Filling metric spaces

TL;DR

This paper establishes a new isoperimetric inequality in metric spaces, linking Hausdorff contents of different dimensions, and applies it to derive systolic inequalities that extend classical results to broader classes of manifolds.

Contribution

It introduces a novel isoperimetric inequality involving Hausdorff content and demonstrates its application to systolic inequalities, extending Gromov's classical results.

Findings

Proves a new isoperimetric inequality for Hausdorff content.

Derives a coarea inequality for Hausdorff content and its variants.

Establishes new systolic inequalities for a wider class of manifolds.

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 $f:Y \rightarrow B$ with $f(Y)$ compact $$\inf_FHC_{m+1}(F(X))\leq c(m)HC_m(f(Y))^{\frac{m+1}{m}},$$ where $HC_m$ denotes the $m$-dimensional Hausdorff content, the infimum is taken over the set of all continuous maps $F:X\longrightarrow B$ 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 $HC_{m+1}$ such that its image lies in the $C(m)HC_m(f(Y))^{1\over m}$-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.

Equations

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

$${\mathcal{F}}_{m+1}(f,X,B)\le const(m){\mathrm{HC}}_m(f(Y){)}^{\frac{m+1}{m}},$$

$${\mathcal{F}}_{m+1}(f,X,N_R(f(Y)))\le const(m){\mathrm{HC}}_m(f(Y){)}^{\frac{m+1}{m}}.$$

$${\mathcal{F}}_{m+1}(f,X,U)\le const(m){\mathrm{HC}}_m(f(Y){)}^{\frac{m+1}{m}}.$$

$$c(l_{\infty }^{n+1},n)=1.$$

$$U{W}_{m-1}(X)\le c(m){\mathrm{HC}}_m^{\frac{1}{m}}(X).$$

$$X$$

$$sy{s}_1(X)\le C(m){\mathrm{HC}}_m^{\frac{1}{m}}(X).(\ast )$$

$${\mathcal{F}}_m(\tau ,X,U)\le m(1+\frac{1}{m}{)}^{m}R{\mathrm{HC}}_{m-1}(Y)\le emR{\mathrm{HC}}_{m-1}(Y).$$

$${\mathcal{F}}_m(\tau ,X,U;W)\le m(1+\frac{1}{m}{)}^{m}R{\mathrm{HC}}_{m-1}(Y;W)\le emR{\mathrm{HC}}_{m-1}(Y;W)$$

$${\mathrm{HC}}_m(F(X);W)\le m(1+\frac{1}{m}{)}^{m}R{\mathrm{HC}}_{m-1}(Y;W)+\delta$$

$${\int }_{R_1}^{\ast R_2}{\mathrm{HC}}_{m-1}(f^{-1}(R);\Gamma )dR\le 2Lip(f){\mathrm{HC}}_m(U;\Gamma ),$$

$${\int }_{R_1}^{R_2}\sum _{i\in I(R)}{r}_i^{m-1}dR={\int }_{R_1}^{R_2}\sum _{i=1}^N{r}_i^{m-1}{\chi }_i(R)dR=\sum _{i=1}^N{r}_i^{m-1}{\int }_{R_1}^{R_2}{\chi }_i(R)dR,$$

$$|{\varrho }_1-{\varrho }_2|\le Lip(f)dist(f^{-1}({\varrho }_1)\cap B_{r_i}(p_i),f^{-1}({\varrho }_2)\cap B_{r_i}(p_i))\le 2Lip(f)r_i$$

$${\int }_{R_1}^{\ast R_2}{\mathrm{HC}}_{m-1}({\Sigma }_R\cap U;W)dR\le 2{\mathrm{HC}}_m(U;W).$$

$${\int }_{R_1}^{\ast R_2}{\mathrm{HC}}_{m-1}(S_R)dR\le 2{\mathrm{HC}}_m(A_{R_1,R_2}),$$

$${\mathcal{F}}_m(f,X,{\mathbb{R}}^n)\le c_1(n){\mathrm{HC}}_{m-1}(f(Y){)}^{\frac{m}{m-1}}$$

$${\mathcal{F}}_m(f,X,N_R(f(Y))\cup Q(R{)}^{(\lceil m\rceil -1)})\le c_1(n){\mathrm{HC}}_{m-1}(f(Y){)}^{\frac{m}{m-1}}$$

$${\mathrm{HC}}_{m-1}(V)\le c_0(k)R^{m-1}$$

$${\Phi }_{p,V}(t,x)=(1-t)x+t{P}_{C,p}(x)$$

$${\mathrm{HC}}_m({\Phi }_{p,V}([0,1]\times V))\le const(k)R{\mathrm{HC}}_{m-1}(V)$$

$$\sum _m{\mathrm{HC}}_{m-1}(V_k\cap C_m)\le const(n){\mathrm{HC}}_{m-1}(V_k)$$

$${\mathcal{F}}_m(f,X,U)\le I_1(m){\mathrm{HC}}_{m-1}(f(Y){)}^{\frac{m}{m-1}}.$$

$${\mathrm{HC}}_m(F(X))\le I_1(m){\mathrm{HC}}_{m-1}(f(Y){)}^{\frac{m}{m-1}}+\epsilon ,$$

$${\mathcal{F}}_m(f,X,P)\le I_1(m){\mathrm{HC}}_{m-1}(f(Y){)}^{\frac{m}{m-1}}.$$

$${\mathrm{HC}}_m(F(X);S^{\prime })\le I_1(m){\mathrm{HC}}_{m-1}(f(Y);S^{\prime }{)}^{\frac{m}{m-1}}+\epsilon ,$$

$${\mathrm{HC}}_m(F(X))={\mathrm{HC}}_m(F_j({\cup }_{j=1}^{N^{\prime }}B(p_j,R_j+\varrho ))),$$

$$\underset{j}{\max }{r}_j\le (1+\frac{1}{m}{)}^{2}A(m){\mathrm{HC}}_m(Y^{\prime };S^{\prime }{)}^{\frac{1}{m}}+\epsilon$$

$${\mathrm{HC}}_m(Y^{\prime }\setminus \bigcup B_j;S^{\prime })\le (1-{\alpha }^m){\mathrm{HC}}_m(Y^{\prime };S^{\prime })+\epsilon$$

$$\sum _j{r}_j{\mathrm{HC}}_{m-1}(\partial B_j\cap Y^{\prime };S^{\prime }{)}^{\frac{m}{m-1}}\le \frac{2004^{\frac{1}{m-1}}m{\alpha }^{m+1}}{A(m{)}^{\frac{1}{m-1}}}{\mathrm{HC}}_m(Y^{\prime };S^{\prime }{)}^{\frac{m+1}{m}}+\epsilon$$

Full text

[TABLE]

where $\operatorname{HC}_{m}$ denotes the $m$-dimensional Hausdorff content, the infimum is taken over the set of all continuous maps $F:X\longrightarrow B$ 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 $\operatorname{HC}_{m+1}$ such that its image lies in the $C(m)\operatorname{HC}_{m}(f(Y))^{1\over m}$-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.

