{"intvolume":"       293","abstract":[{"text":"The Upper Bound Theorem for convex polytopes implies that the p-th Betti number of the Čech complex of any set of N points in ℝ^d and any radius satisfies β_p = O(N^m), with m = min{p+1, ⌈d/2⌉}. We construct sets in even and odd dimensions, which prove that this upper bound is asymptotically tight. For example, we describe a set of N = 2(n+1) points in ℝ³ and two radii such that the first Betti number of the Čech complex at one radius is (n+1)² - 1, and the second Betti number of the Čech complex at the other radius is n². In particular, there is an arrangement of n contruent balls in ℝ³ that enclose a quadratic number of voids, which answers a long-standing open question in computational geometry.","lang":"eng"}],"department":[{"_id":"HeEd"}],"year":"2024","publication_identifier":{"issn":["1868-8969"],"isbn":["9783959773164"]},"date_published":"2024-06-01T00:00:00Z","day":"01","alternative_title":["LIPIcs"],"acknowledgement":"The first author is supported by the European Research Council (ERC), grant no. 788183, and by the DFG Collaborative Research Center TRR 109, Austrian Science Fund (FWF), grant no. {I 02979-N35.} The second author is supported by the European Research Council (ERC), grant \"GeoScape\" and by the Hungarian Science Foundation (NKFIH), grant K-131529. Both authors are supported by the Wittgenstein Prize, Austrian Science Fund (FWF), grant no. Z 342-N31.\r\nThe authors thank Matt Kahle for communicating the question about extremal Čech complexes, Ben Schweinhart for early discussions on the linked circles construction in three dimensions, and Gábor Tardos for helpful remarks and suggestions.","author":[{"orcid":"0000-0002-9823-6833","full_name":"Edelsbrunner, Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","first_name":"Herbert","last_name":"Edelsbrunner"},{"id":"E62E3130-B088-11EA-B919-BF823C25FEA4","full_name":"Pach, János","last_name":"Pach","first_name":"János"}],"tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)","image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)"},"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","file_date_updated":"2024-06-17T08:46:33Z","has_accepted_license":"1","doi":"10.4230/LIPIcs.SoCG.2024.53","date_created":"2024-06-16T22:01:06Z","article_number":"53","article_processing_charge":"No","publisher":"Schloss Dagstuhl - Leibniz-Zentrum für Informatik","_id":"17146","ec_funded":1,"status":"public","citation":{"short":"H. Edelsbrunner, J. Pach, in:, 40th International Symposium on Computational Geometry, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024.","apa":"Edelsbrunner, H., &#38; Pach, J. (2024). Maximum Betti numbers of Čech complexes. In <i>40th International Symposium on Computational Geometry</i> (Vol. 293). Athens, Greece: Schloss Dagstuhl - Leibniz-Zentrum für Informatik. <a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2024.53\">https://doi.org/10.4230/LIPIcs.SoCG.2024.53</a>","ista":"Edelsbrunner H, Pach J. 2024. Maximum Betti numbers of Čech complexes. 40th International Symposium on Computational Geometry. SoCG: Symposium on Computational Geometry, LIPIcs, vol. 293, 53.","ieee":"H. Edelsbrunner and J. Pach, “Maximum Betti numbers of Čech complexes,” in <i>40th International Symposium on Computational Geometry</i>, Athens, Greece, 2024, vol. 293.","mla":"Edelsbrunner, Herbert, and János Pach. “Maximum Betti Numbers of Čech Complexes.” <i>40th International Symposium on Computational Geometry</i>, vol. 293, 53, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024, doi:<a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2024.53\">10.4230/LIPIcs.SoCG.2024.53</a>.","ama":"Edelsbrunner H, Pach J. Maximum Betti numbers of Čech complexes. In: <i>40th International Symposium on Computational Geometry</i>. Vol 293. Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2024. doi:<a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2024.53\">10.4230/LIPIcs.SoCG.2024.53</a>","chicago":"Edelsbrunner, Herbert, and János Pach. “Maximum Betti Numbers of Čech Complexes.” In <i>40th International Symposium on Computational Geometry</i>, Vol. 293. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024. <a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2024.53\">https://doi.org/10.4230/LIPIcs.SoCG.2024.53</a>."},"project":[{"grant_number":"788183","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","name":"Alpha Shape Theory Extended","call_identifier":"H2020"},{"grant_number":"I02979-N35","call_identifier":"FWF","name":"Persistence and stability of geometric complexes","_id":"2561EBF4-B435-11E9-9278-68D0E5697425"},{"_id":"268116B8-B435-11E9-9278-68D0E5697425","name":"The Wittgenstein Prize","call_identifier":"FWF","grant_number":"Z00342"}],"month":"06","publication_status":"published","type":"conference","language":[{"iso":"eng"}],"ddc":["510"],"title":"Maximum Betti numbers of Čech complexes","external_id":{"arxiv":["2310.14801"]},"file":[{"relation":"main_file","file_size":766562,"access_level":"open_access","checksum":"5442d44fb89d77477a87668d6e61aac9","content_type":"application/pdf","date_created":"2024-06-17T08:46:33Z","creator":"dernst","file_id":"17152","date_updated":"2024-06-17T08:46:33Z","success":1,"file_name":"2024_LIPICS_Edelsbrunner.pdf"}],"scopus_import":"1","oa":1,"quality_controlled":"1","conference":{"start_date":"2024-06-11","name":"SoCG: Symposium on Computational Geometry","location":"Athens, Greece","end_date":"2024-06-14"},"date_updated":"2024-06-17T08:47:30Z","publication":"40th International Symposium on Computational Geometry","volume":293,"oa_version":"Published Version"}