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# Barycentric Cuts Through a Convex Body

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## 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 As

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

1. P. Blagojević and R. Karasev. Local multiplicity of continuous maps between manifolds, 2016. Preprint. URL: http://arxiv.org/abs/1603.06723.
2. W. Blaschke. Über affine Geometrie IX: Verschiedene Bemerkungen und Aufgaben. Ber. Verh. Sächs. Akad. Wiss. Leipzig. Math.-Nat. Kl., 69:412-420, 1917.
3. D. Bremner, D. Chen, J. Iacono, S. Langerman, and P. Morin. Output-sensitive algorithms for Tukey depth and related problems. Stat. Comput., 18(3):259-266, 2008. URL: https://doi.org/10.1007/s11222-008-9054-2.
4. T. M. Chan. An optimal randomized algorithm for maximum Tukey depth. In Proceedings of the Fifteenth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 430-436. ACM, New York, 2004.
5. D. Chen, P. Morin, and U. Wagner. Absolute approximation of Tukey depth: theory and experiments. Comput. Geom., 46(5):566-573, 2013. URL: https://doi.org/10.1016/j.comgeo.2012.03.001.
6. H. T. Croft, K. J. Falconer, and R. K. Guy. Unsolved problems in geometry. Problem Books in Mathematics. Springer-Verlag, New York, 1994. Corrected reprint of the 1991 original, Unsolved Problems in Intuitive Mathematics, II.
7. D. L. Donoho. Breakdown properties of multivariate location estimators, 1982. Unpublished qualifying paper, Harvard University.
8. D. L. Donoho and M. Gasko. Breakdown properties of location estimates based on halfspace depth and projected outlyingness. Ann. Statist., 20(4):1803-1827, 1992. URL: https://doi.org/10.1214/aos/1176348890.
9. C. Dupin. Applications de géométrie et de méchanique, a la marine, aux ponts et chaussées, etc., pour faire suite aux Développements de géométrie, par Charles Dupin. Bachelier, successeur de Mme. Ve. Courcier, libraire, 1822.
10. R. Dyckerhoff and P. Mozharovskyi. Exact computation of the halfspace depth. Comput. Statist. Data Anal., 98:19-30, 2016. URL: https://doi.org/10.1016/j.csda.2015.12.011.
11. B. Grünbaum. On some properties of convex sets. Colloq. Math., 8:39-42, 1961. URL: https://doi.org/10.4064/cm-8-1-39-42.
12. B. Grünbaum. Measures of symmetry for convex sets. In Proc. Sympos. Pure Math., Vol. VII, pages 233-270. Amer. Math. Soc., Providence, R.I., 1963.
13. A. Hassairi and O. Regaieg. On the Tukey depth of a continuous probability distribution. Statist. Probab. Lett., 78(15):2308-2313, 2008.
14. A. Hatcher. Algebraic topology. Cambridge University Press, Cambridge, 2002.
15. R. Karasev. Geometric coincidence results from multiplicity of continuous maps, 2011. Preprint. URL: http://arxiv.org/abs/1106.6176.
16. J. Kynčl and P. Valtr, 2019. Personal communication.
17. X. Liu, K. Mosler, and P. Mozharovskyi. Fast computation of Tukey trimmed regions and median in dimension p>2. J. Comput. Graph. Statist., 28(3):682-697, 2019. URL: https://doi.org/10.1080/10618600.2018.1546595.
18. S. Nagy, C. Schütt, and E. M. Werner. Halfspace depth and floating body. Stat. Surv., 13:52-118, 2019.
19. Z. Patáková, M. Tancer, and U. Wagner. Barycentric cuts through a convex body, 2020. Preprint. URL: http://arxiv.org/abs/2003.13536.
20. P. J. Rousseeuw and I. Ruts. The depth function of a population distribution. Metrika, 49(3):213-244, 1999.
21. P. J. Rousseeuw and A. Struyf. Computing location depth and regression depth in higher dimensions. Statistics and Computing, 8(3):193-203, 1998.
22. C. Schütt and E. Werner. Homothetic floating bodies. Geom. Dedicata, 49(3):335-348, 1994.
23. J. Tukey. Mathematics and the picturing of data. In Proceedings of the International Congress of Mathematicians (Vancouver, B. C., 1974), Vol. 2, pages 523-531, 1975.
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