1. Introduction

1.1. Isoperimetric inequalities.

Given a metric space $X$ and a real $m>0$, one can define the $m$-dimensional Hausdorff content $\operatorname{HC}_{m}(U)$ of $U$, $U\subset X$, as the infimum of $\sum_{i}r_{i}^{m}$ among all coverings of $U$ by countably many balls $B(x_{i},r_{i})$ in $X$. The definition of Hausdorff content looks similar to the definition of Hausdorff measure, except that for Hausdorff content we do not take the limit over all coverings with the maximal radius of the ball in the covering tending to [math]. In particular, the $m$-dimensional Hausdorff content of a set $U$ is always less than or equal to the $m$-dimensional Hausdorff measure of $U$. When the Hausdorff dimension of a compact metric space $U$ is greater than $m$ the $m$-dimensional Hausdorff measure is infinite, but $\operatorname{HC}_{m}(U)$ is always finite and can be very small. For integer $m$ the Lebesgue measure (volume) $vol(U)$ of a set with Hausdorff dimension $m$ satisfies the inequality $\operatorname{HC}_{m}(U)\leq{1\over V_{n}(1)}vol(U),$ where $V_{n}(1)={\pi^{m\over 2}\over\Gamma({m\over 2}+1)}$ is the volume of a ball of radius $1$ in $\mathbb{R}^{n}$.

The main result of this paper is an isoperimetric inequality for Hausdorff contents. To state our main theorem it will be convenient to introduce notation for the infimal Hausdorff content of an extension of a map between metric spaces. One can think of it as an analogue of the filling volume function. Given metric spaces $X,B$, subsets $Y\subset X$, $U\subset B$ and a map $f:Y\rightarrow U$ define $\mathcal{F}_{m}(f,X,U)=\inf\{\operatorname{HC}_{m}(F(X))\}$, where the infimum is taken over all maps $F:X\rightarrow U$, such that $F|_{Y}=f$. If no such extension $F$ exists we set $\mathcal{F}_{m}(f,X,U)=\infty$.

Theorem 1.1**.**

For each positive $m$ there exists a constant $const(m)$, such that the following holds. Let $B$ be a Banach space, $X$ metric space, $Y\subset X$ closed subset and $f:Y\longrightarrow B$ a continuous map with $f(Y)$ compact, then

[TABLE]

*where one can take $const(m)=(100m)^{m}$. *

Note that $B$ may be infinite-dimensional, and we do not require $m$ to be an integer. Moreover, we will show (see Theorem 3.1) that the isoperimetric extension in Theorem 1.1 also obeys the following filling radius estimate: Assume $\operatorname{HC}_{m}(f(Y))\neq 0$ and let $R=Const(m)\operatorname{HC}_{m}(f(Y))^{1\over m}$, where one can take $Const(m)=(1500m)^{m}$. In general, for any choice of $Const(m)$ it is not true that $f$ can be extended to map $F$ whose image lies in the neighbourhood $N_{R}(f(Y))$. Indeed, imagine the case when $Y$ is a very long, but very thin “round” $2$-torus embedded in $\mathbb{R}^{3}$, and $X$ is the cone over this torus. The image of $X$ should at least fill the long parallel of the torus.

However, we prove that if $N_{R}(Y)$ is $(\lceil m\rceil-1)$-connected, then

[TABLE]

More generally, if $U$ is a $(\lceil m\rceil-1)$-connected open set that contains $N_{R}(f(Y))$, then

[TABLE]

Also, one can always find $F$ such that $\operatorname{HC}_{m+1}(F(X))\leq const(m)\operatorname{HC}_{m}(f(Y))^{m+1\over m},$ and $F(X)\setminus N_{R}(Y)$ is contained in a $\lceil m\rceil$-dimensional simplicial complex, where $const(m)$ and $Const(m)$ can be chosen as above (see Theorem 3.1 below for the exact statement).

Finally, given a continuous map (e.g. an inclusion) $i:Y\longrightarrow B$ with compact image, $i$ can be “filled” in $Const(m)\operatorname{HC}_{m}(i(Y))^{1\over m}$-neighborhood of $Y$. We define the $m$-filling for such $i$ and each positive $m$ as follows: An $m$-filling of $i$ consists of an open set $O\supset i(Y)$, a $(\lceil m\rceil-1)$-dimensional simplicial complex $K\subset O$, and a continuous map $\phi:Y\longrightarrow K$ such that $i$ is homotopic to $\phi$ inside $O$.

In section 3 we prove that there always exists an $m$-filling of $i$ with $O$ in $Const(m)\operatorname{HC}_{m}^{1\over m}(Y)$-neighborhood of $Y$ such that $dist_{B}(i(y),\phi(y))\leq Const(m)\operatorname{HC}_{m}^{1\over m}(Y)$, and $\operatorname{HC}_{m+1}(O)\leq const(m)\operatorname{HC}_{m}^{m+1\over m}(Y)$. Now the general theory of extendability of continuous maps to topological spaces easily implies that given a map of any $X\supset Y$ to $B$ that extends $i$, this map can be altered on $X\setminus Y$ to a map of $X$ into any contractible or even $(\lceil m\rceil-1)$-connected open set $O^{\prime}$ that contains $O$. One can also get a map of $X$ extending $i$ into the union of $O$ and the image of the cone over $K$ under a continuous map to $B.$

Now observe the following corollary of our results in the classical case of codimension one:

Corollary 1.2**.**

Let $M^{n}$ be a closed connected topological submanifold of $(n+1)$-dimensional Banach space $B^{n+1}$. Let $\Omega$ denote the closure of the bounded connected component of $B^{n+1}\setminus M^{n}$. Then for each $m>0$, $\operatorname{HC}_{m+1}(\Omega)\leq const(m)\operatorname{HC}_{m}(M^{n})^{m+1\over m}$.

Proof.

Let $B=B^{n+1}$, $Y=M^{n}$, $X=\Omega$. Observe that the image of any filling of $M^{n}$ by $\Omega$ in $B^{n+1}$ must contain the whole $\Omega$ (as $M^{n}$ is not null-homologous in $B^{n+1}$ minus any point in the interior of $\Omega$). Now the corollary immediately follows from the previous theorem. ∎

Remarks and questions. Observe, that Corollary 1.2 is an analog of the classical (codimension one) isoperimetric inequality. The inequality in Corollary 1.2 is interesting only for $m\leq n$. (If $m>n$, then both sides of the inequality are equal to [math].) So, assume that $m\leq n$. Now the first question is what is the optimal value of $const(m)$? We also can fix $B^{n+1}$ and $m$ and consider the optimal constant $c(B^{n+1},m)$ such that $\operatorname{HC}_{m+1}(\Omega)\leq c(B^{n+1},m)\operatorname{HC}_{m}(M^{n})^{m+1\over m}$ for all open domains $\Omega\subset B^{n+1}$ bounded by a closed hypersurface $M^{n}$. It is interesting to find the exact values of $c(B^{n+1},m)$ for $B^{n+1}=\mathbb{R}^{n+1}$ or $l_{\infty}^{n+1}$. A naive conjecture is that $c(B^{n+1},m)$ is always equal to $1$. In fact, it is possible that the constant $const(m)$ in Theorem 1.1 and $Const(m)$ in its extension (Theorem 3.1) are both equal to $1$.

The reasoning behind this conjecture is that one expect that the optimal constants in isoperimetric inequalities are achieved for metric balls. (One can explore this conjecture also for infinite-dimensional Banach spaces, especially, $l^{\infty}$.) As a first step in this direction we will demonstrate that:

Proposition 1.3**.**

[TABLE]

In particular, if a closed topological manifold $M^{n}$ bounds a domain $\Omega$ in $l^{n+1}_{\infty}$, then $\operatorname{HC}_{n+1}(\Omega)\leq\operatorname{HC}_{n}(M^{n})^{n+1\over n}$, and if $\Omega$ is a coordinate cube, then this inequality becomes an equality.

