We prove an inequality conjectured by Larry Guth that relates the m-dimensional Hausdorff content of a compact metric space with its (m − 1)- dimensional Urysohn width. As a corollary, we obtain new systolic inequalities that both strengthen the classical Gromov’s systolic inequality for essential Riemannian manifolds and extend this inequality to a wider class of non-simply connected manifolds. We also present a new version of isoperimetric inequality. It asserts that for every positive integer m, Banach space B, compact subset X of B, and a closed subset Y of X there is a filling of Y by a continuous image of X with the (m + 1)-dimensional Hausdorff content bounded in terms of the m-dimensional Hausdorff content of Y. Joint work with Yevgeny Liokumovich, Boris Lishak and Regina Rotman.