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Counting cells of order-k voronoi tessellations in ℝ3 with morse theory

Biswas, R ; Sharif University of Technology | 2021

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  1. Type of Document: Article
  2. DOI: 10.4230/LIPIcs.SoCG.2021.16
  3. Publisher: Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing , 2021
  4. Abstract:
  5. 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 Nk−1 − (k/2)n + n, for 1 ≤ k ≤ n − 1, in which Nk−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. © Ranita Biswas, Sebastiano Cultrera di Montesano, Herbert Edelsbrunner, and Morteza Saghafian; licensed under Creative Commons License CC-BY 4.0 37th International Symposium on Computational Geometry (SoCG 2021)
  6. Keywords:
  7. Computation theory ; Euler characteristic ; Higher-order ; Morse the-ory ; Piece-wise-constant functions ; Sub-level sets ; Voronoi tessellations ; Computational geometry
  8. Source: 37th International Symposium on Computational Geometry, SoCG 2021, 7 June 2021 through 11 June 2021 ; Volume 189 , 2021 ; 18688969 (ISSN) ; 9783959771849 (ISBN)
  9. URL: https://drops.dagstuhl.de/opus/volltexte/2021/13815