---
_id: '87'
abstract:
- lang: eng
text: 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.
author:
- first_name: Herbert
full_name: Edelsbrunner, Herbert
id: 3FB178DA-F248-11E8-B48F-1D18A9856A87
last_name: Edelsbrunner
orcid: 0000-0002-9823-6833
- first_name: Anton
full_name: Nikitenko, Anton
id: 3E4FF1BA-F248-11E8-B48F-1D18A9856A87
last_name: Nikitenko
citation:
ama: Edelsbrunner H, Nikitenko A. Random inscribed polytopes have similar radius
functions as Poisson-Delaunay mosaics. *Annals of Applied Probability*. 2018;28(5):3215-3238.
doi:10.1214/18-AAP1389
apa: Edelsbrunner, H., & Nikitenko, A. (2018). Random inscribed polytopes have
similar radius functions as Poisson-Delaunay mosaics. *Annals of Applied Probability*,
*28*(5), 3215–3238. https://doi.org/10.1214/18-AAP1389
chicago: 'Edelsbrunner, Herbert, and Anton Nikitenko. “Random Inscribed Polytopes
Have Similar Radius Functions as Poisson-Delaunay Mosaics.” *Annals of Applied
Probability* 28, no. 5 (2018): 3215–38. https://doi.org/10.1214/18-AAP1389.'
ieee: H. Edelsbrunner and A. Nikitenko, “Random inscribed polytopes have similar
radius functions as Poisson-Delaunay mosaics,” *Annals of Applied Probability*,
vol. 28, no. 5, pp. 3215–3238, 2018.
ista: Edelsbrunner H, Nikitenko A. 2018. Random inscribed polytopes have similar
radius functions as Poisson-Delaunay mosaics. Annals of Applied Probability. 28(5),
3215–3238.
mla: Edelsbrunner, Herbert, and Anton Nikitenko. “Random Inscribed Polytopes Have
Similar Radius Functions as Poisson-Delaunay Mosaics.” *Annals of Applied Probability*,
vol. 28, no. 5, Institute of Mathematical Statistics, 2018, pp. 3215–38, doi:10.1214/18-AAP1389.
short: H. Edelsbrunner, A. Nikitenko, Annals of Applied Probability 28 (2018) 3215–3238.
date_created: 2018-12-11T11:44:33Z
date_published: 2018-10-01T00:00:00Z
date_updated: 2019-08-13T08:31:21Z
day: '01'
department:
- _id: HeEd
doi: 10.1214/18-AAP1389
external_id:
arxiv:
- '1705.02870'
intvolume: ' 28'
issue: '5'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://arxiv.org/abs/1705.02870
month: '10'
oa_version: Preprint
page: 3215 - 3238
project:
- _id: 2561EBF4-B435-11E9-9278-68D0E5697425
grant_number: I02979-N35
name: Persistence and stability of geometric complexes
publication: Annals of Applied Probability
publication_status: published
publisher: Institute of Mathematical Statistics
publist_id: '7967'
quality_controlled: '1'
related_material:
record:
- id: '6287'
relation: dissertation_contains
status: public
status: public
title: Random inscribed polytopes have similar radius functions as Poisson-Delaunay
mosaics
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 28
year: '2018'
...