{"tmp":{"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","short":"CC BY (4.0)"},"date_published":"2025-06-20T00:00:00Z","publication_identifier":{"eissn":["1868-8969"],"isbn":["9783959773706"]},"corr_author":"1","OA_type":"gold","ddc":["510"],"file":[{"file_name":"2025_LIPIcs.SoCG_Streltsova.pdf","creator":"dernst","content_type":"application/pdf","checksum":"a8f7feb1aa3b896e31195841a989d622","file_id":"20015","success":1,"date_created":"2025-07-14T07:11:04Z","relation":"main_file","access_level":"open_access","date_updated":"2025-07-14T07:11:04Z","file_size":952807}],"article_number":"75","oa_version":"Published Version","OA_place":"publisher","scopus_import":"1","volume":332,"oa":1,"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","abstract":[{"text":"A long-standing conjecture of Eckhoff, Linhart, and Welzl, which would generalize McMullen’s Upper Bound Theorem for polytopes and refine asymptotic bounds due to Clarkson, asserts that for k ⩽ ⌊(n-d-2)/2⌋, the complexity of the (⩽ k)-level in a simple arrangement of n hemispheres in S^d is maximized for arrangements that are polar duals of neighborly d-polytopes. We prove this conjecture in the case n = d+4. By Gale duality, this implies the following result about crossing numbers: In every spherical arc drawing of K_n in S² (given by a set V ⊂ S² of n unit vectors connected by spherical arcs), the number of crossings is at least 1/4 ⌊n/2⌋ ⌊(n-1)/2⌋ ⌊(n-2)/2⌋ ⌊(n-3)/2⌋. This lower bound is attained if every open linear halfspace contains at least ⌊(n-2)/2⌋ of the vectors in V.\r\nMoreover, we determine the space of all linear and affine relations that hold between the face numbers of levels in simple arrangements of n hemispheres in S^d. This completes a long line of research on such relations, answers a question posed by Andrzejak and Welzl in 2003, and generalizes the classical fact that the Dehn-Sommerville relations generate all linear relations between the face numbers of simple polytopes (which correspond to the 0-level).\r\nTo prove these results, we introduce the notion of the g-matrix, which encodes the face numbers of levels in an arrangement and generalizes the classical g-vector of a polytope.","lang":"eng"}],"citation":{"mla":"Streltsova, Elizaveta, and Uli Wagner. “Levels in Arrangements: Linear Relations, the g-Matrix, and Applications to Crossing Numbers.” <i> 41st International Symposium on Computational Geometry</i>, vol. 332, 75, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2025, doi:<a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2025.75\">10.4230/LIPIcs.SoCG.2025.75</a>.","chicago":"Streltsova, Elizaveta, and Uli Wagner. “Levels in Arrangements: Linear Relations, the g-Matrix, and Applications to Crossing Numbers.” 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.2025.75\">https://doi.org/10.4230/LIPIcs.SoCG.2025.75</a>.","ama":"Streltsova E, Wagner U. Levels in arrangements: Linear relations, the g-matrix, and applications to crossing numbers. 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.2025.75\">10.4230/LIPIcs.SoCG.2025.75</a>","ista":"Streltsova E, Wagner U. 2025. Levels in arrangements: Linear relations, the g-matrix, and applications to crossing numbers.  41st International Symposium on Computational Geometry. SoCG: Symposium on Computational Geometry, LIPIcs, vol. 332, 75.","short":"E. Streltsova, U. Wagner, in:,  41st International Symposium on Computational Geometry, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2025.","apa":"Streltsova, E., &#38; Wagner, U. (2025). Levels in arrangements: Linear relations, the g-matrix, and applications to crossing numbers. 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.2025.75\">https://doi.org/10.4230/LIPIcs.SoCG.2025.75</a>","ieee":"E. Streltsova and U. Wagner, “Levels in arrangements: Linear relations, the g-matrix, and applications to crossing numbers,” in <i> 41st International Symposium on Computational Geometry</i>, Kanazawa, Japan, 2025, vol. 332."},"title":"Levels in arrangements: Linear relations, the g-matrix, and applications to crossing numbers","department":[{"_id":"UlWa"}],"file_date_updated":"2025-07-14T07:11:04Z","date_updated":"2025-07-14T07:19:19Z","publication":" 41st International Symposium on Computational Geometry","arxiv":1,"quality_controlled":"1","_id":"20004","day":"20","doi":"10.4230/LIPIcs.SoCG.2025.75","publication_status":"published","author":[{"id":"57a170da-dc96-11ea-b7c8-ab3565071bf7","first_name":"Elizaveta","full_name":"Streltsova, Elizaveta","last_name":"Streltsova"},{"orcid":"0000-0002-1494-0568","id":"36690CA2-F248-11E8-B48F-1D18A9856A87","first_name":"Uli","full_name":"Wagner, Uli","last_name":"Wagner"}],"article_processing_charge":"Yes","date_created":"2025-07-13T22:01:22Z","year":"2025","alternative_title":["LIPIcs"],"status":"public","month":"06","language":[{"iso":"eng"}],"has_accepted_license":"1","type":"conference","publisher":"Schloss Dagstuhl - Leibniz-Zentrum für Informatik","intvolume":"       332","conference":{"name":"SoCG: Symposium on Computational Geometry","start_date":"2025-06-23","location":"Kanazawa, Japan","end_date":"2025-06-27"},"external_id":{"arxiv":["2504.07752","2504.07770"]}}