Similarly, one can investigate the dependence of the constant $C(B,m)=Const(m)$ in the upper bound $Const(m)\ \operatorname{HC}_{m}(Y)^{1\over m}$ for the radius $R$ of $N_{R}(f(Y))$. The case when this Banach space is isomorphic to $l^{\infty}$ is especially interesting. As the metric balls in $l^{\infty}$ are cubes, the most natural conjecture is that $c(l^{\infty},m)=C(l^{\infty},m)=1$. We can be even more specific.

Conjecture 1.4**.**

Let $Y$ be a compact subset of $l^{\infty}$. For each positive $m$ and each positive $\varepsilon$ there exists an $m$-filling $(O,K,\phi)$ of (the inclusion of ) $Y$ in $B$ such that

(a) $\operatorname{HC}_{m+1}(O)\leq\operatorname{HC}_{m}(Y)^{m+1\over m}+\varepsilon$;

*(b) For each $y\in Y$ $\|y-\phi(y)\|_{B}\leq\operatorname{HC}_{m}^{1\over m}(Y)$. *

This conjecture might even be true for all ambient Banach spaces. The complex $K$ might arise as the nerve of a “good” covering of $Y$ by metric balls (or by smaller convex sets). To get a flavour of this conjecture, consider, for example, a round $2$-sphere of radius $1$ in $B=\mathbb{R}^{3}$. If one tries to cover it by very small balls, as one does in order to determine the $2$-dimensional Hausdorff measure, the resulting upper bound for $\operatorname{HC}_{2}$ will be $4$ (the area of the sphere divided by the area of the $2$-disc of radius 1.) However, the covering by one ball of radius $1$ yields an upper bound equal to $1$. And, of course, the sphere contracts inside this ball to a lower dimensional complex (e.g. the center of the ball).

If this conjecture is true, one can use the Kuratowski embedding to conclude that for each integer $m$ and each compact metric space $Y$ there exists a continuous map of $Y$ into an $(m-1)$-dimensional complex $K$ such that the inverse image of each point can be covered by a metric ball of radius $\leq 2\operatorname{HC}_{m}(Y)^{1\over m}$ and that $UW_{m-1}(Y)\leq 2\operatorname{HC}_{m}^{1\over m}(Y)$. This would imply that a closed essential Riemannian manifold $M^{n}$ satisfies the inequality $sys_{1}(M^{n})\leq 4\operatorname{HC}_{n}^{1\over n}(Y)\leq{4\over V_{n}(1)^{1\over n}}vol^{1\over n}(M^{n})=\sqrt{8\over\pi e}\sqrt{n}(1+o(1))vol^{1\over n}(M^{n})$ (cf. [Gr] or [N]). Here $sys_{1}(M^{n})$ denotes the minimal length of a non-contractible closed curve in $M^{n}$. We give Gromov’s definition of essential manifolds in section 1.3 below. Not only the dependence on $n$ is optimal here, but looking at $\mathbb{R}\mathbb{P}^{n}$ with the canonical metric, we see that asymptotically the right hand side this inequality is within the factor of $\pi$ from the optimal value. Thus, optimal (or even nearly optima) “strengthened” isoperimetric inequalities for $l^{\infty}$ and $\operatorname{HC}_{m}$ would lead to a resolution of a long-standing conjecture in systolic geometry that asserts that the dimensional constant $c(n)$ in the right hand side of Gromov’s systolic inequality behaves as $\sim\sqrt{n}$, when $n\longrightarrow\infty$. The combinatorial nature of the problem and the extremely simple form of the conjectured optimal constants suggest that optimal isoperimetric inequalities for $\operatorname{HC}_{m}$ in $L^{\infty}$ might be easier to establish than their classical analogs for the Lebesgue measure.

1.2. Inequalities relating Hausdorff content with Urysohn width.

Let $X$ be a metric space, and $q$ a non-negative integer. One defines $q$-dimensional Urysohn width of $X$ as the infimum of $W$ such that there exist a $q$-dimensional simplicial complex $K$ and a continuous map $\pi:X\rightarrow K$ so that every fiber $\pi^{-1}(s)$ has diameter $\leq W$ in $X$. Intuitively, the Urysohn width $UW_{m-1}(X)$ measures how well metric space $X$ can be approximated by an $(m-1)$-dimensional space.

As a part of our proof of Theorem 1.1 we establish that:

Theorem 1.5**.**

For each $m$ there exists a constant $c(m)$ such that each compact metric space $X$ satisfies the inequality

[TABLE]

This result was first conjectured by L. Guth in [Gu17, Question 5.2]. In fact, Guth conjectured a stronger assertion that was established in the first version of this paper ([LLNR]). Recall, that metric space $X$ is called boundedly compact, if each closed metric ball in $X$ is compact or, equivalently, each closed and bounded subset of $X$ is compact.

Theorem 1.6**.**

For each positive integer $m$ there exists $\varepsilon_{m}>0$, such that the following holds. If $X$ is a boundedly compact metric space and there exists a radius $R$, such that every ball of radius $R$ in $X$ has $m$-dimensional Hausdorff content less than $\varepsilon_{m}R^{m}$, then $UW_{m-1}(X)\leq R$.

As observed by Guth in [Gu17], Theorem 1.5 can be viewed as a quantitative version of the classical Szpilrajn theorem asserting that each compact metric space with zero $m$-dimensional Hausdorff measure has Lebesgue covering dimension $\leq m-1$. Indeed, if the $m$-dimensional Hausdorff measure of $X$ is equal to zero, then, as $\operatorname{HC}_{m}$ does not exceed the $m$-dimensional Hausdorff measure, $\operatorname{HC}_{m}(X)$ also must be equal to zero. Now Theorem 1.5 implies that $UW_{m-1}(X)=0$, which implies that the covering dimension of $X$ is at most $m-1$ (see the proof of Lemma 0.8 in [Gu17]). Also, note that if $X$ is a closed $m$-dimensional Riemannian manifold, then Theorem 1.5 improves the well-known Gromov’s inequality relating the volume of a closed Riemannian manifold and its filling radius (as according to Gromov the filling radius does not exceed ${1\over 2}UW_{m-1}$ – see [Gr], pp. 128-129, where $UW_{m-1}$ is denoted as $Diam^{\prime}_{m-1}$).

Theorem 1.6 generalizes a result of Guth in [Gu17], where $X$ is assumed to be an $m$-dimensional Riemannian manifold, and $\operatorname{HC}_{m}$ replaced by the volume. This result has been previously conjectured by M. Gromov, and for a long time had been an open problem. Guth’s proof is based on a clever construction of a covering by balls with controlled overlap from his previous paper [Gu11] and also uses S. Wenger’s simplified version ([W]) of Gromov’s proof of J.Michael-L.Simon isoperimetric inequality and its generalizations ([Gr]).

If $X$ is compact, then choosing $R=({\operatorname{HC}_{m}(X)\over\varepsilon_{m}^{\prime}})^{1\over m}$ and denoting ${1\over(\varepsilon_{m}^{\prime})^{1\over m}}$ as $c(m)$ we see that Theorem 1.6 implies Theorem 1.5.

The proof of Theorem 1.6 in [LLNR] is considerably longer than the proofs of Theorems 1.1 and 1.5 in the present paper. Soon after [LLNR] was posted on arXiv, Panos Papasoglu learned about it from the talk of one of the authors and our conversations with him during a conference at Barcelona. Later he came up with a much shorter proof of Theorem 1.6 than ours ([P]). While his proof uses some technical and philosophical ideas from our work, its central idea is different. He replaced Gromov’s “chopping off of thin fingers” argument (that we use in the present paper) by an adapatation of Schoen-Yau style approach. (Note that Schoen-Yau approach was first introduced to systolic geometry by Guth; [Gu10]). A later paper of one of the authors (see section 3 of [N]) contains a simplification of Papasoglu’s proof that not only proves Theorem 1.6 on three pages but also yields an explicit linear upper bound for $\varepsilon_{m}^{1\over m}$ (while our work or [P] yielded only exponential upper bounds for $\varepsilon_{m}^{1\over m}$). This makes our original proof of Theorem 1.6 obsolete, and we decided to exclude from the present paper all sections that are not required for the proof of Theorem 1.1 (the proof of Theorem 1.5 is embedded in the proof of Theorem 1.1).

