---
_id: '7992'
abstract:
- lang: eng
  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.'
alternative_title:
- LIPIcs
article_number: 62:1 - 62:16
article_processing_charge: No
arxiv: 1
author:
- first_name: Zuzana
  full_name: Patakova, Zuzana
  id: 48B57058-F248-11E8-B48F-1D18A9856A87
  last_name: Patakova
  orcid: 0000-0002-3975-1683
- first_name: Martin
  full_name: Tancer, Martin
  id: 38AC689C-F248-11E8-B48F-1D18A9856A87
  last_name: Tancer
  orcid: 0000-0002-1191-6714
- first_name: Uli
  full_name: Wagner, Uli
  id: 36690CA2-F248-11E8-B48F-1D18A9856A87
  last_name: Wagner
  orcid: 0000-0002-1494-0568
citation:
  ama: '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>'
  apa: 'Patakova, Z., Tancer, M., &#38; Wagner, U. (2020). Barycentric cuts through
    a convex body. In <i>36th International Symposium on Computational Geometry</i>
    (Vol. 164). Zürich, Switzerland: Schloss Dagstuhl - Leibniz-Zentrum für Informatik.
    <a href="https://doi.org/10.4230/LIPIcs.SoCG.2020.62">https://doi.org/10.4230/LIPIcs.SoCG.2020.62</a>'
  chicago: Patakova, Zuzana, Martin Tancer, and Uli Wagner. “Barycentric Cuts through
    a Convex Body.” In <i>36th International Symposium on Computational Geometry</i>,
    Vol. 164. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. <a href="https://doi.org/10.4230/LIPIcs.SoCG.2020.62">https://doi.org/10.4230/LIPIcs.SoCG.2020.62</a>.
  ieee: Z. Patakova, M. Tancer, and U. Wagner, “Barycentric cuts through a convex
    body,” in <i>36th International Symposium on Computational Geometry</i>, Zürich,
    Switzerland, 2020, vol. 164.
  ista: 'Patakova Z, Tancer M, Wagner U. 2020. Barycentric cuts through a convex body.
    36th International Symposium on Computational Geometry. SoCG: Symposium on Computational
    Geometry, LIPIcs, vol. 164, 62:1-62:16.'
  mla: Patakova, Zuzana, et al. “Barycentric Cuts through a Convex Body.” <i>36th
    International Symposium on Computational Geometry</i>, vol. 164, 62:1-62:16, 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>.
  short: Z. Patakova, M. Tancer, U. Wagner, in:, 36th International Symposium on Computational
    Geometry, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020.
conference:
  end_date: 2020-06-26
  location: Zürich, Switzerland
  name: 'SoCG: Symposium on Computational Geometry'
  start_date: 2020-06-22
corr_author: '1'
date_created: 2020-06-22T09:14:20Z
date_published: 2020-06-01T00:00:00Z
date_updated: 2025-07-10T11:54:57Z
day: '01'
ddc:
- '510'
department:
- _id: UlWa
doi: 10.4230/LIPIcs.SoCG.2020.62
external_id:
  arxiv:
  - '2003.13536'
file:
- access_level: open_access
  checksum: ce1c9194139a664fb59d1efdfc88eaae
  content_type: application/pdf
  creator: dernst
  date_created: 2020-06-23T06:45:52Z
  date_updated: 2020-07-14T12:48:06Z
  file_id: '8004'
  file_name: 2020_LIPIcsSoCG_Patakova.pdf
  file_size: 750318
  relation: main_file
file_date_updated: 2020-07-14T12:48:06Z
has_accepted_license: '1'
intvolume: '       164'
language:
- iso: eng
month: '06'
oa: 1
oa_version: Published Version
publication: 36th International Symposium on Computational Geometry
publication_identifier:
  isbn:
  - '9783959771436'
  issn:
  - 1868-8969
publication_status: published
publisher: Schloss Dagstuhl - Leibniz-Zentrum für Informatik
quality_controlled: '1'
scopus_import: '1'
status: public
title: Barycentric cuts through a convex body
tmp:
  image: /images/cc_by.png
  legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
  name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
  short: CC BY (4.0)
type: conference
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 164
year: '2020'
...