---
_id: '9465'
abstract:
- lang: eng
  text: "Given a locally finite set \U0001D44B⊆ℝ\U0001D451 and an integer \U0001D458≥0,
    we consider the function \U0001D430\U0001D458:Del\U0001D458(\U0001D44B)→ℝ on the
    dual of the order-k Voronoi tessellation, whose sublevel sets generalize the notion
    of alpha shapes from order-1 to order-k (Edelsbrunner et al. in IEEE Trans Inf
    Theory IT-29:551–559, 1983; Krasnoshchekov and Polishchuk in Inf Process Lett
    114:76–83, 2014). While this function is not necessarily generalized discrete
    Morse, in the sense of Forman (Adv Math 134:90–145, 1998) and Freij (Discrete
    Math 309:3821–3829, 2009), we prove that it satisfies similar properties so that
    its increments can be meaningfully classified into critical and non-critical steps.
    This result extends to the case of weighted points and sheds light on k-fold covers
    with balls in Euclidean space."
article_number: '15'
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: Anton
  full_name: Nikitenko, Anton
  id: 3E4FF1BA-F248-11E8-B48F-1D18A9856A87
  last_name: Nikitenko
- first_name: Georg F
  full_name: Osang, Georg F
  id: 464B40D6-F248-11E8-B48F-1D18A9856A87
  last_name: Osang
citation:
  ama: Edelsbrunner H, Nikitenko A, Osang GF. A step in the Delaunay mosaic of order
    k. <i>Journal of Geometry</i>. 2021;112(1). doi:<a href="https://doi.org/10.1007/s00022-021-00577-4">10.1007/s00022-021-00577-4</a>
  apa: Edelsbrunner, H., Nikitenko, A., &#38; Osang, G. F. (2021). A step in the Delaunay
    mosaic of order k. <i>Journal of Geometry</i>. Springer Nature. <a href="https://doi.org/10.1007/s00022-021-00577-4">https://doi.org/10.1007/s00022-021-00577-4</a>
  chicago: Edelsbrunner, Herbert, Anton Nikitenko, and Georg F Osang. “A Step in the
    Delaunay Mosaic of Order K.” <i>Journal of Geometry</i>. Springer Nature, 2021.
    <a href="https://doi.org/10.1007/s00022-021-00577-4">https://doi.org/10.1007/s00022-021-00577-4</a>.
  ieee: H. Edelsbrunner, A. Nikitenko, and G. F. Osang, “A step in the Delaunay mosaic
    of order k,” <i>Journal of Geometry</i>, vol. 112, no. 1. Springer Nature, 2021.
  ista: Edelsbrunner H, Nikitenko A, Osang GF. 2021. A step in the Delaunay mosaic
    of order k. Journal of Geometry. 112(1), 15.
  mla: Edelsbrunner, Herbert, et al. “A Step in the Delaunay Mosaic of Order K.” <i>Journal
    of Geometry</i>, vol. 112, no. 1, 15, Springer Nature, 2021, doi:<a href="https://doi.org/10.1007/s00022-021-00577-4">10.1007/s00022-021-00577-4</a>.
  short: H. Edelsbrunner, A. Nikitenko, G.F. Osang, Journal of Geometry 112 (2021).
date_created: 2021-06-06T22:01:29Z
date_published: 2021-04-01T00:00:00Z
date_updated: 2022-05-12T11:41:45Z
day: '01'
ddc:
- '510'
department:
- _id: HeEd
doi: 10.1007/s00022-021-00577-4
file:
- access_level: open_access
  checksum: e52a832f1def52a2b23d21bcc09e646f
  content_type: application/pdf
  creator: kschuh
  date_created: 2021-06-11T13:16:26Z
  date_updated: 2021-06-11T13:16:26Z
  file_id: '9544'
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  file_size: 694706
  relation: main_file
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file_date_updated: 2021-06-11T13:16:26Z
has_accepted_license: '1'
intvolume: '       112'
issue: '1'
language:
- iso: eng
month: '04'
oa: 1
oa_version: Published Version
publication: Journal of Geometry
publication_identifier:
  eissn:
  - '14208997'
  issn:
  - '00472468'
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
scopus_import: '1'
status: public
title: A step in the Delaunay mosaic of order k
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: 112
year: '2021'
...