On the Size of Chromatic Delaunay Mosaics

TL;DR

This paper introduces the chromatic Delaunay mosaic, a geometric structure representing point color interactions, and provides bounds on its size for various point distributions, highlighting its potential for practical applications.

Contribution

The paper defines the chromatic Delaunay mosaic and establishes size bounds for different point sets, including finite, Delone, and Poisson processes, under various coloring schemes.

Findings

Expected size is $O(n^{ ceil d/2 ceil})$ for random colorings of finite sets.

Expected number of cells in Delone and Poisson sets is proportional to the number of points.

In $ eal^2$, dense point sets have linear-sized chromatic Delaunay mosaics.

Abstract

Given a locally finite set $A \subseteq \mathbb{R}^d$ and a coloring $\chi \colon A \to \{0,1,\ldots,s\}$, we introduce the chromatic Delaunay mosaic of $\chi$, which is a Delaunay mosaic in $\mathbb{R}^{s+d}$ 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 $O(n^{\lceil{d/2}\rceil})$ cells in expectation. In contrast, for Delone sets and Poisson point processes in $\mathbb{R}^d$, the expected number of cells within a closed ball is only a constant times the number of points in this ball. Furthermore, in $\mathbb{R}^2$ 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.