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Barycentric Cuts Through a Convex Body

Authors Zuzana Patáková , Martin Tancer, Uli Wagner



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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

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.

Cite AsGet 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

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.

Subject Classification

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

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References

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