Schloss Dagstuhl - Leibniz-Zentrum für Informatik GmbH Schloss Dagstuhl - Leibniz-Zentrum für Informatik GmbH scholarly article en Biswas, Ranita; Cultrera di Montesano, Sebastiano; Edelsbrunner, Herbert; Saghafian, Morteza https://www.dagstuhl.de/lipics License: Creative Commons Attribution 4.0 license (CC BY 4.0)
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URN: urn:nbn:de:0030-drops-138152
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Counting Cells of Order-k Voronoi Tessellations in ℝ³ with Morse Theory

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Abstract

Generalizing Lee’s inductive argument for counting the cells of higher order Voronoi tessellations in ℝ² to ℝ³, we get precise relations in terms of Morse theoretic quantities for piecewise constant functions on planar arrangements. Specifically, we prove that for a generic set of n ≥ 5 points in ℝ³, the number of regions in the order-k Voronoi tessellation is N_{k-1} - binom(k,2)n + n, for 1 ≤ k ≤ n-1, in which N_{k-1} is the sum of Euler characteristics of these function’s first k-1 sublevel sets. We get similar expressions for the vertices, edges, and polygons of the order-k Voronoi tessellation.

BibTeX - Entry

@InProceedings{biswas_et_al:LIPIcs.SoCG.2021.16,
  author =	{Biswas, Ranita and Cultrera di Montesano, Sebastiano and Edelsbrunner, Herbert and Saghafian, Morteza},
  title =	{{Counting Cells of Order-k Voronoi Tessellations in \mathbb{R}³ with Morse Theory}},
  booktitle =	{37th International Symposium on Computational Geometry (SoCG 2021)},
  pages =	{16:1--16:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-184-9},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{189},
  editor =	{Buchin, Kevin and Colin de Verdi\`{e}re, \'{E}ric},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2021/13815},
  URN =		{urn:nbn:de:0030-drops-138152},
  doi =		{10.4230/LIPIcs.SoCG.2021.16},
  annote =	{Keywords: Voronoi tessellations, Delaunay mosaics, arrangements, convex polytopes, Morse theory, counting}
}

Keywords: Voronoi tessellations, Delaunay mosaics, arrangements, convex polytopes, Morse theory, counting
Seminar: 37th International Symposium on Computational Geometry (SoCG 2021)
Issue date: 2021
Date of publication: 02.06.2021


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