{"author":[{"orcid":"0000-0002-5372-7890","last_name":"Biswas","first_name":"Ranita","id":"3C2B033E-F248-11E8-B48F-1D18A9856A87","full_name":"Biswas, Ranita"},{"id":"34D2A09C-F248-11E8-B48F-1D18A9856A87","full_name":"Cultrera di Montesano, Sebastiano","last_name":"Cultrera di Montesano","orcid":"0000-0001-6249-0832","first_name":"Sebastiano"},{"first_name":"Ondrej","last_name":"Draganov","full_name":"Draganov, Ondrej","id":"2B23F01E-F248-11E8-B48F-1D18A9856A87"},{"id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","full_name":"Edelsbrunner, Herbert","orcid":"0000-0002-9823-6833","last_name":"Edelsbrunner","first_name":"Herbert"},{"first_name":"Morteza","last_name":"Saghafian","full_name":"Saghafian, Morteza","id":"f86f7148-b140-11ec-9577-95435b8df824"}],"publication":"arXiv","language":[{"iso":"eng"}],"day":"06","main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/2212.03121"}],"publication_status":"submitted","date_created":"2024-03-08T09:54:20Z","month":"12","year":"2022","external_id":{"arxiv":["2212.03121"]},"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."}],"license":"https://creativecommons.org/licenses/by/4.0/","oa":1,"status":"public","date_published":"2022-12-06T00:00:00Z","_id":"15090","ec_funded":1,"citation":{"mla":"Biswas, Ranita, et al. “On the Size of Chromatic Delaunay Mosaics.” <i>ArXiv</i>, 2212.03121.","ista":"Biswas R, Cultrera di Montesano S, Draganov O, Edelsbrunner H, Saghafian M. On the size of chromatic Delaunay mosaics. arXiv, 2212.03121.","ieee":"R. Biswas, S. Cultrera di Montesano, O. Draganov, H. Edelsbrunner, and M. Saghafian, “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.","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>.","ama":"Biswas R, Cultrera di Montesano S, Draganov O, Edelsbrunner H, Saghafian M. On the size of chromatic Delaunay mosaics. <i>arXiv</i>.","short":"R. Biswas, S. Cultrera di Montesano, O. Draganov, H. Edelsbrunner, M. Saghafian, ArXiv (n.d.)."},"title":"On the size of chromatic Delaunay mosaics","related_material":{"record":[{"id":"15094","status":"public","relation":"dissertation_contains"}]},"article_processing_charge":"No","type":"preprint","date_updated":"2024-03-20T09:36:56Z","article_number":"2212.03121","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)","image":"/images/cc_by.png"},"user_id":"8b945eb4-e2f2-11eb-945a-df72226e66a9","project":[{"name":"Alpha Shape Theory Extended","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","grant_number":"788183","call_identifier":"H2020"},{"name":"Discretization in Geometry and Dynamics","_id":"0aa4bc98-070f-11eb-9043-e6fff9c6a316","grant_number":"I4887"},{"_id":"268116B8-B435-11E9-9278-68D0E5697425","name":"The Wittgenstein Prize","grant_number":"Z00342","call_identifier":"FWF"}],"oa_version":"Preprint","department":[{"_id":"HeEd"}]}