[{"department":[{"_id":"HeEd"}],"corr_author":"1","year":"2025","language":[{"iso":"eng"}],"author":[{"first_name":"Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","last_name":"Edelsbrunner","full_name":"Edelsbrunner, Herbert","orcid":"0000-0002-9823-6833"},{"full_name":"Garber, Alexey","first_name":"Alexey","last_name":"Garber"},{"first_name":"Morteza","last_name":"Saghafian","id":"f86f7148-b140-11ec-9577-95435b8df824","full_name":"Saghafian, Morteza"}],"ddc":["510"],"status":"public","conference":{"end_date":"2025-06-27","start_date":"2025-06-23","location":"Kanazawa, Japan","name":"SoCG: Symposium on Computational Geometry"},"alternative_title":["LIPIcs"],"file_date_updated":"2025-07-14T07:24:22Z","oa":1,"publisher":"Schloss Dagstuhl - Leibniz-Zentrum für Informatik","date_created":"2025-07-13T22:01:22Z","has_accepted_license":"1","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)","image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode"},"project":[{"name":"Mathematics, Computer Science","call_identifier":"FWF","grant_number":"Z00342","_id":"268116B8-B435-11E9-9278-68D0E5697425"},{"call_identifier":"FWF","name":"Persistence and stability of geometric complexes","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","grant_number":"I02979-N35"}],"OA_place":"publisher","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","volume":332,"publication_identifier":{"eissn":["1868-8969"],"isbn":["9783959773706"]},"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","arxiv":1,"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."}],"oa_version":"Published Version","title":"On spheres with k points inside","type":"conference","scopus_import":"1","publication":"41st International Symposium on Computational Geometry","day":"20","external_id":{"arxiv":["2410.21204"]},"file":[{"relation":"main_file","checksum":"b5313ed8575ea87913c71a6e3c7513c8","file_id":"20016","success":1,"creator":"dernst","file_name":"2025_LIPIcs.SoCG_Edelsbrunner.pdf","date_created":"2025-07-14T07:24:22Z","access_level":"open_access","file_size":661893,"content_type":"application/pdf","date_updated":"2025-07-14T07:24:22Z"}],"citation":{"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.","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>.","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>.","short":"H. Edelsbrunner, A. Garber, M. Saghafian, in:, 41st International Symposium on Computational Geometry, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2025.","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>","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."},"article_processing_charge":"Yes","OA_type":"gold","article_number":"43","doi":"10.4230/LIPIcs.SoCG.2025.43","date_updated":"2025-07-14T07:26:14Z","publication_status":"published","_id":"20005","quality_controlled":"1","month":"06","intvolume":"       332","date_published":"2025-06-20T00:00:00Z"}]