article
Random inscribed polytopes have similar radius functions as Poisson-Delaunay mosaics
published
yes
Herbert
Edelsbrunner
author 3FB178DA-F248-11E8-B48F-1D18A9856A870000-0002-9823-6833
Anton
Nikitenko
author 3E4FF1BA-F248-11E8-B48F-1D18A9856A87
HeEd
department
Persistence and stability of geometric complexes
project
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.
Institute of Mathematical Statistics2018
eng
Annals of Applied Probability
1705.0287010.1214/18-AAP1389
2853215 - 3238
https://research-explorer.app.ist.ac.at/record/6287
H. Edelsbrunner, A. Nikitenko, Annals of Applied Probability 28 (2018) 3215–3238.
Edelsbrunner H, Nikitenko A. Random inscribed polytopes have similar radius functions as Poisson-Delaunay mosaics. <i>Annals of Applied Probability</i>. 2018;28(5):3215-3238. doi:<a href="https://doi.org/10.1214/18-AAP1389">10.1214/18-AAP1389</a>
Edelsbrunner H, Nikitenko A. 2018. Random inscribed polytopes have similar radius functions as Poisson-Delaunay mosaics. Annals of Applied Probability. 28(5), 3215–3238.
Edelsbrunner, Herbert, and Anton Nikitenko. “Random Inscribed Polytopes Have Similar Radius Functions as Poisson-Delaunay Mosaics.” <i>Annals of Applied Probability</i>. Institute of Mathematical Statistics, 2018. <a href="https://doi.org/10.1214/18-AAP1389">https://doi.org/10.1214/18-AAP1389</a>.
Edelsbrunner, Herbert, and Anton Nikitenko. “Random Inscribed Polytopes Have Similar Radius Functions as Poisson-Delaunay Mosaics.” <i>Annals of Applied Probability</i>, vol. 28, no. 5, Institute of Mathematical Statistics, 2018, pp. 3215–38, doi:<a href="https://doi.org/10.1214/18-AAP1389">10.1214/18-AAP1389</a>.
Edelsbrunner, H., & Nikitenko, A. (2018). Random inscribed polytopes have similar radius functions as Poisson-Delaunay mosaics. <i>Annals of Applied Probability</i>. Institute of Mathematical Statistics. <a href="https://doi.org/10.1214/18-AAP1389">https://doi.org/10.1214/18-AAP1389</a>
H. Edelsbrunner and A. Nikitenko, “Random inscribed polytopes have similar radius functions as Poisson-Delaunay mosaics,” <i>Annals of Applied Probability</i>, vol. 28, no. 5. Institute of Mathematical Statistics, pp. 3215–3238, 2018.
872018-12-11T11:44:33Z2021-01-12T08:20:36Z