On the other hand, the approach of [P] does not seem applicable to prove our isoperimetric inequality, Theorem 1.1. Moreover, one of the authors tried to combine the idea of [P] with the ideas of the present paper in order to either simplify our proof or obtain a better value of the constant $const(m)$ in Theorem 1.1, but was not able to do that.

1.3. New systolic

inequalities.

We observed that Theorem 1.5 has the following corollary.

We first give the following definition, which generalizes the notion of “essential” polyhedra on p. 139 of [Gr].

Definition 1.7**.**

A CW-complex $X$ is $m$-essential if the classifying map $Q:X\longrightarrow K(\pi_{1}(X),1)$ is not homotopic to a map $Q^{\prime}$ that factors through a map to an $(m-1)$-dimensional $CW$-complex.

[TABLE]

An $n$-dimensional manifold $M$ is called essential, as defined in [Gr], if there exists a coefficient group $G$, such that the homomorphism $Q_{*n}:H_{n}(M^{n};G)\longrightarrow H_{i}(K(\pi_{1}(M^{n}),1);G)$ induced by the classifying map $Q:X\longrightarrow K(\pi_{1}(X),1)$ is non-trivial (that is, has a non-zero image). According to definition above every essential manifold is $n$-essential. Also, to prove that a manifold is $m$-essential it is enough to find a coefficient group $G$ and $i\geq m$, such that one of the homomorphisms $Q_{*i}:H_{i}(X;G)\longrightarrow H_{i}(K(\pi_{1}(X),1);G)$ induced by the classifying map $Q:X\longrightarrow K(\pi_{1}(X),1)$ is non-trivial.

Recall that a metric is called a length metric, if the distance between each pair of points is equal to the infimum of lengths of paths connecting these points, and a metric space such that its metric is a length metric is called a length space.

Theorem 1.8**.**

For each positive integer $m$ there exist a constant $C(m)$ with the following property. Let $X$ be an $m$-essential finite CW-complex endowed with a length metric. Then

[TABLE]

Remarks.

1. Compact $m$-essential Riemannian manifolds with or without boundary of dimension $n\geq m$ constitute the most obvious example of path metric spaces satisfying the conditions of the theorem.

2. If $m$ is the dimension of $X$, then this inequality improves Theorem $B^{\prime}_{1}$ on p. 139 of [Gr] (as $\operatorname{HC}_{n}(X)\leq V_{n}(1)vol(X)$). When $X$ is a closed $m$-dimensional Riemannian manifold, this inequality is a strengthening of the famous Gromov systolic inequality $sys_{1}(M^{n})\leq c(n)vol^{1\over n}(M^{n}).$

3. Observe that if $k<m$ then $HC_{k}^{1\over k}(X)\geq HC_{m}^{\frac{1}{m}}(X),$ (see Lemma 1.10 below). Therefore, disregarding the constants $c(m)$, these inequalities become stronger as $m$ increases. On the other hand, the assumption that $X$ is $m$-essential also becomes stronger when $m$ increases.

We are going to prove this theorem in section 4. The proof combines the ideas from [Gr] with Theorem 1.5.

Examples.

1. If $E^{m}$ is an essential closed manifold (in the sense of [Gr]), then any product $E^{m}\times N^{n-m}$ with a closed manifold, or a connected sum $E^{m}\times N_{1}^{n-m}\#N_{2}^{n}$ is $m$-essential and satisfies the systolic inequality from the previous theorem. For example, if $M^{8}$ is a Riemannian manifold diffeomorphic to $T^{3}\times S^{5}\#S^{4}\times S^{4}$, then $sys_{1}(M^{8})\leq c(3)\operatorname{HC}_{3}(M^{8})^{1\over 3}.$

2. Observe that if $X$ is not $2$-essential, then the classifying map factors through some $1$-complex, and, therefore, $\pi_{1}(X)$ is free. Therefore, if $X$ is a nonsimply-connected compact length space homeomorphic to a finite $CW$-complex, and $\pi_{1}(X)$ is not free, then $sys_{1}(M^{n})\leq C(2)\sqrt{\operatorname{HC}_{2}(M^{n})}.$

The inequalities (*) can be restated as the assertion that there exists a constant $b(m)>0$ such that if $\operatorname{HC}_{m}(X)\leq b(m)$ for an $m$-essential $X$, then $sys_{1}(X)\leq 1$. Of course, Theorem 1.6 implies more, namely, that the assumption about the Hausdorff content of the whole manifold can be replaced by the assumption about all metric balls of radius $1$: Also, as Theorem 1.6 holds for boundedly compact metric spaces, the previous theorem can be immediately generalized for locally finite CW-complexes. In particular, it holds for $m$-essential complete non-compact Riemannian manifolds with or without boundary.

Theorem 1.9**.**

For each integer $m\geq 2$ there exists $b(m)>0$ with the following property: Let $X$ be an $m$-essential boundedly compact length space homeomorphic to a CW-complex. If for some $R>0$ for each metric ball $B$ of radius $R$ in $X$ $\operatorname{HC}_{m}(B)\leq b(m)R^{m}$, then $sys_{1}(X)\leq R$.

Example. Consider a complete Riemannian metric on $T^{2}\times\mathbb{R}^{N}$. If $\operatorname{HC}_{2}$ of each metric ball of radius $1$ does not exceed $b(2)$, then there exists a non-contractible closed curve of length $\leq 1$.

1.4. Hausdorff content: basic properties and some examples.

In this section we list some properties of the Hausdorff content for compact metric spaces that immediately follow from the definition:

1. (1)

Subadditivity. $\operatorname{HC}_{m}(\bigcup_{i}A_{i})\leq\sum_{i}\operatorname{HC}_{m}(A_{i}).$ 2. (2)

Good behaviour under Lipschitz maps. If $f:X\longrightarrow Z$ is an $L$-Lipschitz map, and $Y\subset X$, then $\operatorname{HC}_{m}(f(Y))\leq L^{m}\operatorname{HC}_{m}(Y).$

Applying this to the inclusion map of $Y$ into $X$ (which is, obviously, $1$-Lipschitz) we obtain: 3. (3)

Monotonicity. If $Y\subset X$, then $\operatorname{HC}_{m}(Y)\leq\operatorname{HC}_{m}(X)$. 4. (4)

Rescaling. If $A$ is a subset of a Banach space, and $\lambda$ is a scalar, then $\operatorname{HC}_{m}(\lambda A)=|\lambda|^{m}\operatorname{HC}_{m}(A).$ 5. (5)

For each $m$ we have $\operatorname{HC}_{m}(X)\leq rad^{m}(X)\leq diam^{m}(X)$.

Indeed, a compact metric space $X$ can always be covered by one metric ball of radius $diam(X)$. 6. (6)

Note the following basic inequality relating Hausdorff contents in different dimensions:

Lemma 1.10**.**

If $m>k$ then $\operatorname{HC}_{k}^{1\over k}(X)\geq\operatorname{HC}_{m}^{1\over m}(X)$.

