Pach’s Selection Theorem Does Not Admit a Topological Extension

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

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

2 Scopus citations

Abstract

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.

Original languageAmerican English
Pages (from-to)420-429
Number of pages10
JournalDiscrete and Computational Geometry
Volume60
Issue number2
DOIs
StatePublished - 1 Sep 2018

Bibliographical note

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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