[{"date_published":"2021-04-01T00:00:00Z","article_type":"original","citation":{"mla":"Edelsbrunner, Herbert, et al. “A Step in the Delaunay Mosaic of Order K.” Journal of Geometry, vol. 112, no. 1, 15, Springer Nature, 2021, doi:10.1007/s00022-021-00577-4.","short":"H. Edelsbrunner, A. Nikitenko, G.F. Osang, Journal of Geometry 112 (2021).","chicago":"Edelsbrunner, Herbert, Anton Nikitenko, and Georg F Osang. “A Step in the Delaunay Mosaic of Order K.” Journal of Geometry. Springer Nature, 2021. https://doi.org/10.1007/s00022-021-00577-4.","ista":"Edelsbrunner H, Nikitenko A, Osang GF. 2021. A step in the Delaunay mosaic of order k. Journal of Geometry. 112(1), 15.","apa":"Edelsbrunner, H., Nikitenko, A., & Osang, G. F. (2021). A step in the Delaunay mosaic of order k. Journal of Geometry. Springer Nature. https://doi.org/10.1007/s00022-021-00577-4","ieee":"H. Edelsbrunner, A. Nikitenko, and G. F. Osang, “A step in the Delaunay mosaic of order k,” Journal of Geometry, vol. 112, no. 1. Springer Nature, 2021."},"publication":"Journal of Geometry","article_processing_charge":"Yes (via OA deal)","uri_base":"https://research-explorer.ista.ac.at","has_accepted_license":"1","day":"01","scopus_import":"1","dc":{"description":["Given a locally finite set 𝑋⊆ℝ𝑑 and an integer 𝑘≥0, we consider the function 𝐰𝑘:Del𝑘(𝑋)→ℝ on the dual of the order-k Voronoi tessellation, whose sublevel sets generalize the notion of alpha shapes from order-1 to order-k (Edelsbrunner et al. in IEEE Trans Inf Theory IT-29:551–559, 1983; Krasnoshchekov and Polishchuk in Inf Process Lett 114:76–83, 2014). While this function is not necessarily generalized discrete Morse, in the sense of Forman (Adv Math 134:90–145, 1998) and Freij (Discrete Math 309:3821–3829, 2009), we prove that it satisfies similar properties so that its increments can be meaningfully classified into critical and non-critical steps. This result extends to the case of weighted points and sheds light on k-fold covers with balls in Euclidean space."],"identifier":["https://research-explorer.ista.ac.at/record/9465","https://research-explorer.ista.ac.at/download/9465/9544"],"type":["info:eu-repo/semantics/article","doc-type:article","text","http://purl.org/coar/resource_type/c_6501"],"creator":["Edelsbrunner, Herbert","Nikitenko, Anton","Osang, Georg F"],"relation":["info:eu-repo/semantics/altIdentifier/doi/10.1007/s00022-021-00577-4","info:eu-repo/semantics/altIdentifier/issn/00472468","info:eu-repo/semantics/altIdentifier/issn/14208997"],"subject":["ddc:510"],"publisher":["Springer Nature"],"title":["A step in the Delaunay mosaic of order k"],"rights":["https://creativecommons.org/licenses/by/4.0/","info:eu-repo/semantics/openAccess"],"source":["Edelsbrunner H, Nikitenko A, Osang GF. A step in the Delaunay mosaic of order k. Journal of Geometry. 2021;112(1). doi:10.1007/s00022-021-00577-4"],"language":["eng"],"date":["2021"]},"file":[{"checksum":"e52a832f1def52a2b23d21bcc09e646f","success":1,"date_created":"2021-06-11T13:16:26Z","date_updated":"2021-06-11T13:16:26Z","relation":"main_file","file_id":"9544","file_size":694706,"content_type":"application/pdf","creator":"kschuh","access_level":"open_access","file_name":"2021_Geometry_Edelsbrunner.pdf"}],"oa_version":"Published Version","intvolume":" 112","status":"public","ddc":[],"_id":"9465","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","issue":"1","abstract":[{"lang":"eng"}],"type":"journal_article","language":[{}],"quality_controlled":"1","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"},"oa":1,"publication_identifier":{"eissn":[],"issn":[]},"month":"04","volume":112,"date_created":"2021-06-06T22:01:29Z","dini_type":"doc-type:article","date_updated":"2022-05-12T11:41:45Z","author":[{"orcid":"0000-0002-9823-6833","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","last_name":"Edelsbrunner","first_name":"Herbert"},{"id":"3E4FF1BA-F248-11E8-B48F-1D18A9856A87","first_name":"Anton","last_name":"Nikitenko"},{"id":"464B40D6-F248-11E8-B48F-1D18A9856A87","last_name":"Osang","first_name":"Georg F"}],"department":[{"tree":[{"_id":"ResearchGroups"},{"_id":"IST"}],"_id":"HeEd"}],"publication_status":"published","creator":{"id":"3E5EF7F0-F248-11E8-B48F-1D18A9856A87","login":"kschuh"},"file_date_updated":"2021-06-11T13:16:26Z","article_number":"15"}]