Indeed, choose a finite covering of $X$ by metric balls with radii $r_{i}$ so that $\sum_{i}r_{i}^{k}\leq\operatorname{HC}_{k}(X)+\varepsilon$ for an arbitrarily small $\varepsilon$. Now $\sum_{i}r_{i}^{m}=\sum_{i}(r_{i}^{k})^{m\over k}\leq(\sum_{i}r_{i}^{k})^{m\over k}\leq(\operatorname{HC}_{k}(X)+\varepsilon)^{m\over k}$. Now the desired inequality follows when we take $\varepsilon\longrightarrow 0$. 7. (7)

Consider the Euclidean $2$-ball $B$ of radius $1$. We already know that its $\operatorname{HC}_{1}$ cannot exceed $1$, but, in fact, it is equal to $1$, as the sum of radii of any collection of balls covering a diameter of $B$ (that has length $2$) should be at least $1$. 8. (8)

$\operatorname{HC}_{m}$ is not additive. Indeed, cut the ball $B$ in the previous example into two halves $H_{1}$, $H_{2}$ along a diameter. Note, that $\operatorname{HC}_{1}(H_{i})$ is still equal to $1$. So, when we remove one of these halves, the values of $\operatorname{HC}_{1}$ does not decrease. Moreover, if we will take the union of the remaining half with a metric ball of arbitrarily small radius centered at one of two points of the diameter, the value of $\operatorname{HC}_{1}$ of the union will, in fact, become greater than $1$. This example illustrates difficulties that one can encounter trying to make the value of $\operatorname{HC}_{m}$ of a metric space smaller by cutting out its subsets and replacing them by subsets with a smaller value of $\operatorname{HC}_{m}$. 9. (9)

Let $B_{K}$ be a two-dimensional metric ball of radius $1$ in the hyperbolic space with constant negative curvature $K<<-1$. The area of $B_{K}$ behaves as $\exp(\sqrt{-K})>>1$, yet $\operatorname{HC}_{2}(B_{K})\leq 1<<vol(B_{K}),$ as $B_{K}$ can be covered by just one metric ball of radius $1$. It is obvious that $\operatorname{HC}_{2}(B_{K})=1$, and that if we will cut a concentric metric ball of a small radius $\varepsilon$ out of $B_{K}$, the value of $\operatorname{HC}_{2}$ will not change. (This is another example of non-additivity of $\operatorname{HC}_{m}$, this time in the situation when $m=dim(X)$.) 10. (10)

Let $l^{n}_{\infty}$ denote the $n$-dimensional linear space endowed with the $\max$-norm.

Lemma 1.11**.**

Consider the unit cube $C=[0,1]^{n}$ in $l^{n}_{\infty}$. For each $m\leq n$ $\operatorname{HC}_{m}(C)={1\over 2^{m}}.$

Indeed, observe that the cube is the metric ball of radius ${1\over 2}$ centered at $({1\over 2},\ldots,{1\over 2})$, and so $\operatorname{HC}_{m}(C)\leq({1\over 2})^{m}$.

To prove the opposite inequality consider a covering of $C$ by metric balls with radii $r_{i}$ so that $\sum_{i}r_{i}^{m}\leq\operatorname{HC}_{m}(C)+\varepsilon$ for an arbitrarily small $\varepsilon$. Observe that these metric balls are cubes with side length $2r_{i}$. As the Lebesgue measure of their union cannot be less than the Lebesgue measure of $C$, we conclude that $\sum_{i}(2r_{i})^{n}\geq 1$, whence ${1\over 2^{n}}\leq\sum_{i}(r_{i}^{m})^{n\over m}\leq(\sum_{i}r_{i}^{m})^{n\over m}\leq(\operatorname{HC}_{m}(C)+\varepsilon)^{n\over m}.$ When $\varepsilon\longrightarrow 0$, we obtain the desired inequality.

Corollary 1.12**.**

If $P=[0,r_{1}]\times\ldots\times[0,r_{n}]$, is an $n$-dimensional parallelepiped in $l^{n}_{\infty}$, where $r_{1}\leq\ldots\leq r_{n}$, and $m\leq n$, then $\operatorname{HC}_{m}(P)\geq({r_{1}\over 2})^{m}.$

Indeed, by monotonicity, $\operatorname{HC}_{m}(P)$ is not less than $\operatorname{HC}_{m}$ of the $n$-cube $C\subset P$ with side length $r_{1}$, for which we have $\operatorname{HC}_{m}(C)=({r_{1}\over 2})^{m}.$

1.5. Plan of the paper.

Our proof of the isoperimetric inequality follows Gromov’s approach ([Gr]) with simplifications of Wenger ([W]) and modifications made by Guth ([Gu13]).

The additivity of volume (or, equivalently, Hausdorff measure) is used many times throughout Gromov’s, or Wenger’s, or Guth’s proofs. Our biggest technical problem that the additivity property of the Hausdorff measure does not hold anymore for Hausdorff contents. Of course, the sole reason for failure of additivity for $\operatorname{HC}_{m}$ is the fact that given a cover of the union of two disjoint sets $A$ and $B$ by metric balls, the same metric ball can cover both a part of $A$ and a part of $B$ providing “savings” compared to the situation, when $A$ and $B$ are covered separately.

Our first remedy involves the following observation: Assume that disjoint metric balls $B_{1},\ldots,B_{k}$ have comparable radii (say, all their radii are in the interval $[{s\over 2},s]$ for some $s$), and $\operatorname{HC}_{m}$ of their union is very small in comparison with $s^{m}$ (say, $<({s\over 1000})^{m}$). This means that metric balls used for an almost optimal (from the point of view of $\operatorname{HC}_{m}$) covering of $\bigcup_{i}B_{i}$ can contain only metric balls of radius $\leq{s\over 1000}.$ Now we can just throw away all metric balls in this covering that intersect more than one of the balls $B_{i}$, and observe that the remaining balls still cover $\bigcup_{i}(1-{1\over 250})B_{i}$, and no remaining ball can intersect more than one $B_{i}$. As a result, we can conclude that $\operatorname{HC}_{m}(\bigcup_{i}B_{i})$ is at least as large as $\sum_{i}\operatorname{HC}_{m}(0.996B_{i})$. If, in addition, we know that there is some $\theta>0$ such that $\operatorname{HC}_{m}(0.996B_{i})\geq\theta\operatorname{HC}_{m}(B_{i}),$ then we have the inequality $\operatorname{HC}_{m}(\bigcup_{i}B_{i})\geq\theta\sum_{i}\operatorname{HC}_{m}(B_{i})$ that provides a replacement for the missing additivity. A version of this argument is used in the proof of Proposition 3.3 below.

The proof of the main theorem continues only in section 3. Section 2 is devoted to the proof of the coarea inequality, the cone inequality and a version of Federer-Fleming deformation theorem for compact metric space, continuous functions and the Hausdorff content. These inequalities will then be used in section 3 to prove all our main results.

The cone inequality involves two metric spaces $X$ and $Y\subset X$ inside of a ball of radius $R$ in a Banach space. One wants to construct a continuous map $\psi$ of $X$ into $B$ so that this map remains the identity map on $Y$, yet $\operatorname{HC}_{m+1}(\psi(X))\leq c(m)R\operatorname{HC}_{m}(Y).$ The obvious idea is to try to map $X$ into the cone $CY$ over $Y$ with the tip of the cone at the center of $B(R)$. Yet how can one map $X$ into $CY$? The mapping to the cone would involve mapping at least some neighbourhood of $Y$ in $X$ to $Y$, yet, there does not seem to be any general way to do this. As a result, our “cone” is not $CY$ (although it lies within a small neighbourhood of it).

