Counting Cells of Order-k Voronoi Tessellations in ℝ³ with Morse Theory

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

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Ranita Biswas
  • IST Austria (Institute of Science and Technology Austria), Klosterneuburg, Austria
Sebastiano Cultrera di Montesano
  • IST Austria (Institute of Science and Technology Austria), Klosterneuburg, Austria
Herbert Edelsbrunner
  • IST Austria (Institute of Science and Technology Austria), Klosterneuburg, Austria
Morteza Saghafian
  • Department of Mathematical Sciences, Sharif University of Technology, Tehran, Iran

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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)


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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • Voronoi tessellations
  • Delaunay mosaics
  • arrangements
  • convex polytopes
  • Morse theory
  • counting


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