Filling metric spaces

Abstract

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 $f:Y\to B$ with $f(Y)$ compact

$$\underset{F}{inf}{\mathrm{HC}}_{m+1}(F(X))\le c(m){\mathrm{HC}}_m{(f(Y))}^{\frac{m+1}{m}},$$

where ${\mathrm{HC}}_m$ denotes the m-dimensional Hausdorff content, the infimum is taken over the set of all continuous maps $F:X⟶B$ such that $F(y)=f(y)$ for all $y\in Y$, and $c(m)$ depends only on m. Moreover, one can find F with a nearly minimal ${\mathrm{HC}}_{m+1}$ such that its image lies in the $C(m){\mathrm{HC}}_m{(f(Y))}^{\frac{1}{m}}$-neighborhood of $f(Y)$ with the exception of a subset with zero $(m+1)$-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 $(m-1)$-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.