{"day":"11","quality_controlled":"1","intvolume":"        99","date_updated":"2023-09-07T13:29:00Z","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png"},"abstract":[{"lang":"eng","text":"Given a locally finite X ⊆ ℝd and a radius r ≥ 0, the k-fold cover of X and r consists of all points in ℝd that have k or more points of X within distance r. We consider two filtrations - one in scale obtained by fixing k and increasing r, and the other in depth obtained by fixing r and decreasing k - and we compute the persistence diagrams of both. While standard methods suffice for the filtration in scale, we need novel geometric and topological concepts for the filtration in depth. In particular, we introduce a rhomboid tiling in ℝd+1 whose horizontal integer slices are the order-k Delaunay mosaics of X, and construct a zigzag module from Delaunay mosaics that is isomorphic to the persistence module of the multi-covers. "}],"status":"public","related_material":{"record":[{"relation":"later_version","id":"9317","status":"public"},{"id":"9056","status":"public","relation":"dissertation_contains"}]},"acknowledgement":"This work is partially supported by the DFG Collaborative Research Center TRR 109, ‘Discretization in Geometry and Dynamics’, through grant no. I02979-N35 of the Austrian Science Fund (FWF).","volume":99,"project":[{"_id":"2561EBF4-B435-11E9-9278-68D0E5697425","name":"Persistence and stability of geometric complexes","grant_number":"I02979-N35","call_identifier":"FWF"}],"has_accepted_license":"1","file":[{"date_created":"2018-12-18T09:27:22Z","file_name":"2018_LIPIcs_Edelsbrunner_Osang.pdf","file_size":528018,"date_updated":"2020-07-14T12:45:19Z","content_type":"application/pdf","relation":"main_file","access_level":"open_access","file_id":"5738","creator":"dernst","checksum":"d8c0533ad0018eb4ed1077475eb8fc18"}],"ddc":["516"],"year":"2018","date_published":"2018-06-11T00:00:00Z","alternative_title":["LIPIcs"],"article_number":"34","date_created":"2018-12-11T11:45:05Z","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","department":[{"_id":"HeEd"}],"file_date_updated":"2020-07-14T12:45:19Z","language":[{"iso":"eng"}],"author":[{"first_name":"Herbert","orcid":"0000-0002-9823-6833","full_name":"Edelsbrunner, Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","last_name":"Edelsbrunner"},{"orcid":"0000-0002-8882-5116","first_name":"Georg F","full_name":"Osang, Georg F","last_name":"Osang","id":"464B40D6-F248-11E8-B48F-1D18A9856A87"}],"publist_id":"7732","conference":{"end_date":"2018-06-14","start_date":"2018-06-11","location":"Budapest, Hungary","name":"SoCG: Symposium on Computational Geometry"},"publication_status":"published","publisher":"Schloss Dagstuhl - Leibniz-Zentrum für Informatik","oa":1,"scopus_import":1,"oa_version":"Published Version","type":"conference","citation":{"short":"H. Edelsbrunner, G.F. Osang, in:, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2018.","apa":"Edelsbrunner, H., &#38; Osang, G. F. (2018). The multi-cover persistence of Euclidean balls (Vol. 99). Presented at the SoCG: Symposium on Computational Geometry, Budapest, Hungary: Schloss Dagstuhl - Leibniz-Zentrum für Informatik. <a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2018.34\">https://doi.org/10.4230/LIPIcs.SoCG.2018.34</a>","ista":"Edelsbrunner H, Osang GF. 2018. The multi-cover persistence of Euclidean balls. SoCG: Symposium on Computational Geometry, LIPIcs, vol. 99, 34.","ama":"Edelsbrunner H, Osang GF. The multi-cover persistence of Euclidean balls. In: Vol 99. Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2018. doi:<a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2018.34\">10.4230/LIPIcs.SoCG.2018.34</a>","mla":"Edelsbrunner, Herbert, and Georg F. Osang. <i>The Multi-Cover Persistence of Euclidean Balls</i>. Vol. 99, 34, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2018, doi:<a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2018.34\">10.4230/LIPIcs.SoCG.2018.34</a>.","ieee":"H. Edelsbrunner and G. F. Osang, “The multi-cover persistence of Euclidean balls,” presented at the SoCG: Symposium on Computational Geometry, Budapest, Hungary, 2018, vol. 99.","chicago":"Edelsbrunner, Herbert, and Georg F Osang. “The Multi-Cover Persistence of Euclidean Balls,” Vol. 99. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2018. <a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2018.34\">https://doi.org/10.4230/LIPIcs.SoCG.2018.34</a>."},"_id":"187","doi":"10.4230/LIPIcs.SoCG.2018.34","month":"06","title":"The multi-cover persistence of Euclidean balls"}