{"quality_controlled":"1","day":"01","status":"public","abstract":[{"lang":"eng","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."}],"intvolume":"       164","date_updated":"2021-01-12T08:16:23Z","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png"},"volume":164,"external_id":{"arxiv":["2003.13536"]},"has_accepted_license":"1","article_processing_charge":"No","date_published":"2020-06-01T00:00:00Z","ddc":["510"],"file":[{"date_updated":"2020-07-14T12:48:06Z","content_type":"application/pdf","date_created":"2020-06-23T06:45:52Z","file_name":"2020_LIPIcsSoCG_Patakova.pdf","file_size":750318,"relation":"main_file","access_level":"open_access","file_id":"8004","checksum":"ce1c9194139a664fb59d1efdfc88eaae","creator":"dernst"}],"year":"2020","date_created":"2020-06-22T09:14:20Z","alternative_title":["LIPIcs"],"publication":"36th International Symposium on Computational Geometry","article_number":"62:1 - 62:16","department":[{"_id":"UlWa"}],"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","author":[{"id":"48B57058-F248-11E8-B48F-1D18A9856A87","last_name":"Patakova","full_name":"Patakova, Zuzana","first_name":"Zuzana","orcid":"0000-0002-3975-1683"},{"full_name":"Tancer, Martin","orcid":"0000-0002-1191-6714","first_name":"Martin","last_name":"Tancer","id":"38AC689C-F248-11E8-B48F-1D18A9856A87"},{"full_name":"Wagner, Uli","orcid":"0000-0002-1494-0568","first_name":"Uli","last_name":"Wagner","id":"36690CA2-F248-11E8-B48F-1D18A9856A87"}],"file_date_updated":"2020-07-14T12:48:06Z","language":[{"iso":"eng"}],"conference":{"start_date":"2020-06-22","location":"Zürich, Switzerland","end_date":"2020-06-26","name":"SoCG: Symposium on Computational Geometry"},"publication_status":"published","publisher":"Schloss Dagstuhl - Leibniz-Zentrum für Informatik","scopus_import":1,"oa":1,"type":"conference","citation":{"ama":"Patakova Z, Tancer M, Wagner U. Barycentric cuts through a convex body. In: <i>36th International Symposium on Computational Geometry</i>. Vol 164. Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2020. doi:<a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2020.62\">10.4230/LIPIcs.SoCG.2020.62</a>","ista":"Patakova Z, Tancer M, Wagner U. 2020. Barycentric cuts through a convex body. 36th International Symposium on Computational Geometry. SoCG: Symposium on Computational Geometry, LIPIcs, vol. 164, 62:1-62:16.","chicago":"Patakova, Zuzana, Martin Tancer, and Uli Wagner. “Barycentric Cuts through a Convex Body.” In <i>36th International Symposium on Computational Geometry</i>, Vol. 164. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. <a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2020.62\">https://doi.org/10.4230/LIPIcs.SoCG.2020.62</a>.","mla":"Patakova, Zuzana, et al. “Barycentric Cuts through a Convex Body.” <i>36th International Symposium on Computational Geometry</i>, vol. 164, 62:1-62:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020, doi:<a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2020.62\">10.4230/LIPIcs.SoCG.2020.62</a>.","ieee":"Z. Patakova, M. Tancer, and U. Wagner, “Barycentric cuts through a convex body,” in <i>36th International Symposium on Computational Geometry</i>, Zürich, Switzerland, 2020, vol. 164.","short":"Z. Patakova, M. Tancer, U. Wagner, in:, 36th International Symposium on Computational Geometry, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020.","apa":"Patakova, Z., Tancer, M., &#38; Wagner, U. (2020). Barycentric cuts through a convex body. In <i>36th International Symposium on Computational Geometry</i> (Vol. 164). Zürich, Switzerland: Schloss Dagstuhl - Leibniz-Zentrum für Informatik. <a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2020.62\">https://doi.org/10.4230/LIPIcs.SoCG.2020.62</a>"},"oa_version":"Published Version","month":"06","title":"Barycentric cuts through a convex body","_id":"7992","doi":"10.4230/LIPIcs.SoCG.2020.62","publication_identifier":{"issn":["18688969"],"isbn":["9783959771436"]}}