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The Crossing Tverberg Theorem

Authors Radoslav Fulek , Bernd Gärtner, Andrey Kupavskii, Pavel Valtr , Uli Wagner

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Subject Classification

ACM Subject Classification
  • Mathematics of computing → Combinatoric problems
  • Theory of computation → Computational geometry
Keywords
  • Discrete geometry
  • Tverberg theorem
  • Crossing Tverberg theorem

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Abstract

The Tverberg theorem is one of the cornerstones of discrete geometry. It states that, given a set X of at least (d+1)(r-1)+1 points in R^d, one can find a partition X=X_1 cup ... cup X_r of X, such that the convex hulls of the X_i, i=1,...,r, all share a common point. In this paper, we prove a strengthening of this theorem that guarantees a partition which, in addition to the above, has the property that the boundaries of full-dimensional convex hulls have pairwise nonempty intersections. Possible generalizations and algorithmic aspects are also discussed. 
As a concrete application, we show that any n points in the plane in general position span floor[n/3] vertex-disjoint triangles that are pairwise crossing, meaning that their boundaries have pairwise nonempty intersections; this number is clearly best possible. A previous result of Alvarez-Rebollar et al. guarantees floor[n/6] pairwise crossing triangles. Our result generalizes to a result about simplices in R^d,d >=2.

Cite As Get BibTex

Radoslav Fulek, Bernd Gärtner, Andrey Kupavskii, Pavel Valtr, and Uli Wagner. The Crossing Tverberg Theorem. In 35th International Symposium on Computational Geometry (SoCG 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 129, pp. 38:1-38:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019) https://doi.org/10.4230/LIPIcs.SoCG.2019.38

Author Details

Radoslav Fulek
  • IST Austria, Klosterneuburg, Austria
Bernd Gärtner
  • Department of Computer Science, ETH Zürich, Switzerland
Andrey Kupavskii
  • University of Oxford, UK
  • Moscow Institute of Physics and Technology, Russia
Pavel Valtr
  • Department of Applied Mathematics, Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic
  • Department of Computer Science, ETH Zürich, Switzerland
Uli Wagner
  • IST Austria, Klosterneuburg, Austria

Funding

  • Fulek, Radoslav: The author gratefully acknowledges support from Austrian Science Fund (FWF), Project M2281-N35.
  • Kupavskii, Andrey: The research was supported by the Advanced Postdoc.Mobility grant no. P300P2_177839 of the Swiss National Science Foundation.
  • Valtr, Pavel: Research by P. Valtr was supported by the grant no. 18-19158S of the Czech Science Foundation (GAČR).

Acknowledgements

Part of the research leading to this paper was done during the 16th Gremo Workshop on Open Problems (GWOP), Waltensburg, Switzerland, June 12-16, 2018. We thank Patrick Schnider for suggesting the problem, and Stefan Felsner, Malte Milatz and Emo Welzl for fruitful discussions during the workshop. We also thank Stefan Felsner and Manfred Scheucher for finding and communicating the example from Section 3.2. We thank Dömötör Pálvölgyi and the SoCG reviewers for various helpful comments.

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