On the Size of Chromatic Delaunay Mosaics

12/06/2022

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by   Ranita Biswas, et al.

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Given a locally finite set A ⊆ℝ^d and a coloring χ A →{0,1,…,s}, we introduce the chromatic Delaunay mosaic of χ, which is a Delaunay mosaic in ℝ^s+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 ℝ^d, the expected number of cells within a closed ball is only a constant times the number of points in this ball. Furthermore, in ℝ^2 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.