https://arxiv.org/abs/1905.06522Filling 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