functions as Poisson–Delaunay mosaics
Herbert Edelsbrunner, Anton Nikitenko
Ann. Appl. Probab. 28(5): 3215-3238 (October 2018). DOI: 10.1214/18-AAP1389


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.


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Herbert Edelsbrunner. Anton Nikitenko. "Random inscribed polytopes have similar radius functions as Poisson–Delaunay mosaics." Ann. Appl. Probab. 28 (5) 3215 - 3238, October 2018.


Received: 1 May 2017; Revised: 1 February 2018; Published: October 2018
First available in Project Euclid: 28 August 2018

zbMATH: 06974778
MathSciNet: MR3847986
Digital Object Identifier: 10.1214/18-AAP1389

Primary: 60D05
Secondary: 68U05

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

Rights: Copyright © 2018 Institute of Mathematical Statistics

Vol.28 • No. 5 • October 2018
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