Open Collections

Pach's selection theorem does not admit a topological extension.

Banff International Research Station for Mathematical Innovation and Discovery

Pach's selection theorem asserts that for any positive integer $d$ there exists a constant $c_d > 0$ such that for any positive integer $n$ and any finite sets $X_1, ..., X_{d+1} in \mathbb{R}^d$ each with $n$ points there exist disjoint subsets $Z_1, ..., Z_{d+1}, Z_i$ is a subset of $X_i$ and a point $z$ such that $z$ belongs to any rainbow $(Z_1, ..., Z_{d+1})$-simplex; that is, a convex hull of points $z_1, ..., z_{d+1}$ where $z_i$ belongs to $Z_i$. Although the topological method is a valuable tool for improving the bounds for certain selection theorems (introduced by Gromov), we prove that Pach's theorem does not admit a topological extension. Joint work with Imre B\'ar\'any, Roy Meshulam and Eran Nevo.

Mathematics; Convex and discrete geometry; Algebraic topology; Combinatorics

video/mp4

Author affiliation: Charles University in Prague

2017-06-07

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/61858

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/