{"oa":1,"oa_version":"Preprint","user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","date_created":"2022-02-20T23:01:35Z","publisher":"Springer Nature","scopus_import":"1","quality_controlled":"1","title":"Barycentric cuts through a convex body","status":"public","type":"journal_article","day":"01","article_processing_charge":"No","month":"12","date_updated":"2023-08-02T14:38:58Z","publication_status":"published","date_published":"2022-12-01T00:00:00Z","department":[{"_id":"UlWa"}],"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"}],"main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/2003.13536"}],"intvolume":"        68","volume":68,"year":"2022","isi":1,"acknowledgement":"The work by Zuzana Patáková has been partially supported by Charles University Research Center Program No. UNCE/SCI/022, and part of it was done during her research stay at IST Austria. The work by Martin Tancer is supported by the GAČR Grant 19-04113Y and by the Charles University Projects PRIMUS/17/SCI/3 and UNCE/SCI/004.","page":"1133-1154","doi":"10.1007/s00454-021-00364-7","publication":"Discrete and Computational Geometry","author":[{"id":"48B57058-F248-11E8-B48F-1D18A9856A87","full_name":"Patakova, Zuzana","first_name":"Zuzana","last_name":"Patakova","orcid":"0000-0002-3975-1683"},{"full_name":"Tancer, Martin","last_name":"Tancer","first_name":"Martin"},{"orcid":"0000-0002-1494-0568","last_name":"Wagner","first_name":"Uli","id":"36690CA2-F248-11E8-B48F-1D18A9856A87","full_name":"Wagner, Uli"}],"external_id":{"isi":["000750681500001"],"arxiv":["2003.13536"]},"publication_identifier":{"issn":["0179-5376"],"eissn":["1432-0444"]},"citation":{"ama":"Patakova Z, Tancer M, Wagner U. Barycentric cuts through a convex body. <i>Discrete and Computational Geometry</i>. 2022;68:1133-1154. doi:<a href=\"https://doi.org/10.1007/s00454-021-00364-7\">10.1007/s00454-021-00364-7</a>","chicago":"Patakova, Zuzana, Martin Tancer, and Uli Wagner. “Barycentric Cuts through a Convex Body.” <i>Discrete and Computational Geometry</i>. Springer Nature, 2022. <a href=\"https://doi.org/10.1007/s00454-021-00364-7\">https://doi.org/10.1007/s00454-021-00364-7</a>.","ista":"Patakova Z, Tancer M, Wagner U. 2022. Barycentric cuts through a convex body. Discrete and Computational Geometry. 68, 1133–1154.","apa":"Patakova, Z., Tancer, M., &#38; Wagner, U. (2022). Barycentric cuts through a convex body. <i>Discrete and Computational Geometry</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s00454-021-00364-7\">https://doi.org/10.1007/s00454-021-00364-7</a>","short":"Z. Patakova, M. Tancer, U. Wagner, Discrete and Computational Geometry 68 (2022) 1133–1154.","mla":"Patakova, Zuzana, et al. “Barycentric Cuts through a Convex Body.” <i>Discrete and Computational Geometry</i>, vol. 68, Springer Nature, 2022, pp. 1133–54, doi:<a href=\"https://doi.org/10.1007/s00454-021-00364-7\">10.1007/s00454-021-00364-7</a>.","ieee":"Z. Patakova, M. Tancer, and U. Wagner, “Barycentric cuts through a convex body,” <i>Discrete and Computational Geometry</i>, vol. 68. Springer Nature, pp. 1133–1154, 2022."},"article_type":"original","language":[{"iso":"eng"}],"_id":"10776"}