Barycentric cuts through a convex body LIPIcs Patakova, Zuzana ; https://orcid.org/0000-0002-3975-1683 Tancer, Martin ; https://orcid.org/0000-0002-1191-6714 Wagner, Uli ; https://orcid.org/0000-0002-1494-0568 ddc:510 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. Schloss Dagstuhl - Leibniz-Zentrum für Informatik 2020 info:eu-repo/semantics/conferenceObject doc-type:conferenceObject text http://purl.org/coar/resource_type/c_5794 https://research-explorer.ista.ac.at/record/7992 https://research-explorer.ista.ac.at/download/7992/8004 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> eng info:eu-repo/semantics/altIdentifier/doi/10.4230/LIPIcs.SoCG.2020.62 info:eu-repo/semantics/altIdentifier/issn/1868-8969 info:eu-repo/semantics/altIdentifier/isbn/9783959771436 info:eu-repo/semantics/altIdentifier/arxiv/2003.13536 https://creativecommons.org/licenses/by/4.0/ info:eu-repo/semantics/openAccess