{"ddc":["510"],"oa_version":"Published Version","month":"06","doi":"10.4230/LIPIcs.SoCG.2020.62","title":"Barycentric cuts through a convex body","day":"01","volume":164,"external_id":{"arxiv":["2003.13536"]},"status":"public","language":[{"iso":"eng"}],"article_processing_charge":"No","author":[{"first_name":"Zuzana","last_name":"Patakova","id":"48B57058-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-3975-1683","full_name":"Patakova, Zuzana"},{"id":"38AC689C-F248-11E8-B48F-1D18A9856A87","last_name":"Tancer","first_name":"Martin","orcid":"0000-0002-1191-6714","full_name":"Tancer, Martin"},{"id":"36690CA2-F248-11E8-B48F-1D18A9856A87","first_name":"Uli","last_name":"Wagner","orcid":"0000-0002-1494-0568","full_name":"Wagner, Uli"}],"alternative_title":["LIPIcs"],"type":"conference","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","date_created":"2020-06-22T09:14:20Z","has_accepted_license":"1","date_published":"2020-06-01T00:00:00Z","tmp":{"image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)"},"article_number":"62:1 - 62:16","corr_author":"1","quality_controlled":"1","abstract":[{"text":"Let K be a convex body in ℝⁿ (i.e., a compact convex set with nonempty interior). Given a point p in the interior of K, a hyperplane h passing through p is called barycentric if p is the barycenter of K ∩ h. In 1961, Grünbaum raised the question whether, for every K, there exists an interior point p through which there are at least n+1 distinct barycentric hyperplanes. Two years later, this was seemingly resolved affirmatively by showing that this is the case if p=p₀ is the point of maximal depth in K. However, while working on a related question, we noticed that one of the auxiliary claims in the proof is incorrect. Here, we provide a counterexample; this re-opens Grünbaum’s question. It follows from known results that for n ≥ 2, there are always at least three distinct barycentric cuts through the point p₀ ∈ K of maximal depth. Using tools related to Morse theory we are able to improve this bound: four distinct barycentric cuts through p₀ are guaranteed if n ≥ 3.","lang":"eng"}],"license":"https://creativecommons.org/licenses/by/4.0/","year":"2020","scopus_import":"1","publication":"36th International Symposium on Computational Geometry","conference":{"location":"Zürich, Switzerland","start_date":"2020-06-22","end_date":"2020-06-26","name":"SoCG: Symposium on Computational Geometry"},"department":[{"_id":"UlWa"}],"publication_identifier":{"isbn":["9783959771436"],"issn":["1868-8969"]},"file_date_updated":"2020-07-14T12:48:06Z","publication_status":"published","date_updated":"2025-04-10T11:12:38Z","oa":1,"file":[{"file_size":750318,"file_id":"8004","relation":"main_file","checksum":"ce1c9194139a664fb59d1efdfc88eaae","access_level":"open_access","date_updated":"2020-07-14T12:48:06Z","creator":"dernst","date_created":"2020-06-23T06:45:52Z","content_type":"application/pdf","file_name":"2020_LIPIcsSoCG_Patakova.pdf"}],"intvolume":"       164","arxiv":1,"_id":"7992","publisher":"Schloss Dagstuhl - Leibniz-Zentrum für Informatik"}