On the size of chromatic Delaunay mosaics

Biswas, Ranita; Cultrera di Montesano, Sebastiano; Draganov, Ondrej; Edelsbrunner, Herbert; Saghafian, Morteza

On the size of chromatic Delaunay mosaics

Biswas R, Cultrera di Montesano S, Draganov O, Edelsbrunner H, Saghafian M. On the size of chromatic Delaunay mosaics. arXiv, 2212.03121.

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Author

Biswas, Ranita^ISTA ; Cultrera di Montesano, Sebastiano^ISTA ; Draganov, Ondrej^ISTA; Edelsbrunner, Herbert^ISTA ; Saghafian, Morteza^ISTA

Department

Edelsbrunner Group

Grant

Alpha Shape Theory Extended
Discretization in Geometry and Dynamics
The Wittgenstein Prize

Abstract

Given a locally finite set A⊆Rd and a coloring χ:A→{0,1,…,s}, we introduce the chromatic Delaunay mosaic of χ, which is a Delaunay mosaic in Rs+d that represents how points of different colors mingle. Our main results are bounds on the size of the chromatic Delaunay mosaic, in which we assume that d and s are constants. For example, if A is finite with n=#A, and the coloring is random, then the chromatic Delaunay mosaic has O(n⌈d/2⌉) cells in expectation. In contrast, for Delone sets and Poisson point processes in Rd, the expected number of cells within a closed ball is only a constant times the number of points in this ball. Furthermore, in R2 all colorings of a dense set of n points have chromatic Delaunay mosaics of size O(n). This encourages the use of chromatic Delaunay mosaics in applications.

Publishing Year

2022

Date Published

2022-12-06

Journal Title

arXiv

Article Number

2212.03121

IST-REx-ID

15090

Cite this

Biswas R, Cultrera di Montesano S, Draganov O, Edelsbrunner H, Saghafian M. On the size of chromatic Delaunay mosaics. arXiv.

Biswas, R., Cultrera di Montesano, S., Draganov, O., Edelsbrunner, H., & Saghafian, M. (n.d.). On the size of chromatic Delaunay mosaics. arXiv.

Biswas, Ranita, Sebastiano Cultrera di Montesano, Ondrej Draganov, Herbert Edelsbrunner, and Morteza Saghafian. “On the Size of Chromatic Delaunay Mosaics.” ArXiv, n.d.

R. Biswas, S. Cultrera di Montesano, O. Draganov, H. Edelsbrunner, and M. Saghafian, “On the size of chromatic Delaunay mosaics,” arXiv. .

Biswas R, Cultrera di Montesano S, Draganov O, Edelsbrunner H, Saghafian M. On the size of chromatic Delaunay mosaics. arXiv, 2212.03121.

Biswas, Ranita, et al. “On the Size of Chromatic Delaunay Mosaics.” ArXiv, 2212.03121.

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