# Local-to-global Urysohn width estimates

@article{Balitskiy2020LocaltoglobalUW,
title={Local-to-global Urysohn width estimates},
author={Alexey Balitskiy and Aleksandr Berdnikov},
journal={Journal f{\"u}r die reine und angewandte Mathematik (Crelles Journal)},
year={2020},
volume={2021},
pages={265 - 274}
}
• Published 18 August 2020
• Mathematics
• Journal für die reine und angewandte Mathematik (Crelles Journal)
Abstract The notion of the Urysohn d-width measures to what extent a metric space can be approximated by a d-dimensional simplicial complex. We investigate how local Urysohn width bounds on a Riemannian manifold affect its global width. We bound the 1-width of a Riemannian manifold in terms of its first homology and the supremal width of its unit balls. Answering a question of Larry Guth, we give examples of n-manifolds of considerable (n-1){(n-1)}-width in which all unit balls have arbitrarily…
2 Citations

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If (Mn,g) is a closed Riemannian manifold where every unit ball has volume at most ϵn (a sufficiently small constant), then the (n − 1)-dimensional Uryson width of (Mn,g) is at most 1.