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Friday, May 17th, 2019

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    Filling metric spaces
    Yevgeny Liokumovich, Boris Lishak, Alexander Nabutovsky, Regina Rotman
    (Submitted on 16 May 2019)
    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.
    The paper also contains a new version of isoperimetric inequality: Given an integer m≥2, a continuous map f:X⟶B from a compact metric space X to a Banach space B, and a closed subset Y⊂X, there exists a continuous map F:X⟶B that coincides with f on Y, such that the m-dimensional Hausdorff content of F(X) does not exceed C(m)HCm−1(f(Y))mm−1

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