eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2019-06-11
38:1
38:13
10.4230/LIPIcs.SoCG.2019.38
article
The Crossing Tverberg Theorem
Fulek, Radoslav
1
https://orcid.org/0000-0001-8485-1774
Gärtner, Bernd
2
Kupavskii, Andrey
3
4
Valtr, Pavel
5
2
https://orcid.org/0000-0002-3102-4166
Wagner, Uli
1
https://orcid.org/0000-0002-3435-0100
IST Austria, Klosterneuburg, Austria
Department of Computer Science, ETH Zürich, Switzerland
University of Oxford, UK
Moscow Institute of Physics and Technology, Russia
Department of Applied Mathematics, Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic
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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol129-socg2019/LIPIcs.SoCG.2019.38/LIPIcs.SoCG.2019.38.pdf
Discrete geometry
Tverberg theorem
Crossing Tverberg theorem