Also, in section 2 we present a concise proof of a very general version of the coarea inequality for Lipschitz maps in metric spaces and Hausdorff content. Moreover, it is applicable to coverings by arbitrary restricted families of balls, for example, balls with centers in a certain set, or balls of radius bounded above. We believe that our coarea inequality for Hausdorff content and its modifications has applications that go beyond the present paper.

Classical Federer-Fleming deformation theorem allows one to construct a filling of a Lipschitz cycle with controlled mass using repeated coning and projection into skeleta of a fixed cubical grid. It also follows from the construction that the filling lies in a neighbourhood of the cycle of controlled size. We prove a version of this result for extensions of arbitrary continuous map from a compact metric space and Hausdorff content instead of mass. A closely related results can be found in papers of Robert Young ([Y]) and Guth ([Gu13]). The (not very significant) difference between our result and these results is that these authors look for fillings of cycles by arbitrary chains, while we are looking for a filling by a map from a fixed domain.

Section 3 contains our main results, and, in particular, an adaptation of the Michael-Simon-Gromov isoperimetric inequality for Hausdorff contents. The original J. Michael-L. Simon isoperimetric inequality [MS] asserts that an $(n-1)$-dimensional cycle $Y$ in $\mathbb{R}^{N}$ bounds an $n$-chain with $n$-dimensional volume $\leq c(n)vol_{n-1}(Y)^{n\over n-1}$. Gromov proved this using a different method that makes it possible to prove this inequality for $L^{\infty}$ and other Banach spaces instead of $\mathbb{R}^{N}$ ([Gr]). Wenger simplified Gromov’s proof and improved the value of the constant ([W]). In [Gu17] Guth adopted Wenger’s proof to prove a version of the Michael-Simon inequality for maps, where given a $n$-manifold $X$ with boundary $Y$ and its map $f$ into $\mathbb{R}^{N}$ one wants to alter $f$ to a new map $F$ that coincides with $f$ on $Y$ and satisfies the inequality $vol_{n}(F(X))\leq c(n)vol(f(Y))^{n\over n-1}$. Yet a similar theorem valid for maps to all Banach spaces and with a concrete value for constant $c(n)$ is stated (without proof) already in [Gr] in section (A*′′′*) of Appendix 2. In this theorem stated by Gromov both $X$ and $Y$ are assumed to be polyhedra rather than manifolds. In our Theorem 3.1 sets $X$ and $Y$ are assumed to be compact metric spaces, and we use Hausdorff contents instead of Hausdorff measures. Proofs of Gromov, Wenger and Guth use induction on $n$. Our proof also uses induction on $m$, despite the fact that unlike the situation in all these papers, our $m$ has nothing to do with the dimension of the compact metric space $Y$.

Our proof also provides an estimate of how far the filling of $f(Y)$ by $F(X)$ will lie from $f(Y)$ (an analogue of Gromov’s filling radius estimate). Note, that it is not true that for arbitrary $Y$ and $X$ the filling will be in $R$-neighbourhood of $f(Y)$ for $R=const(m)\operatorname{HC}_{m}(Y)^{\frac{1}{m}}$. Yet we prove that a filling by $X$ is possible in any $(\lceil m\rceil-1)$-connected open set containing the $R$-neighbourhood of $f(Y)$ and, moreover, $\operatorname{HC}_{m+1}(F(X))\leq const(m)\operatorname{HC}_{m}(f(Y))^{m+1\over m}$.

We want to follow the approach of [W], [Gu17] that involves a sequence of local improvements of the cycle $Y$ that we want to fill. Assuming that $Y$ is a subset of $B$, and $f$ is the inclusion, we construct a sequence $Y=Y_{1},Y_{2},\ldots,$ of compact metric spaces in $B$ and maps $\psi_{k}:Y_{k}\longrightarrow Y_{k+1}$. The distance between $y\in Y_{k}$ and $\psi_{k}(y)$ is bounded by $const(m)\operatorname{HC}_{m}(Y_{k})^{1\over m}$, and $\operatorname{HC}_{m}(Y_{k})$ exponentially decreases with $k$. Therefore, for each $y\in Y$ the distance between $y$ and the image of $y$ under the composition of $\psi_{i},i\in\{1,\ldots,k\}$ is uniformly bounded by $Const(m)\operatorname{HC}_{m}^{1\over m}(Y)$.

If the ambient space $B$ has a finite dimension $N$, then once $\operatorname{HC}_{m}(Y_{k})$ becomes less than a certain $\varepsilon$ that is allowed to depend also on $N$, one performs a version of Federer-Fleming argument to find the last map $\psi_{\infty}:Y_{k}\longrightarrow Y_{\infty}$ into a $(\lceil m\rceil-1)$-dimensional CW-complex $Y_{\infty}$ so that each $y\in Y_{k}$ and its image under $\psi_{\infty}$ are close.

We explicitly construct extensions of the inclusion $f:Y\to B$ to parts of $X$ so that the images of these maps will fill the “gaps” between $Y_{i}$ and $Y_{i+1}$. The image of one of these parts of $X$ will partially fill between $Y_{k}$ and $Y_{\infty}$. The remaining part of $X$ will be mapped to the cone over $Y_{\infty}$. While we like the constructive nature of this extension process, and it does not make our proof significantly more complicated, we could have just forgotten about $X$, construct an $m$-filling of $Y$ (defined in section 1.1) and use the theory of absolute extensors to conclude that a desired extension to any $X$ always exists.

In this case, $Y_{\infty}$ would play the role of $K$ in the definition of $m$-filling, the composition of all $\psi_{i}$ the role of $\phi$, and an arbitrarily small open neighbourhood of an image of a homotopy between the inclusion $i$ of $Y$ and $\phi$ that naturally appears in our proof the role of $O$.

To “improve” $Y$ (and then $Y_{2}$, $Y_{3}$, etc.) one finds a certain collection of disjoint metric balls, cuts them out, and replaces them by chains with smaller volumes. On the first glance this approach seems absolutely bound to fail in our situation. Indeed, consider examples 8 and 9 in the previous subsection. In both cases we cut out a large piece from some $Y$, yet $\operatorname{HC}$ does not decrease, and, moreover, can increase once one adds a set with an arbitrarily small Hausdorff content.

Yet a remedy exists: We combine the idea explained in (1) that involves using only the metric balls with $\operatorname{HC}_{m}$ that is much less than the $m$th power of the radius with a powerful new idea: Fix an almost optimal (from the point of view of $\operatorname{HC}_{m}$) finite covering $Q$ of $Y$ by metric balls $B_{i}$. We introduce the concept of $m$-dimensional Hausdorff content, $\widetilde{\operatorname{HC}}_{m}$, with respect to the covering $Q$ for subsets of $Y$. $\widetilde{\operatorname{HC}}_{m}$ is defined as $\operatorname{HC}_{m}$ but with the following important difference: This infimum of $\sum_{i}r^{m}$ is taken only over coverings by metric balls from $Q$. (That is, each ball of any covering of a subset of $Y$ that we are allowed to consider must be one of the balls $B_{i}$.) Observe that: (a) $\operatorname{HC}_{m}\leq\widetilde{\operatorname{HC}}_{m}$, so an upper bound for $\widetilde{\operatorname{HC}}_{m}$ is automatically an upper bound for $\operatorname{HC}_{m}$; (b) For $Y$ $\widetilde{\operatorname{HC}}_{m}$ and $\operatorname{HC}_{m}$ are almost the same. Now the idea is to run Wenger’s argument by removing (and then replacing) only metric balls with $\widetilde{\operatorname{HC}}_{m}\sim({r\over A(m)})^{m},$ where $r$ denotes the radius of the considered ball, and $A(m)$ is a sufficiently large constant. (It is not difficult to see that if there are not sufficiently many metric balls with this property to run this argument, then $Y$ has a “round shape” (meaning that $\operatorname{HC}_{m}(Y)^{1\over m}\sim$ the diameter of $Y$ in the ambient metric), so that the isoperimetric inequality for $Y$ follows from our version of the cone inequality.) The basic idea here is that now we are throwing out only “important” balls. Moreover, for each ball $B$ that is being replaced, the balls from $Q$ in the optimal covering of $B$ yielding the value of $\widetilde{\operatorname{HC}}_{m}(B)$ are also important (despite being very small): removal of each of those balls from $Y$ reduces $\widetilde{\operatorname{HC}}_{m}(Y)$ and, therefore, its $\operatorname{HC}_{m}$.

