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