@inproceedings{9604,
  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.},
  author       = {Biswas, Ranita and Cultrera di Montesano, Sebastiano and Edelsbrunner, Herbert and Saghafian, Morteza},
  booktitle    = {Leibniz International Proceedings in Informatics},
  isbn         = {9783959771849},
  issn         = {18688969},
  location     = {Online},
  publisher    = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik},
  title        = {{Counting cells of order-k voronoi tessellations in ℝ<sup>3</sup> with morse theory}},
  doi          = {10.4230/LIPIcs.SoCG.2021.16},
  volume       = {189},
  year         = {2021},
}

@inproceedings{9605,
  abstract     = {Given a finite set A ⊂ ℝ^d, let Cov_{r,k} denote the set of all points within distance r to at least k points of A. Allowing r and k to vary, we obtain a 2-parameter family of spaces that grow larger when r increases or k decreases, called the multicover bifiltration. Motivated by the problem of computing the homology of this bifiltration, we introduce two closely related combinatorial bifiltrations, one polyhedral and the other simplicial, which are both topologically equivalent to the multicover bifiltration and far smaller than a Čech-based model considered in prior work of Sheehy. Our polyhedral construction is a bifiltration of the rhomboid tiling of Edelsbrunner and Osang, and can be efficiently computed using a variant of an algorithm given by these authors as well. Using an implementation for dimension 2 and 3, we provide experimental results. Our simplicial construction is useful for understanding the polyhedral construction and proving its correctness. },
  author       = {Corbet, René and Kerber, Michael and Lesnick, Michael and Osang, Georg F},
  booktitle    = {Leibniz International Proceedings in Informatics},
  isbn         = {9783959771849},
  issn         = {18688969},
  location     = {Online},
  publisher    = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik},
  title        = {{Computing the multicover bifiltration}},
  doi          = {10.4230/LIPIcs.SoCG.2021.27},
  volume       = {189},
  year         = {2021},
}