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DOI: 10.4230/LIPIcs.SoCG.2021.16
Counting Cells of Order-k Voronoi Tessellations in ℝ³ with Morse Theory

Authors: Ranita Biswas, Sebastiano Cultrera di Montesano, Herbert Edelsbrunner, and Morteza Saghafian

Published in: LIPIcs, Volume 189, 37th International Symposium on Computational Geometry (SoCG 2021)

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.
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Ranita Biswas, Sebastiano Cultrera di Montesano, Herbert Edelsbrunner, and Morteza Saghafian. Counting Cells of Order-k Voronoi Tessellations in ℝ³ with Morse Theory. In 37th International Symposium on Computational Geometry (SoCG 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 189, pp. 16:1-16:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2021.16},
  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}
}
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