10.1214/18-AAP1389
Edelsbrunner, Herbert
Herbert
Edelsbrunner0000-0002-9823-6833
Nikitenko, Anton
Anton
Nikitenko
Random inscribed polytopes have similar radius functions as Poisson-Delaunay mosaics
Institute of Mathematical Statistics
2018
2018-12-11T11:44:33Z
2019-08-13T08:31:21Z
journal_article
https://research-explorer.app.ist.ac.at/record/87
https://research-explorer.app.ist.ac.at/record/87.json
1705.02870
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.