---
res:
  bibo_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.@eng'
  bibo_authorlist:
  - foaf_Person:
      foaf_givenName: Zuzana
      foaf_name: Patakova, Zuzana
      foaf_surname: Patakova
      foaf_workInfoHomepage: http://www.librecat.org/personId=48B57058-F248-11E8-B48F-1D18A9856A87
    orcid: 0000-0002-3975-1683
  - foaf_Person:
      foaf_givenName: Martin
      foaf_name: Tancer, Martin
      foaf_surname: Tancer
      foaf_workInfoHomepage: http://www.librecat.org/personId=38AC689C-F248-11E8-B48F-1D18A9856A87
    orcid: 0000-0002-1191-6714
  - foaf_Person:
      foaf_givenName: Uli
      foaf_name: Wagner, Uli
      foaf_surname: Wagner
      foaf_workInfoHomepage: http://www.librecat.org/personId=36690CA2-F248-11E8-B48F-1D18A9856A87
    orcid: 0000-0002-1494-0568
  bibo_doi: 10.4230/LIPIcs.SoCG.2020.62
  bibo_volume: 164
  dct_date: 2020^xs_gYear
  dct_isPartOf:
  - http://id.crossref.org/issn/1868-8969
  - http://id.crossref.org/issn/9783959771436
  dct_language: eng
  dct_publisher: Schloss Dagstuhl - Leibniz-Zentrum für Informatik@
  dct_title: Barycentric cuts through a convex body@
...