License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.SoCG.2019.38
URN: urn:nbn:de:0030-drops-104423
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2019/10442/
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Fulek, Radoslav ; Gärtner, Bernd ; Kupavskii, Andrey ; Valtr, Pavel ; Wagner, Uli

The Crossing Tverberg Theorem

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LIPIcs-SoCG-2019-38.pdf (0.5 MB)

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.

BibTeX - Entry

@InProceedings{fulek_et_al:LIPIcs:2019:10442,
  author =	{Radoslav Fulek and Bernd G{\"a}rtner and Andrey Kupavskii and Pavel Valtr and Uli Wagner},
  title =	{{The Crossing Tverberg Theorem}},
  booktitle =	{35th International Symposium on Computational Geometry (SoCG 2019)},
  pages =	{38:1--38:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-104-7},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{129},
  editor =	{Gill Barequet and Yusu Wang},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2019/10442},
  URN =		{urn:nbn:de:0030-drops-104423},
  doi =		{10.4230/LIPIcs.SoCG.2019.38},
  annote =	{Keywords: Discrete geometry, Tverberg theorem, Crossing Tverberg theorem}
}

Keywords: Discrete geometry, Tverberg theorem, Crossing Tverberg theorem
Collection: 35th International Symposium on Computational Geometry (SoCG 2019)
Issue Date: 2019
Date of publication: 11.06.2019

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