There is an additional difficulty in the proof that arises only in the case when the ambient Banach space is infinite-dimensional. The problem is that our version of Federer-Fleming deformation theorem is proven only for finite-dimensional ambient Banach spaces. Therefore, in order to prove the filling radius estimate we need to apply Federer-Fleming deformation theorem to a part of $Y$ of very small Hausdorff content by projecting it onto a finite dimensional subspace $S^{\prime}$ of the Banach space. By Kadec-Snobar theorem we can bound the Hausdorff content of the projection in terms of the dimension of $S^{\prime}$. For the argument to work we need to have an a priori bound for the dimension of $S^{\prime}$ and also know that the small part of $Y$ will lie close to $S^{\prime}$. To accomplish this we fix an almost optimal covering of $Y$ in the beginning of the argument and then choose $S^{\prime}$ that contains all centres of balls in the covering. We then carry out the inductive construction for a version of Hausdorff content measured using ball with centres in $S^{\prime}$.

Finally, we observe that our proof of Theorem 1.1 yields not only Theorem 1.1 but also Theorem 1.5.

The last section contains the proof of the inequality $sys_{1}(M^{n})\leq 3UW_{m-1}(M^{n})$ for $m$-essential manifolds that combined together with Theorem 1.5 yields Theorem 1.8. Our proof is similar to Gromov’s proof of the inequality $sys_{1}(M^{n})\leq 6FillRad(M^{n})$ in [Gr].

2. Cone, coarea and Federer-Fleming inequalities for Hausdorff content

Recall the notation for the infimal Hausdorff content of an extension of a map between metric spaces we introduced earlier. Given metric spaces $X,Y$, subsets $A\subset X$, $U\subset Y$ and a map $f:A\rightarrow U$ define $\mathcal{F}_{m}(f,X,U)=\inf\{\operatorname{HC}_{m}(F(X))\}$, where the infimum is taken over all maps $F:X\rightarrow U$, such that $F|_{A}=f$. If no such extension $F$ exists we set $\mathcal{F}_{m}(f,X,U)=\infty$.

We will also need a version of $\mathcal{F}$ where Hausdorff content is measured with respect to a collection of balls whose centers are contained in some fixed subset $W$. Let $V,W\subset X$ be subsets of a metric space $X$. Define $\operatorname{HC}_{m}(V;W)$ to be the infimum of $\sum_{i}r_{i}^{m}$ among all coverings of $V$ by countably many balls $B(x_{i},r_{i})$ with $x_{i}\in W$. We set $\mathcal{F}_{m}(f,X,U;W)=\inf\{\operatorname{HC}_{m}(F(X);W)\}$, where the infimum is taken over all maps $F:X\rightarrow U$, such that $F|_{A}=f$, and $\mathcal{F}_{m}(f,X,U;W)=\infty$ if there is no extension.

Lemma 2.1** (Cone inequality).**

Let $B=B(p,R)$ be a closed metric ball of radius $R$ in a Banach space $S$, $W$ a linear subspace of $S$, $U$ a closed convex subset of $B(p,R)$ with $p\in W\cap U$. Suppose $X$ is a metric space, $Y\subset X$ closed subset and $\tau:Y\longrightarrow U$ is a continuous map with $\tau(Y)$ compact. Then

[TABLE]

More generally,

[TABLE]

Moreover, if $X\subset B$ and $\tau:Y\longrightarrow U$ is the inclusion map, $\varrho>0$, then for every $\delta>0$ there exists a map $F:X\longrightarrow U$ with

[TABLE]

and $F(X\setminus N_{\varrho}(Y))\subset\{p\}$.

Here $e=\lim_{m\rightarrow\infty}(1+\frac{1}{m})^{m}=2.718\ldots$ is the Euler’s number.

Proof.

We will prove the more general upper bound for $\mathcal{F}_{m}(\tau,X,U;W)$. By taking $W=S$ we get the bound for $\mathcal{F}_{m}(\tau,X,U)$.

First we observe that the Lemma follows from its special case when $X$ is a subset of $U$, and $\tau$ is the inclusion map (so $Y$ is a compact subset of $U$). Indeed, since $U$ is convex, by a generalization of Tietze extension theorem proven by Dugundji (see [Du, Theorem 4.1]) there exists an extension $F^{\prime}:X\longrightarrow U$ of $\tau:Y\longrightarrow U$. If for every $\varepsilon>0$ there exists a map $\Psi:F^{\prime}(X)\longrightarrow U$ that is equal to the identity on the set $\tau(Y)$ and with $\operatorname{HC}_{m}(\Psi(F^{\prime}(X)))\leq L+\varepsilon$, then by considering composition $\Psi\circ F^{\prime}$ we obtain $\mathcal{F}_{m}(\tau,X,U)\leq L$. Hence, without any loss of generality we may assume $Y\subset X\subset U$ and that $Y$ is compact.

Let $\{B_{i}(r_{i})\}_{i\in I}$ be a finite covering of $Y$ by closed metric balls with centers in $W$, such that $\sum_{i}r_{i}^{m-1}\leq\operatorname{HC}_{m-1}(Y;W)+\varepsilon$ for some $\varepsilon$ that can be taken arbitrarily small. Without any loss of generality we can assume that the union of these balls covers not only $Y$ but an open neighbourhood $N_{r}(Y)$ of $Y$ for a very small positive radius $r\leq\min_{i}r_{i}$. We take $r<\varrho$ from the statement of the theorem and $r<\varepsilon$.

For each set $V$ let $Cone_{p}(V)$ denote the union of the closed straight line segments in the ambient Banach space connecting $p$ with all points of $V$.

Observe, that for each metric ball $B_{i}(r_{i})$ the set $Cone_{p}(B_{i}(r_{i}))$ can be covered by at most $m{R\over r_{i}}$ metric balls of radius $(1+{1\over m})r_{i}$ with centers in $W$. Indeed, the first of these balls is $B_{i}((1+{1\over m})r_{i})$; the centers of all subsequent balls are spaced along the segment connecting the center $q_{i}$ of $B_{i}(r_{i})$ and $p$ at distances ${r_{i}\over m}$ apart. For every $t\in[0,1]$ the distance between $tq_{i}$ and the center point $y$ of one of these balls is at most ${r_{i}\over 2m}$. By triangle inequality, if $x\in B_{i}(r_{i})$ then $dist(tx,y)\leq tr_{i}+{r_{i}\over 2m}\leq(1+{1\over m})r_{i}$.

