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Uryson Width and Volume
Geometric and Functional Analysis  (IF2.148),  Pub Date : 2020-03-28, DOI: 10.1007/s00039-020-00533-5
Panos Papasoglu

We 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 $$C>0$$ there is a Riemannian metric g on a 3-sphere such that $${\hbox {vol}}(S^3,g)=1$$ and for any map $$f:S^3\rightarrow {\mathbb {R}}^2$$ there is some $$x\in {\mathbb {R}}^2$$ for which $$\text {diam}(f^{-1}(x))>C$$, answering a question of Guth.