Random inscribed polytopes have similar radius functions as Poisson-Delaunay mosaics Journal Article

Author(s): Edelsbrunner, Herbert; Nikitenko, Anton
Article Title: Random inscribed polytopes have similar radius functions as Poisson-Delaunay mosaics
Affiliation IST Austria
Abstract: 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 Rn+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.
Keywords: intervals; Poisson point process; discrete Morse theory; critical simplices; stochastic geometry; Delaunay mosaics; Blaschke–Petkantschin formula; Fisher information metric; Inscribed polytopes; Voronoi tessellations
Journal Title: Annals of Applied Probability
Volume: 28
Issue 5
ISSN: 1050-5164
Publisher: Institute of Mathematical Statistics  
Date Published: 2018-10-01
Start Page: 3215
End Page: 3238
Sponsor: Austrian Science Fund (FWF) Grant I02979-N35
DOI: 10.1214/18-AAP1389
Open access: no
IST Austria Authors