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Barycentric Cuts Through a Convex Body Patáková, Zuzana Tancer, Martin Wagner, Uli convex body barycenter Tukey depth smooth manifold critical points 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 Zuzana Patáková and Martin Tancer and Uli Wagner 2020 Is Part Of LIPIcs, Volume 164, 36th International Symposium on Computational Geometry (SoCG 2020) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.SoCG.2020.62 urn:nbn:de:0030-drops-122201 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2020.62 eng https://creativecommons.org/licenses/by/3.0/legalcode