Uryson width and volume (Q776887)

The \(k\)-Uryson width measures how far a metric space is from being a \(k\)-dimensional simplicial complex. In [J. Topol. Anal. 9, 195--219 (2017; Zbl 1368.53026)], \textit{L. Guth} demonstrated a relationship between the volumes of balls in a Riemannian manifold and the Uryson width of the manifold which was conjectured by Gromov: that there exists \(\epsilon_n > 0\) so that, if \((M^n,g)\) is a closed Riemannian manifold and there is a radius \(r\) so that every ball of radius \(r\) in \((M^n,g)\) has volume at most \(\epsilon_n r^n\), then the Uryson width \(UW_{n-1}(M^n,g) \leq r\). This result has also been generalised to metric spaces. The current paper gives a more direct proof of these results which relies only on the co-area inequality and, in fact, gives a weaker sufficient condition for a space to have small Uryson width.