---
_id: '718'
abstract:
- lang: eng
  text: Mapping every simplex in the Delaunay mosaic of a discrete point set to the
    radius of the smallest empty circumsphere gives a generalized discrete Morse function.
    Choosing the points from a Poisson point process in ℝ n , we study the expected
    number of simplices in the Delaunay mosaic as well as the expected number of critical
    simplices and nonsingular intervals in the corresponding generalized discrete
    gradient. Observing connections with other probabilistic models, we obtain precise
    expressions for the expected numbers in low dimensions. In particular, we obtain
    the expected numbers of simplices in the Poisson–Delaunay mosaic in dimensions
    n ≤ 4.
arxiv: 1
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
  orcid: 0000-0002-0659-3201
- first_name: Matthias
  full_name: Reitzner, Matthias
  last_name: Reitzner
citation:
  ama: Edelsbrunner H, Nikitenko A, Reitzner M. Expected sizes of poisson Delaunay
    mosaics and their discrete Morse functions. <i>Advances in Applied Probability</i>.
    2017;49(3):745-767. doi:<a href="https://doi.org/10.1017/apr.2017.20">10.1017/apr.2017.20</a>
  apa: Edelsbrunner, H., Nikitenko, A., &#38; Reitzner, M. (2017). Expected sizes
    of poisson Delaunay mosaics and their discrete Morse functions. <i>Advances in
    Applied Probability</i>. Cambridge University Press. <a href="https://doi.org/10.1017/apr.2017.20">https://doi.org/10.1017/apr.2017.20</a>
  chicago: Edelsbrunner, Herbert, Anton Nikitenko, and Matthias Reitzner. “Expected
    Sizes of Poisson Delaunay Mosaics and Their Discrete Morse Functions.” <i>Advances
    in Applied Probability</i>. Cambridge University Press, 2017. <a href="https://doi.org/10.1017/apr.2017.20">https://doi.org/10.1017/apr.2017.20</a>.
  ieee: H. Edelsbrunner, A. Nikitenko, and M. Reitzner, “Expected sizes of poisson
    Delaunay mosaics and their discrete Morse functions,” <i>Advances in Applied Probability</i>,
    vol. 49, no. 3. Cambridge University Press, pp. 745–767, 2017.
  ista: Edelsbrunner H, Nikitenko A, Reitzner M. 2017. Expected sizes of poisson Delaunay
    mosaics and their discrete Morse functions. Advances in Applied Probability. 49(3),
    745–767.
  mla: Edelsbrunner, Herbert, et al. “Expected Sizes of Poisson Delaunay Mosaics and
    Their Discrete Morse Functions.” <i>Advances in Applied Probability</i>, vol.
    49, no. 3, Cambridge University Press, 2017, pp. 745–67, doi:<a href="https://doi.org/10.1017/apr.2017.20">10.1017/apr.2017.20</a>.
  short: H. Edelsbrunner, A. Nikitenko, M. Reitzner, Advances in Applied Probability
    49 (2017) 745–767.
date_created: 2018-12-11T11:48:07Z
date_published: 2017-09-01T00:00:00Z
date_updated: 2023-09-07T12:07:12Z
day: '01'
department:
- _id: HeEd
doi: 10.1017/apr.2017.20
ec_funded: 1
external_id:
  arxiv:
  - '1607.05915'
intvolume: '        49'
issue: '3'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://arxiv.org/abs/1607.05915
month: '09'
oa: 1
oa_version: Preprint
page: 745 - 767
project:
- _id: 255D761E-B435-11E9-9278-68D0E5697425
  call_identifier: FP7
  grant_number: '318493'
  name: Topological Complex Systems
- _id: 2561EBF4-B435-11E9-9278-68D0E5697425
  call_identifier: FWF
  grant_number: I02979-N35
  name: Persistence and stability of geometric complexes
publication: Advances in Applied Probability
publication_identifier:
  issn:
  - '00018678'
publication_status: published
publisher: Cambridge University Press
publist_id: '6962'
quality_controlled: '1'
related_material:
  record:
  - id: '6287'
    relation: dissertation_contains
    status: public
scopus_import: 1
status: public
title: Expected sizes of poisson Delaunay mosaics and their discrete Morse functions
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 49
year: '2017'
...