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.

中文翻译：

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乌里森宽度和体积

我们给出了有关球体积和Uryson宽度的古斯定理的简短证明。相同的方法适用于Hausdorff内容，暗示Liokumovich–Lishak–Nabutovsky–Rotman的最新结果。我们还表明，对于任何\（C> 0 \），在3个球面上都有一个黎曼度量g，这样\（{\ hbox {vol}}（S ^ 3，g）= 1 \）和任何地图\（f：S ^ 3 \ rightarrow {\ mathbb {R}} ^ 2 \）有一些\（x \ in {\ mathbb {R}} ^ 2 \）为此，\（\ text {diam}（f ^ {-1}（x））> C \），回答古斯的问题。