article
A geometric proof of the colored Tverberg theorem
published
Jiří
Matoušek
author
Martin
Tancer
author 38AC689C-F248-11E8-B48F-1D18A9856A870000-0002-1191-6714
Uli
Wagner
author 36690CA2-F248-11E8-B48F-1D18A9856A870000-0002-1494-0568
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.
Springer2012
Discrete & Computational Geometry10.1007/s00454-011-9368-2
472245 - 265
yes
J. Matoušek, M. Tancer, U. Wagner, Discrete & Computational Geometry 47 (2012) 245–265.
J. Matoušek, M. Tancer, and U. Wagner, “A geometric proof of the colored Tverberg theorem,” <i>Discrete & Computational Geometry</i>, vol. 47, no. 2. Springer, pp. 245–265, 2012.
Matoušek J, Tancer M, Wagner U. A geometric proof of the colored Tverberg theorem. <i>Discrete & Computational Geometry</i>. 2012;47(2):245-265. doi:<a href="https://doi.org/10.1007/s00454-011-9368-2">10.1007/s00454-011-9368-2</a>
Matoušek, Jiří, Martin Tancer, and Uli Wagner. “A Geometric Proof of the Colored Tverberg Theorem.” <i>Discrete & Computational Geometry</i>. Springer, 2012. <a href="https://doi.org/10.1007/s00454-011-9368-2">https://doi.org/10.1007/s00454-011-9368-2</a>.
Matoušek J, Tancer M, Wagner U. 2012. A geometric proof of the colored Tverberg theorem. Discrete & Computational Geometry. 47(2), 245–265.
Matoušek, J., Tancer, M., & Wagner, U. (2012). A geometric proof of the colored Tverberg theorem. <i>Discrete & Computational Geometry</i>. Springer. <a href="https://doi.org/10.1007/s00454-011-9368-2">https://doi.org/10.1007/s00454-011-9368-2</a>
Matoušek, Jiří, et al. “A Geometric Proof of the Colored Tverberg Theorem.” <i>Discrete & Computational Geometry</i>, vol. 47, no. 2, Springer, 2012, pp. 245–65, doi:<a href="https://doi.org/10.1007/s00454-011-9368-2">10.1007/s00454-011-9368-2</a>.
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