Uryson Width and Volume

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 vol ( S 3 , g ) = 1 and for any map f : S 3 → R 2 there is some x ∈ R 2 for which diam ( f - 1 ( x ) ) > C , answering a question of Guth.