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Barycentric Cuts Through a Convex Body

Authors Zuzana Patáková , Martin Tancer, Uli Wagner

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ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • convex body
  • barycenter
  • Tukey depth
  • smooth manifold
  • critical points

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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.

Cite As Get BibTex

Zuzana Patáková, Martin Tancer, and Uli Wagner. Barycentric Cuts Through a Convex Body. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 62:1-62:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.SoCG.2020.62

Author Details

Zuzana Patáková
  • Computer Science Institute, Charles University, Prague, Czech Republic
  • IST Austria, Klosterneuburg, Austria
Martin Tancer
  • Department of Applied Mathematics, Charles University, Prague, Czech Republic
Uli Wagner
  • IST Austria, Klosterneuburg, Austria

Funding

  • Patáková, Zuzana: The research stay at IST Austria is funded by the project Improvement of internationalization (CZ.02.2.69/0.0/0.0/17_050/0008466) in the field of research and development at Charles University, through the support of quality projects MSCA-IF.
  • Tancer, Martin: Supported by the GAČR grant 19-04113Y and by the Charles University projects PRIMUS/17/SCI/3 and UNCE/SCI/004.

Acknowledgements

We thank Stanislav Nagy for introducing us to Grünbaum’s questions, for useful discussions on the topic, for providing us with many references, and for comments on a preliminary version of this paper. We thank Jan Kynčl and Pavel Valtr for letting us know about a more general counterexample they found, and Roman Karasev for pointing us to related work [R. Karasev, 2011; P. Blagojević and R. Karasev, 2016] and for comments on a preliminary version of this paper. Finally, we thank an anonymous referee for many comments on a preliminary version of the paper which, in particular, yielded an important correction in Section 4.

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