Uryson Width and Volume

Abstract

AbstractWe give a short proof of a theorem of Guth relating volume of balls and Uryson width. The same approach applies to Hausdorff content implying a recent result of Liokumovich–Lishak–Nabutovsky–Rotman. We show also that for any $$C>0$$C>0 there is a Riemannian metric g on a 3-sphere such that $${\hbox {vol}}(S^3,g)=1$$vol(S3,g)=1 and for any map $$f:S^3\rightarrow {\mathbb {R}}^2$$f:S3→R2 there is some $$x\in {\mathbb {R}}^2$$x∈R2 for which $$\text {diam}(f^{-1}(x))>C$$diam(f-1(x))>C, answering a question of Guth.