---
OA_place: repository
_id: '15090'
abstract:
- lang: eng
  text: 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.
article_number: '2212.03121'
article_processing_charge: No
arxiv: 1
author:
- first_name: Ranita
  full_name: Biswas, Ranita
  id: 3C2B033E-F248-11E8-B48F-1D18A9856A87
  last_name: Biswas
  orcid: 0000-0002-5372-7890
- first_name: Sebastiano
  full_name: Cultrera di Montesano, Sebastiano
  id: 34D2A09C-F248-11E8-B48F-1D18A9856A87
  last_name: Cultrera di Montesano
  orcid: 0000-0001-6249-0832
- first_name: Ondrej
  full_name: Draganov, Ondrej
  id: 2B23F01E-F248-11E8-B48F-1D18A9856A87
  last_name: Draganov
  orcid: 0000-0003-0464-3823
- first_name: Herbert
  full_name: Edelsbrunner, Herbert
  id: 3FB178DA-F248-11E8-B48F-1D18A9856A87
  last_name: Edelsbrunner
  orcid: 0000-0002-9823-6833
- first_name: Morteza
  full_name: Saghafian, Morteza
  id: f86f7148-b140-11ec-9577-95435b8df824
  last_name: Saghafian
citation:
  ama: Biswas R, Cultrera di Montesano S, Draganov O, Edelsbrunner H, Saghafian M.
    On the size of chromatic Delaunay mosaics. <i>arXiv</i>.
  apa: Biswas, R., Cultrera di Montesano, S., Draganov, O., Edelsbrunner, H., &#38;
    Saghafian, M. (n.d.). On the size of chromatic Delaunay mosaics. <i>arXiv</i>.
  chicago: Biswas, Ranita, Sebastiano Cultrera di Montesano, Ondrej Draganov, Herbert
    Edelsbrunner, and Morteza Saghafian. “On the Size of Chromatic Delaunay Mosaics.”
    <i>ArXiv</i>, n.d.
  ieee: R. Biswas, S. Cultrera di Montesano, O. Draganov, H. Edelsbrunner, and M.
    Saghafian, “On the size of chromatic Delaunay mosaics,” <i>arXiv</i>. .
  ista: Biswas R, Cultrera di Montesano S, Draganov O, Edelsbrunner H, Saghafian M.
    On the size of chromatic Delaunay mosaics. arXiv, 2212.03121.
  mla: Biswas, Ranita, et al. “On the Size of Chromatic Delaunay Mosaics.” <i>ArXiv</i>,
    2212.03121.
  short: R. Biswas, S. Cultrera di Montesano, O. Draganov, H. Edelsbrunner, M. Saghafian,
    ArXiv (n.d.).
corr_author: '1'
date_created: 2024-03-08T09:54:20Z
date_published: 2022-12-06T00:00:00Z
date_updated: 2026-04-07T12:58:47Z
day: '06'
department:
- _id: HeEd
ec_funded: 1
external_id:
  arxiv:
  - '2212.03121'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://arxiv.org/abs/2212.03121
month: '12'
oa: 1
oa_version: Preprint
project:
- _id: 266A2E9E-B435-11E9-9278-68D0E5697425
  call_identifier: H2020
  grant_number: '788183'
  name: Alpha Shape Theory Extended
- _id: 0aa4bc98-070f-11eb-9043-e6fff9c6a316
  grant_number: I4887
  name: Persistent Homology, Algorithms and Stochastic Geometry
- _id: 268116B8-B435-11E9-9278-68D0E5697425
  call_identifier: FWF
  grant_number: Z00342
  name: Mathematics, Computer Science
publication: arXiv
publication_status: draft
related_material:
  record:
  - id: '20456'
    relation: later_version
    status: public
  - id: '15094'
    relation: dissertation_contains
    status: public
status: public
title: On the size of chromatic Delaunay mosaics
tmp:
  image: /images/cc_by.png
  legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
  name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
  short: CC BY (4.0)
type: preprint
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
year: '2022'
...