{"publication":"Advances in Mathematics","year":"2025","language":[{"iso":"eng"}],"acknowledgement":"Work by the first and third authors is partially supported by the European Research Council (ERC), grant no. 788183, by the Wittgenstein Prize, Austrian Science Fund (FWF), grant no. Z 342-N31, and by the DFG Collaborative Research Center TRR 109, Austrian Science Fund (FWF), grant no. I 02979-N35. Work by the second author is partially supported by the Alexander von Humboldt Foundation.","day":"01","title":"Order-2 Delaunay triangulations optimize angles","date_updated":"2025-02-27T12:37:48Z","department":[{"_id":"HeEd"}],"month":"02","doi":"10.1016/j.aim.2024.110055","article_number":"110055","publication_status":"published","volume":461,"publication_identifier":{"issn":["0001-8708"],"eissn":["1090-2082"]},"publisher":"Elsevier","oa":1,"author":[{"last_name":"Edelsbrunner","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-9823-6833","full_name":"Edelsbrunner, Herbert","first_name":"Herbert"},{"full_name":"Garber, Alexey","first_name":"Alexey","last_name":"Garber"},{"last_name":"Saghafian","id":"f86f7148-b140-11ec-9577-95435b8df824","full_name":"Saghafian, Morteza","first_name":"Morteza"}],"external_id":{"isi":["001370682500001"],"arxiv":["2310.18238"]},"abstract":[{"text":"The local angle property of the (order-1) Delaunay triangulations of a generic set in R2\r\n asserts that the sum of two angles opposite a common edge is less than π. This paper extends this property to higher order and uses it to generalize two classic properties from order-1 to order-2: (1) among the complete level-2 hypertriangulations of a generic point set in R2, the order-2 Delaunay triangulation lexicographically maximizes the sorted angle vector; (2) among the maximal level-2 hypertriangulations of a generic point set in R2, the order-2 Delaunay triangulation is the only one that has the local angle property. We also use our method of establishing (2) to give a new short proof of the angle vector optimality for the (order-1) Delaunay triangulation. For order-1, both properties have been instrumental in numerous applications of Delaunay triangulations, and we expect that their generalization will make order-2 Delaunay triangulations more attractive to applications as well.","lang":"eng"}],"article_processing_charge":"No","scopus_import":"1","OA_type":"green","corr_author":"1","intvolume":"       461","_id":"18626","oa_version":"Preprint","article_type":"original","date_created":"2024-12-08T23:01:54Z","ec_funded":1,"citation":{"chicago":"Edelsbrunner, Herbert, Alexey Garber, and Morteza Saghafian. “Order-2 Delaunay Triangulations Optimize Angles.” <i>Advances in Mathematics</i>. Elsevier, 2025. <a href=\"https://doi.org/10.1016/j.aim.2024.110055\">https://doi.org/10.1016/j.aim.2024.110055</a>.","ama":"Edelsbrunner H, Garber A, Saghafian M. Order-2 Delaunay triangulations optimize angles. <i>Advances in Mathematics</i>. 2025;461. doi:<a href=\"https://doi.org/10.1016/j.aim.2024.110055\">10.1016/j.aim.2024.110055</a>","mla":"Edelsbrunner, Herbert, et al. “Order-2 Delaunay Triangulations Optimize Angles.” <i>Advances in Mathematics</i>, vol. 461, 110055, Elsevier, 2025, doi:<a href=\"https://doi.org/10.1016/j.aim.2024.110055\">10.1016/j.aim.2024.110055</a>.","ieee":"H. Edelsbrunner, A. Garber, and M. Saghafian, “Order-2 Delaunay triangulations optimize angles,” <i>Advances in Mathematics</i>, vol. 461. Elsevier, 2025.","ista":"Edelsbrunner H, Garber A, Saghafian M. 2025. Order-2 Delaunay triangulations optimize angles. Advances in Mathematics. 461, 110055.","short":"H. Edelsbrunner, A. Garber, M. Saghafian, Advances in Mathematics 461 (2025).","apa":"Edelsbrunner, H., Garber, A., &#38; Saghafian, M. (2025). Order-2 Delaunay triangulations optimize angles. <i>Advances in Mathematics</i>. Elsevier. <a href=\"https://doi.org/10.1016/j.aim.2024.110055\">https://doi.org/10.1016/j.aim.2024.110055</a>"},"main_file_link":[{"url":"https://doi.org/10.48550/arXiv.2310.18238","open_access":"1"}],"type":"journal_article","isi":1,"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","quality_controlled":"1","status":"public","project":[{"name":"Alpha Shape Theory Extended","grant_number":"788183","call_identifier":"H2020","_id":"266A2E9E-B435-11E9-9278-68D0E5697425"},{"_id":"268116B8-B435-11E9-9278-68D0E5697425","call_identifier":"FWF","name":"Wittgenstein Award - Herbert Edelsbrunner","grant_number":"Z00342"},{"name":"Persistence and stability of geometric complexes","grant_number":"I02979-N35","call_identifier":"FWF","_id":"2561EBF4-B435-11E9-9278-68D0E5697425"}],"date_published":"2025-02-01T00:00:00Z","OA_place":"repository","arxiv":1}