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 language | American English |
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Pages (from-to) | 420-429 |
Number of pages | 10 |
Journal | Discrete and Computational Geometry |
Volume | 60 |
Issue number | 2 |
DOIs | |
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