Author(s):

Edelsbrunner, Herbert; Nikitenko, Anton

Article Title: 
Random inscribed polytopes have similar radius functions as PoissonDelaunay mosaics

Affiliation 
IST Austria 
Abstract: 
Using the geodesic distance on the ndimensional 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 ndimensional 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 nsphere is isometric to the standard nsimplex equipped with the Fisher information metric. It follows that the latter space has similar stochastic properties as the ndimensional 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:

10505164

Publisher:

Institute of Mathematical Statistics

Date Published:

20181001

Start Page: 
3215

End Page:

3238

Sponsor: 
Austrian Science Fund (FWF) Grant I02979N35

DOI: 
10.1214/18AAP1389

Open access: 
no 