## Abstract

Let U_{1}, … , U_{d} _{+} _{1} be n-element sets in R^{d}. Pach’s selection theorem says that there exist subsets Z_{1}⊂ U_{1}, … , Z_{d} _{+} _{1}⊂ U_{d} _{+} _{1} and a point u∈ R^{d} such that each | Z_{i}| ≥ c_{1}(d) n and u∈ conv { z_{1}, … , z_{d} _{+} _{1}} for every choice of z_{1}∈ Z_{1}, … , 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 |
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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