Pach’s Selection Theorem Does Not Admit a Topological Extension

Imre Bárány; Roy Meshulam; Eran Nevo; Martin Tancer

doi:10.1007/s00454-018-9998-8

Pach’s Selection Theorem Does Not Admit a Topological Extension

Imre Bárány, Roy Meshulam^*, Eran Nevo, Martin Tancer

^*Corresponding author for this work

Einstein Institute of Mathematics

Research output: Contribution to journal › Article › peer-review

2 Scopus citations

Abstract

Let U₁, … , U_d ₊ ₁ be n-element sets in R^d. Pach’s selection theorem says that there exist subsets Z₁⊂ U₁, … , Z_d ₊ ₁⊂ U_d ₊ ₁ and a point u∈ R^d such that each | Z_i| ≥ c₁(d) n and u∈ conv { z₁, … , z_d ₊ ₁} for every choice of z₁∈ Z₁, … , z_d ₊ ₁∈ Z_d ₊ ₁. Here we show that this theorem does not admit a topological extension with linear size sets Z_i. However, there is a topological extension where each | Z_i| is of order (log n) ¹ ^/ ^d.

Original language	American English
Pages (from-to)	420-429
Number of pages	10
Journal	Discrete and Computational Geometry
Volume	60
Issue number	2
DOIs	https://doi.org/10.1007/s00454-018-9998-8
State	Published - 1 Sep 2018

Bibliographical note

Funding Information:
Acknowledgements This research was supported by ERC Advanced Research Grant no 267165 (DIS-CONV). Imre Bárány is partially supported by Hungarian National Research Grant K 111827. Roy Meshulam is partially supported by ISF grant 326/16 and GIF grant 1261/14, Eran Nevo by ISF grant 1695/15 and Martin Tancer by GACˇ R grant 16-01602Y.

Publisher Copyright:
© 2018, Springer Science+Business Media, LLC, part of Springer Nature.

Keywords

Gromov’s Overlap Theorem
Pach’s Selection Theorem

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title = "Pach{\textquoteright}s Selection Theorem Does Not Admit a Topological Extension",

abstract = "Let U1, … , Ud + 1 be n-element sets in Rd. Pach{\textquoteright}s selection theorem says that there exist subsets Z1⊂ U1, … , Zd + 1⊂ Ud + 1 and a point u∈ Rd such that each | Zi| ≥ c1(d) n and u∈ conv { z1, … , zd + 1} for every choice of z1∈ Z1, … , zd + 1∈ Zd + 1. Here we show that this theorem does not admit a topological extension with linear size sets Zi. However, there is a topological extension where each | Zi| is of order (log n) 1 / d.",

keywords = "Gromov{\textquoteright}s Overlap Theorem, Pach{\textquoteright}s Selection Theorem",

author = "Imre B{\'a}r{\'a}ny and Roy Meshulam and Eran Nevo and Martin Tancer",

note = "Funding Information: Acknowledgements This research was supported by ERC Advanced Research Grant no 267165 (DIS-CONV). Imre B{\'a}r{\'a}ny is partially supported by Hungarian National Research Grant K 111827. Roy Meshulam is partially supported by ISF grant 326/16 and GIF grant 1261/14, Eran Nevo by ISF grant 1695/15 and Martin Tancer by GACˇ R grant 16-01602Y. Publisher Copyright: {\textcopyright} 2018, Springer Science+Business Media, LLC, part of Springer Nature.",

year = "2018",

month = sep,

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doi = "10.1007/s00454-018-9998-8",

language = "American English",

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AU - Bárány, Imre

AU - Meshulam, Roy

AU - Nevo, Eran

AU - Tancer, Martin

N1 - Funding Information: Acknowledgements This research was supported by ERC Advanced Research Grant no 267165 (DIS-CONV). Imre Bárány is partially supported by Hungarian National Research Grant K 111827. Roy Meshulam is partially supported by ISF grant 326/16 and GIF grant 1261/14, Eran Nevo by ISF grant 1695/15 and Martin Tancer by GACˇ R grant 16-01602Y. Publisher Copyright: © 2018, Springer Science+Business Media, LLC, part of Springer Nature.

PY - 2018/9/1

Y1 - 2018/9/1

N2 - Let U1, … , Ud + 1 be n-element sets in Rd. Pach’s selection theorem says that there exist subsets Z1⊂ U1, … , Zd + 1⊂ Ud + 1 and a point u∈ Rd such that each | Zi| ≥ c1(d) n and u∈ conv { z1, … , zd + 1} for every choice of z1∈ Z1, … , zd + 1∈ Zd + 1. Here we show that this theorem does not admit a topological extension with linear size sets Zi. However, there is a topological extension where each | Zi| is of order (log n) 1 / d.

AB - Let U1, … , Ud + 1 be n-element sets in Rd. Pach’s selection theorem says that there exist subsets Z1⊂ U1, … , Zd + 1⊂ Ud + 1 and a point u∈ Rd such that each | Zi| ≥ c1(d) n and u∈ conv { z1, … , zd + 1} for every choice of z1∈ Z1, … , zd + 1∈ Zd + 1. Here we show that this theorem does not admit a topological extension with linear size sets Zi. However, there is a topological extension where each | Zi| is of order (log n) 1 / d.

KW - Gromov’s Overlap Theorem

KW - Pach’s Selection Theorem

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JO - Discrete and Computational Geometry

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ER -

Pach’s Selection Theorem Does Not Admit a Topological Extension

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