---
_id: '14345'
abstract:
- lang: eng
  text: For a locally finite set in R2, the order-k Brillouin tessellations form an
    infinite sequence of convex face-to-face tilings of the plane. If the set is coarsely
    dense and generic, then the corresponding infinite sequences of minimum and maximum
    angles are both monotonic in k. As an example, a stationary Poisson point process
    in R2  is locally finite, coarsely dense, and generic with probability one. For
    such a set, the distributions of angles in the Voronoi tessellations, Delaunay
    mosaics, and Brillouin tessellations are independent of the order and can be derived
    from the formula for angles in order-1 Delaunay mosaics given by Miles (Math.
    Biosci. 6, 85–127 (1970)).
acknowledgement: Work by all authors but A. Garber is supported by the European Research
  Council (ERC), Grant No. 788183, by the Wittgenstein Prize, Austrian Science Fund
  (FWF), Grant No. Z 342-N31, and by the DFG Collaborative Research Center TRR 109,
  Austrian Science Fund (FWF), Grant No. I 02979-N35. Work by A. Garber is partially
  supported by the Alexander von Humboldt Foundation.
article_processing_charge: Yes (via OA deal)
article_type: original
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: Alexey
  full_name: Garber, Alexey
  last_name: Garber
- first_name: Mohadese
  full_name: Ghafari, Mohadese
  last_name: Ghafari
- first_name: Teresa
  full_name: Heiss, Teresa
  id: 4879BB4E-F248-11E8-B48F-1D18A9856A87
  last_name: Heiss
  orcid: 0000-0002-1780-2689
- first_name: Morteza
  full_name: Saghafian, Morteza
  id: f86f7148-b140-11ec-9577-95435b8df824
  last_name: Saghafian
citation:
  ama: Edelsbrunner H, Garber A, Ghafari M, Heiss T, Saghafian M. On angles in higher
    order Brillouin tessellations and related tilings in the plane. <i>Discrete and
    Computational Geometry</i>. 2024;72:29-48. doi:<a href="https://doi.org/10.1007/s00454-023-00566-1">10.1007/s00454-023-00566-1</a>
  apa: Edelsbrunner, H., Garber, A., Ghafari, M., Heiss, T., &#38; Saghafian, M. (2024).
    On angles in higher order Brillouin tessellations and related tilings in the plane.
    <i>Discrete and Computational Geometry</i>. Springer Nature. <a href="https://doi.org/10.1007/s00454-023-00566-1">https://doi.org/10.1007/s00454-023-00566-1</a>
  chicago: Edelsbrunner, Herbert, Alexey Garber, Mohadese Ghafari, Teresa Heiss, and
    Morteza Saghafian. “On Angles in Higher Order Brillouin Tessellations and Related
    Tilings in the Plane.” <i>Discrete and Computational Geometry</i>. Springer Nature,
    2024. <a href="https://doi.org/10.1007/s00454-023-00566-1">https://doi.org/10.1007/s00454-023-00566-1</a>.
  ieee: H. Edelsbrunner, A. Garber, M. Ghafari, T. Heiss, and M. Saghafian, “On angles
    in higher order Brillouin tessellations and related tilings in the plane,” <i>Discrete
    and Computational Geometry</i>, vol. 72. Springer Nature, pp. 29–48, 2024.
  ista: Edelsbrunner H, Garber A, Ghafari M, Heiss T, Saghafian M. 2024. On angles
    in higher order Brillouin tessellations and related tilings in the plane. Discrete
    and Computational Geometry. 72, 29–48.
  mla: Edelsbrunner, Herbert, et al. “On Angles in Higher Order Brillouin Tessellations
    and Related Tilings in the Plane.” <i>Discrete and Computational Geometry</i>,
    vol. 72, Springer Nature, 2024, pp. 29–48, doi:<a href="https://doi.org/10.1007/s00454-023-00566-1">10.1007/s00454-023-00566-1</a>.
  short: H. Edelsbrunner, A. Garber, M. Ghafari, T. Heiss, M. Saghafian, Discrete
    and Computational Geometry 72 (2024) 29–48.
corr_author: '1'
date_created: 2023-09-17T22:01:10Z
date_published: 2024-07-01T00:00:00Z
date_updated: 2025-04-23T08:41:59Z
day: '01'
ddc:
- '510'
department:
- _id: HeEd
doi: 10.1007/s00454-023-00566-1
ec_funded: 1
external_id:
  arxiv:
  - '2204.01076'
  isi:
  - '001060727600004'
  pmid:
  - '39610762'
file:
- access_level: open_access
  checksum: b207b4e00f904e8ea8a30e24f0251f79
  content_type: application/pdf
  creator: dernst
  date_created: 2024-07-22T09:43:19Z
  date_updated: 2024-07-22T09:43:19Z
  file_id: '17301'
  file_name: 2024_DiscreteComputGeom_Edelsbrunner.pdf
  file_size: 892019
  relation: main_file
  success: 1
file_date_updated: 2024-07-22T09:43:19Z
has_accepted_license: '1'
intvolume: '        72'
isi: 1
language:
- iso: eng
month: '07'
oa: 1
oa_version: Published Version
page: 29-48
pmid: 1
project:
- _id: 266A2E9E-B435-11E9-9278-68D0E5697425
  call_identifier: H2020
  grant_number: '788183'
  name: Alpha Shape Theory Extended
- _id: 268116B8-B435-11E9-9278-68D0E5697425
  call_identifier: FWF
  grant_number: Z00342
  name: Mathematics, Computer Science
- _id: 2561EBF4-B435-11E9-9278-68D0E5697425
  call_identifier: FWF
  grant_number: I02979-N35
  name: Persistence and stability of geometric complexes
publication: Discrete and Computational Geometry
publication_identifier:
  eissn:
  - 1432-0444
  issn:
  - 0179-5376
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
scopus_import: '1'
status: public
title: On angles in higher order Brillouin tessellations and related tilings in the
  plane
tmp:
  image: /images/cc_by.png
  legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
  name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
  short: CC BY (4.0)
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 72
year: '2024'
...