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 $$\underset{F}{\inf }H{C}_{m+1}(F(X))\le c(m)H{C}_m(f(Y){)}^{\frac{m+1}{m}},$$ where $H{C}_m$ 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 $H{C}_{m+1}$ such that its image lies in the $C(m)H{C}_m(f(Y){)}^{\frac{1}{m}}$-neighbourhood 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.