@inproceedings{7992,
  abstract     = {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.},
  author       = {Patakova, Zuzana and Tancer, Martin and Wagner, Uli},
  booktitle    = {36th International Symposium on Computational Geometry},
  isbn         = {9783959771436},
  issn         = {1868-8969},
  location     = {Zürich, Switzerland},
  publisher    = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik},
  title        = {{Barycentric cuts through a convex body}},
  doi          = {10.4230/LIPIcs.SoCG.2020.62},
  volume       = {164},
  year         = {2020},
}