---
_id: '9317'
abstract:
- lang: eng
  text: Given a locally finite X⊆Rd and a radius r≥0, the k-fold cover of X and r
    consists of all points in Rd that have k or more points of X within distance r.
    We consider two filtrations—one in scale obtained by fixing k and increasing r,
    and the other in depth obtained by fixing r and decreasing k—and we compute the
    persistence diagrams of both. While standard methods suffice for the filtration
    in scale, we need novel geometric and topological concepts for the filtration
    in depth. In particular, we introduce a rhomboid tiling in Rd+1 whose horizontal
    integer slices are the order-k Delaunay mosaics of X, and construct a zigzag module
    of Delaunay mosaics that is isomorphic to the persistence module of the multi-covers.
acknowledgement: "This project has received funding from the European Research Council
  (ERC) under the European Union’s Horizon 2020 research and innovation programme
  (Grant Agreement No. 78818 Alpha), and by the DFG Collaborative Research Center
  TRR 109, ‘Discretization in Geometry and Dynamics’, through Grant No. I02979-N35
  of the Austrian Science Fund (FWF)\r\nOpen Access funding provided by the Institute
  of Science and Technology (IST Austria)."
article_processing_charge: Yes (via OA deal)
article_type: original
author:
- first_name: Herbert
  full_name: Edelsbrunner, Herbert
  id: 3FB178DA-F248-11E8-B48F-1D18A9856A87
  last_name: Edelsbrunner
  orcid: 0000-0002-9823-6833
- first_name: Georg F
  full_name: Osang, Georg F
  id: 464B40D6-F248-11E8-B48F-1D18A9856A87
  last_name: Osang
  orcid: 0000-0002-8882-5116
citation:
  ama: Edelsbrunner H, Osang GF. The multi-cover persistence of Euclidean balls. <i>Discrete
    and Computational Geometry</i>. 2021;65:1296–1313. doi:<a href="https://doi.org/10.1007/s00454-021-00281-9">10.1007/s00454-021-00281-9</a>
  apa: Edelsbrunner, H., &#38; Osang, G. F. (2021). The multi-cover persistence of
    Euclidean balls. <i>Discrete and Computational Geometry</i>. Springer Nature.
    <a href="https://doi.org/10.1007/s00454-021-00281-9">https://doi.org/10.1007/s00454-021-00281-9</a>
  chicago: Edelsbrunner, Herbert, and Georg F Osang. “The Multi-Cover Persistence
    of Euclidean Balls.” <i>Discrete and Computational Geometry</i>. Springer Nature,
    2021. <a href="https://doi.org/10.1007/s00454-021-00281-9">https://doi.org/10.1007/s00454-021-00281-9</a>.
  ieee: H. Edelsbrunner and G. F. Osang, “The multi-cover persistence of Euclidean
    balls,” <i>Discrete and Computational Geometry</i>, vol. 65. Springer Nature,
    pp. 1296–1313, 2021.
  ista: Edelsbrunner H, Osang GF. 2021. The multi-cover persistence of Euclidean balls.
    Discrete and Computational Geometry. 65, 1296–1313.
  mla: Edelsbrunner, Herbert, and Georg F. Osang. “The Multi-Cover Persistence of
    Euclidean Balls.” <i>Discrete and Computational Geometry</i>, vol. 65, Springer
    Nature, 2021, pp. 1296–1313, doi:<a href="https://doi.org/10.1007/s00454-021-00281-9">10.1007/s00454-021-00281-9</a>.
  short: H. Edelsbrunner, G.F. Osang, Discrete and Computational Geometry 65 (2021)
    1296–1313.
corr_author: '1'
date_created: 2021-04-11T22:01:15Z
date_published: 2021-03-31T00:00:00Z
date_updated: 2025-06-12T06:36:54Z
day: '31'
ddc:
- '516'
department:
- _id: HeEd
doi: 10.1007/s00454-021-00281-9
ec_funded: 1
external_id:
  isi:
  - '000635460400001'
  pmid:
  - '34720303'
file:
- access_level: open_access
  checksum: 59b4e1e827e494209bcb4aae22e1d347
  content_type: application/pdf
  creator: cchlebak
  date_created: 2021-12-01T10:56:53Z
  date_updated: 2021-12-01T10:56:53Z
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  file_name: 2021_DisCompGeo_Edelsbrunner_Osang.pdf
  file_size: 677704
  relation: main_file
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file_date_updated: 2021-12-01T10:56:53Z
has_accepted_license: '1'
intvolume: '        65'
isi: 1
language:
- iso: eng
month: '03'
oa: 1
oa_version: Published Version
page: 1296–1313
pmid: 1
project:
- _id: 266A2E9E-B435-11E9-9278-68D0E5697425
  call_identifier: H2020
  grant_number: '788183'
  name: Alpha Shape Theory Extended
- _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'
related_material:
  record:
  - id: '187'
    relation: earlier_version
    status: public
scopus_import: '1'
status: public
title: The multi-cover persistence of Euclidean balls
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: 65
year: '2021'
...