[{"date_created":"2025-07-13T22:01:22Z","OA_place":"publisher","acknowledgement":"Herbert Edelsbrunner: partially supported by the Wittgenstein Prize, Austrian Science\r\nFund (FWF), grant no. Z 342-N31, and by the DFG Collaborative Research Center TRR 109,\r\nAustrian Science Fund (FWF), grant no. I 02979-N35.\r\nAlexey Garber: partially supported by the Simons Foundation.\r\nMorteza Saghafian: partially supported by the Wittgenstein Prize, Austrian Science Fund (FWF),\r\ngrant no. Z 342-N31, and by the DFG Collaborative Research Center TRR 109, Austrian Science\r\nFund (FWF), grant no. I 02979-N35","file":[{"date_created":"2025-07-14T07:24:22Z","relation":"main_file","file_name":"2025_LIPIcs.SoCG_Edelsbrunner.pdf","creator":"dernst","content_type":"application/pdf","access_level":"open_access","file_id":"20016","file_size":661893,"checksum":"b5313ed8575ea87913c71a6e3c7513c8","success":1,"date_updated":"2025-07-14T07:24:22Z"}],"fulldoi":"https://doi.org/10.4230/LIPIcs.SoCG.2025.43","abstract":[{"lang":"eng","text":"We generalize a classical result by Boris Delaunay that introduced Delaunay triangulations. In particular, we prove that for a locally finite and coarsely dense generic point set A in ℝ^d, every generic point of ℝ^d belongs to exactly binom(d+k,d) simplices whose vertices belong to A and whose circumspheres enclose exactly k points of A. We extend this result to the cases in which the points are weighted, and when A contains only finitely many points in ℝ^d or in 𝕊^d. Furthermore, we use the result to give a new geometric proof for the fact that volumes of hypersimplices are Eulerian numbers."}],"publication_identifier":{"isbn":["9783959773706"],"eissn":["1868-8969"]},"article_number":"43","ddc":["510"],"OA_type":"gold","author":[{"orcid":"0000-0002-9823-6833","first_name":"Herbert","full_name":"Edelsbrunner, Herbert","last_name":"Edelsbrunner","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87"},{"last_name":"Garber","full_name":"Garber, Alexey","first_name":"Alexey"},{"first_name":"Morteza","full_name":"Saghafian, Morteza","last_name":"Saghafian","id":"f86f7148-b140-11ec-9577-95435b8df824"}],"file_date_updated":"2025-07-14T07:24:22Z","oa":1,"title":"On spheres with k points inside","status":"public","volume":332,"arxiv":1,"month":"06","project":[{"name":"Mathematics, Computer Science","_id":"268116B8-B435-11E9-9278-68D0E5697425","grant_number":"Z00342","call_identifier":"FWF"},{"name":"Persistence and stability of geometric complexes","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","grant_number":"I02979-N35","call_identifier":"FWF"}],"_id":"20005","date_published":"2025-06-20T00:00:00Z","scopus_import":"1","date_updated":"2025-07-14T07:26:14Z","corr_author":"1","quality_controlled":"1","intvolume":"       332","day":"20","oa_version":"Published Version","external_id":{"arxiv":["2410.21204"]},"conference":{"name":"SoCG: Symposium on Computational Geometry","end_date":"2025-06-27","start_date":"2025-06-23","location":"Kanazawa, Japan"},"citation":{"chicago":"Edelsbrunner, Herbert, Alexey Garber, and Morteza Saghafian. “On Spheres with k Points Inside.” In <i>41st International Symposium on Computational Geometry</i>, Vol. 332. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2025. <a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2025.43\">https://doi.org/10.4230/LIPIcs.SoCG.2025.43</a>.","apa":"Edelsbrunner, H., Garber, A., &#38; Saghafian, M. (2025). On spheres with k points inside. In <i>41st International Symposium on Computational Geometry</i> (Vol. 332). Kanazawa, Japan: Schloss Dagstuhl - Leibniz-Zentrum für Informatik. <a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2025.43\">https://doi.org/10.4230/LIPIcs.SoCG.2025.43</a>","short":"H. Edelsbrunner, A. Garber, M. Saghafian, in:, 41st International Symposium on Computational Geometry, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2025.","ieee":"H. Edelsbrunner, A. Garber, and M. Saghafian, “On spheres with k points inside,” in <i>41st International Symposium on Computational Geometry</i>, Kanazawa, Japan, 2025, vol. 332.","ama":"Edelsbrunner H, Garber A, Saghafian M. On spheres with k points inside. In: <i>41st International Symposium on Computational Geometry</i>. Vol 332. Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2025. doi:<a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2025.43\">10.4230/LIPIcs.SoCG.2025.43</a>","mla":"Edelsbrunner, Herbert, et al. “On Spheres with k Points Inside.” <i>41st International Symposium on Computational Geometry</i>, vol. 332, 43, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2025, doi:<a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2025.43\">10.4230/LIPIcs.SoCG.2025.43</a>.","ista":"Edelsbrunner H, Garber A, Saghafian M. 2025. On spheres with k points inside. 41st International Symposium on Computational Geometry. SoCG: Symposium on Computational Geometry, LIPIcs, vol. 332, 43."},"tmp":{"short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png"},"article_processing_charge":"Yes","publication":"41st International Symposium on Computational Geometry","has_accepted_license":"1","year":"2025","department":[{"_id":"HeEd"}],"type":"conference","doi":"10.4230/LIPIcs.SoCG.2025.43","publisher":"Schloss Dagstuhl - Leibniz-Zentrum für Informatik","publication_status":"published","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","language":[{"iso":"eng"}],"alternative_title":["LIPIcs"]}]