We prove a new version of isoperimetric inequality: Given a positive real m, a Banach space B, a closed subset Y of metric space X, and a continuous map with compact
where denotes the m-dimensional Hausdorff content, the infimum is taken over the set of all continuous maps such that for all , and depends only on m. Moreover, one can find F with a nearly minimal such that its image lies in the -neighborhood of with the exception of a subset with zero -dimensional Hausdorff measure.
The paper also contains a very general coarea inequality for Hausdorff content and its modifications.
As an application we demonstrate an inequality conjectured by Larry Guth that relates the m-dimensional Hausdorff content of a compact metric space with its -dimensional Urysohn width. We show that this result implies 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.
"Filling metric spaces." Duke Math. J. 171 (3) 595 - 632, 15 February 2022. https://doi.org/10.1215/00127094-2021-0039