A Geometric Proof of the Colored Tverberg Theorem
Result description
The colored Tverberg theorem asserts that for every d and r there exists t=t(d,r) such that for every set C in 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_1are red, the points of C_2 blue, etc.), there exist r disjoint sets R_1, R_2, ... ,R_r subset of C that are rainbow, meaning that the size of the intersection of R_i and C_j is at most 1 for every i, j, and whose convex hulls all have a common point. Allknown proofs of this theorem are topological. We present a geometric version of a recent beautiful proof by Blagojevic, Matschke, and Ziegler, avoiding a direct use of topological methods. The purpose of this de-topologization is to make the proof moreconcrete and intuitive, and accessible to a wider audience.
Keywords
ConvexityContinuous motionColored Tverberg theoremTverberg's theoremComplexesRadon's Theorem
The result's identifiers
Result code in IS VaVaI
Result on the web
DOI - Digital Object Identifier
Alternative languages
Result language
angličtina
Original language name
A Geometric Proof of the Colored Tverberg Theorem
Original language description
The colored Tverberg theorem asserts that for every d and r there exists t=t(d,r) such that for every set C in 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_1are red, the points of C_2 blue, etc.), there exist r disjoint sets R_1, R_2, ... ,R_r subset of C that are rainbow, meaning that the size of the intersection of R_i and C_j is at most 1 for every i, j, and whose convex hulls all have a common point. Allknown proofs of this theorem are topological. We present a geometric version of a recent beautiful proof by Blagojevic, Matschke, and Ziegler, avoiding a direct use of topological methods. The purpose of this de-topologization is to make the proof moreconcrete and intuitive, and accessible to a wider audience.
Czech name
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Czech description
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Classification
Type
Jx - Unclassified - Peer-reviewed scientific article (Jimp, Jsc and Jost)
CEP classification
BA - General mathematics
OECD FORD branch
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Result continuities
Project
Continuities
P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)
S - Specificky vyzkum na vysokych skolach
Others
Publication year
2012
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů
Data specific for result type
Name of the periodical
Discrete and Computational Geometry
ISSN
0179-5376
e-ISSN
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Volume of the periodical
47
Issue of the periodical within the volume
2
Country of publishing house
DE - GERMANY
Number of pages
21
Pages from-to
245-265
UT code for WoS article
000299057200002
EID of the result in the Scopus database
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Basic information
Result type
Jx - Unclassified - Peer-reviewed scientific article (Jimp, Jsc and Jost)
CEP
BA - General mathematics
Year of implementation
2012