A step in the Delaunay mosaic of order \(k\)
type: Item
instance of: scholarly article
title: A step in the Delaunay mosaic of order \(k\)
zbMATH Open document ID: 1473.57068
author: Georg Osang
author: Herbert Edelsbrunner
author: Anton V. Nikitenko
published in: Journal of Geometry
publication date: 2021-06-24T00:00:00Z
review text: The main object of study in this paper is the order-\(k\) Delaunay mosaic, denoted \(\textrm{Del}_k(X)\), of a locally finite subset \(X\) of \(\mathbb{R}^d\). Constructed in [\textit{F. Aurenhammer}, Discrete Comput. Geom. 5, No. 3, 243--254 (1990; Zbl 0693.68023)], this is the natural dual of the order-\(k\) Voronoi tessellation determined by \(X\), which in turn is the decomposition of \(\mathbb{R}^d\) into closed convex domains so that each domain is the set of points closest to a fixed \(k\)-element subset of \(X\). Being the dual of this, one can think of cells in \(\textrm{Del}_k(X)\) as representing collections of intersecting domains. There is a natural function \(\textbf{w}_k \colon \textrm{Del}_k(X) \to \mathbb{R}\) given by sending a cell to the minimum \(r\) such that the corresponding intersection of domains contains some ``witness'' point within distance \(r\) of each of the \(k\) nearest neighbors in \(X\). This function is not a (Forman-style) discrete Morse function, nor a so called generalized discrete Morse function, but the main result of the present paper is that it nonetheless behaves similarly to a discrete Morse function in certain ways. More precisely (see Section~3), the authors establish a classification of how the topology of the sublevels changes at both critical and non-critical steps.
reviewed by: Matthew C. B. Zaremsky
Mathematics Subject Classification ID: 57Q70
Mathematics Subject Classification ID: 52C22
zbMATH DE Number: 7362847
zbMATH Keywords: discrete Morse theory
zbMATH Keywords: order-\(k\) Voronoi tessellations
zbMATH Keywords: hyperplane arrangements
zbMATH Keywords: order-\(k\) Delaunay mosaics
zbMATH Keywords: rhomboid tilings
zbMATH Keywords: weighted points
zbMATH Keywords: power distance
MaRDI profile type: Publication
full work available at URL: s00022-021-00577-4
OpenAlex ID: W3137070108
cites work: A new duality result concerning Voronoi diagrams
cites work: The Morse theory of Čech and Delaunay complexes
cites work: Shellable Nonpure Complexes and Posets. I
cites work: On the shape of a set of points in the plane
cites work: Poisson-Delaunay mosaics of order \(k\)
cites work: The Multi-cover Persistence of Euclidean Balls
cites work: Voronoi diagrams and arrangements
cites work: Multiple packing and covering of the plane with circles
cites work: Morse theory for cell complexes
cites work: Equivariant discrete Morse theory
cites work: Combinatorial algebraic topology
cites work: Order-\(k\) \(\alpha\)-hulls and \(\alpha\)-shapes
cites work: Lectures on Polytopes
DOI: 10.1007/S00022-021-00577-4
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description: scientific article; zbMATH DE number 7362847