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