---
_id: '2438'
abstract:
- lang: eng
text: The colored Tverberg theorem asserts that for eve;ry d and r there exists
t=t(d,r) such that for every set C ⊂ ℝ d of cardinality (d + 1)t, partitioned
into t-point subsets C 1, C 2,...,C d+1 (which we think of as color classes; e.
g., the points of C 1 are red, the points of C 2 blue, etc.), there exist r disjoint
sets R 1, R 2,...,R r⊆C that are rainbow, meaning that {pipe}R i∩C j{pipe}≤1 for
every i,j, and whose convex hulls all have a common point. All known proofs of
this theorem are topological. We present a geometric version of a recent beautiful
proof by Blagojević, Matschke, and Ziegler, avoiding a direct use of topological
methods. The purpose of this de-topologization is to make the proof more concrete
and intuitive, and accessible to a wider audience.
acknowledgement: 'We would like to thank Marek Krcál for useful discussions at initial
stages of this research. We also thank Günter M. Ziegler for valuable comments,
and Peter Landweber and two anonymous referees for detailed comments and corrections
that greatly helped to improve the presentation. In particular, we are indebted
to one of the referees for pointing out to us reference [19]. M. Tancer is supported
by the grants SVV-2010-261313 (Discrete Methods and Algorithms) and GAUK 49209.
U. Wagner’s research is supported by the Swiss National Science Foundation (SNF
Projects 200021- 125309 and 200020-125027). '
author:
- first_name: Jiří
full_name: Matoušek, Jiří
last_name: Matoušek
- first_name: Martin
full_name: Martin Tancer
id: 38AC689C-F248-11E8-B48F-1D18A9856A87
last_name: Tancer
orcid: 0000-0002-1191-6714
- first_name: Uli
full_name: Uli Wagner
id: 36690CA2-F248-11E8-B48F-1D18A9856A87
last_name: Wagner
orcid: 0000-0002-1494-0568
citation:
ama: Matoušek J, Tancer M, Wagner U. A geometric proof of the colored Tverberg theorem.
*Discrete & Computational Geometry*. 2012;47(2):245-265. doi:10.1007/s00454-011-9368-2
apa: Matoušek, J., Tancer, M., & Wagner, U. (2012). A geometric proof of the
colored Tverberg theorem. *Discrete & Computational Geometry*, *47*(2),
245–265. https://doi.org/10.1007/s00454-011-9368-2
chicago: 'Matoušek, Jiří, Martin Tancer, and Uli Wagner. “A Geometric Proof of the
Colored Tverberg Theorem.” *Discrete & Computational Geometry* 47, no.
2 (2012): 245–65. https://doi.org/10.1007/s00454-011-9368-2.'
ieee: J. Matoušek, M. Tancer, and U. Wagner, “A geometric proof of the colored Tverberg
theorem,” *Discrete & Computational Geometry*, vol. 47, no. 2, pp. 245–265,
2012.
ista: Matoušek J, Tancer M, Wagner U. 2012. A geometric proof of the colored Tverberg
theorem. Discrete & Computational Geometry. 47(2), 245–265.
mla: Matoušek, Jiří, et al. “A Geometric Proof of the Colored Tverberg Theorem.”
*Discrete & Computational Geometry*, vol. 47, no. 2, Springer, 2012,
pp. 245–65, doi:10.1007/s00454-011-9368-2.
short: J. Matoušek, M. Tancer, U. Wagner, Discrete & Computational Geometry
47 (2012) 245–265.
date_created: 2018-12-11T11:57:39Z
date_published: 2012-03-01T00:00:00Z
date_updated: 2019-04-26T07:22:12Z
day: '01'
doi: 10.1007/s00454-011-9368-2
extern: 1
intvolume: ' 47'
issue: '2'
month: '03'
page: 245 - 265
publication: Discrete & Computational Geometry
publication_status: published
publisher: Springer
publist_id: '4468'
quality_controlled: 0
status: public
title: A geometric proof of the colored Tverberg theorem
type: journal_article
volume: 47
year: '2012'
...