Therefore, $Cone_{p}(N_{r}(Y))$ can be covered by a collection of at most $\sum_{i}m{R\over r_{i}}$ metric balls with $m$ balls of radius $(1+{1\over m})r_{i}$ for each $i$, where $i$ ranges over $I$. Therefore, $\operatorname{HC}_{m}(Cone_{p}(N_{r}(Y);W)\leq\sum_{i}{mR\over r_{i}}((1+{1\over m})r_{i})^{m}\leq m(1+{1\over m})^{m}R(\operatorname{HC}_{m-1}(Y;W)+\varepsilon),$ where $\varepsilon$ is arbitrarily small.

Let $\phi:[0,\infty]\rightarrow[0,1]$ be a continuous monotone function with $\phi(0)=1$ and $\phi(t)=0$ for all $t\geq r$ defined as $1-{1\over r}x$ for $x\in[0,r]$ and zero for all $x\in[r,\infty)$. We define a map $\Phi:U\rightarrow Cone_{p}(N_{r}(Y)\cap U)$ by the formula $\Phi(x)=\phi(dist(x,Y))x+(1-\phi(dist(x,Y))p$.

We set $Z=\Phi(X)$. As $Z\subset Cone_{p}(N_{r}(Y))$, we have $\operatorname{HC}_{m}(Z)\leq\operatorname{HC}_{m}(Cone_{p}(N_{r}(Y);W)\leq m(1+{1\over m})^{m}R(\operatorname{HC}_{m-1}(Y;W)+\varepsilon)$ for arbitrarily small $\varepsilon$. Also, it is clear from the definition of $\Phi$ that $\Phi(x)=p$ if $dist(x,Y)\geq r\geq\varrho$. ∎

Remark. We can improve the inequality to $\mathcal{F}_{m}(\tau,X,U;W)\leq const\ R\ \operatorname{HC}_{m-1}(Y;W)$ where $const$ is an absolute constant that does not depend on $m$ by using balls of radii $(1-{j\over m})(1+{1\over m})r_{i}$, $j=0,\ldots,m-1$ to cover $Cone_{p}(B_{i}(r_{i}))$ instead of the balls with identical radii $(1+{1\over m})r_{i}$. (The new balls are centered at the same points as the old balls.) As the result, we improve multiplicative dimensional constants in the right hand sides of all inequalities in the previous lemma from $m(1+{1\over m})^{m}$ to $2(1+{1\over m})^{m}<2e<6$.

Observe that $Z$ in the proof of Lemma 2.1 is not a cone over $Y$, although it is very close to $Cone_{p}(Y)$. In fact, we can replace the chosen value of $r$ by any smaller positive value, and the proof would still work. So, if desired we could choose $Z$ arbitrarily close to an actual cone over $Y$ in $S$. Below we will be referring to $Z$ as the cone over $Y$, and will call this construction the coning of Y.

We will also need the following co-area formula for Hausdorff content. Let $X$ be a metric space, and $\Gamma$ denote a set of closed metric balls that together cover $X$. Three main examples of $\Gamma$ are: 1) All metric balls; 2) all metric balls with centers contained into some fixed subset $W$ of $X$; 3) A specific covering of $X$ by a family of closed metric balls; 4) All metric balls of radius $\leq R$, where $R$ is some constant. For $V\subset X$ we denote $\operatorname{HC}_{m}(V;\Gamma)$ to be the infimum of $\sum_{i}r_{i}^{m}$ among all coverings of $V$ by countably many balls $B(x_{i},r_{i})$, where $B(x_{i},r_{i})\in\Gamma$. In the case 1) $\operatorname{HC}_{m}(V;\Gamma)=\operatorname{HC}_{m}(V)$, in the case 2) $\operatorname{HC}_{m}(V;\Gamma)=\operatorname{HC}_{m}(V;W)$, in the case 3) when a covering of $X$ is fixed, we will be denoting $\operatorname{HC}_{m}(V;\Gamma)$ as $\widetilde{\operatorname{HC}}_{m}(V)$.

Lemma 2.2**.**

(Co-area inequality) Let $U$ be a compact set in an arbitrary metric space $X$, $\Gamma$ a family of closed metric balls that covers $X$, and $f:U\longrightarrow[R_{1},R_{2}]$ a Lipschitz function with Lipschitz constant $Lip(f)$. Then,

[TABLE]

where $\int^{*}$ denotes the upper Lebesgue integral. Therefore, there exists $R\in[R_{1},R_{2}]$, such that $\operatorname{HC}_{m-1}(f^{-1}(R);\Gamma)\leq\frac{2\ Lip(f)}{R_{2}-R_{1}}\operatorname{HC}_{m}(U;\Gamma)$.

Proof.

Let $\{B_{r_{i}}(p_{i})\}$, $B_{r_{i}}(p_{i})\in\Gamma$, be a covering of $U$ with $\sum_{i}r_{i}^{m}\leq\operatorname{HC}_{m}(U;\Gamma)+\varepsilon$, where $i\in\{1,\ldots,N\}$ for some $N$, and $\varepsilon$ can be chosen arbitrarily small. The desired inequality would follow from the inequality $\int^{*R_{2}}_{R_{1}}\operatorname{HC}_{m-1}(f^{-1}(R);\Gamma)\ dR\leq 2Lip(f)\sum_{i}r_{i}^{m}$. We are going to prove a stronger inequality, where $\operatorname{HC}_{m-1}(f^{-1}(R);\Gamma)$ is replaced by the following quantity that is obviously not less than $\operatorname{HC}_{m-1}(f^{-1}(R);\Gamma)$, namely, $\sum_{i\in I(R)}r_{i}^{m-1}$, where $I(R)$ denotes the set of all indices $i$ such that the intersection of $B_{r_{i}}(p_{i})$ and $f^{-1}(R)$ is not empty. The left hand side of the desired inequality becomes

[TABLE]

where the characteristic function $\chi_{i}(R)$ is equal to $1$ for all $R\in[R_{1},R_{2}]$ such that $f^{-1}(R)$ and $B_{r_{i}}(p_{i})$ have a non-empty intersection, and to [math] otherwise.

Finally, observe that if $\chi_{i}(\varrho_{1})=1$ and $\chi_{i}(\varrho_{2})=1$, then

[TABLE]

It follows that $\int_{R_{1}}^{R_{2}}\chi_{i}(R)dR\leq 2Lip(f)r_{i}$, which implies the desired inequality. ∎

In particular, for a family of equidistant surfaces $\{\Sigma_{R}\}$, $dist(\Sigma_{t},\Sigma_{s})=|t-s|$, in a finite dimensional Banach space we obtain the following version of co-area inequality.

Lemma 2.3**.**

(Co-area inequality 2) Let $\{\Sigma_{R}\}$ be a family of equidistant surfaces in a Banach space $S$, $W\subset S$ any subset, $U\subset S$ a compact set. Then,

[TABLE]

Therefore, there exists $R\in[R_{1},R_{2}]$, such that $\operatorname{HC}_{m-1}(\Sigma_{R}\cap U;W)\leq\frac{2}{R_{2}-R_{1}}\operatorname{HC}_{m}(U;W)$.

Example. Consider the case, when $\Sigma_{R}$ are metric spheres $S_{R}$ (= sets of points at the distance $R$ from a point $p\in X$). Taking $W=X$, we see that

[TABLE]

where $A_{R_{1},R_{2}}$ denotes the set of points $x$ such that $dist(x,p)\in[R_{1},R_{2}]$.

We will also need a version of the Federer-Fleming deformation theorem from [FF] for Hausdorff content. Instead of constructing an extension of $f$ into a cone over $Y$ this argument gives an extension of $f$ into a portion of the cone between $Y$ and a lower dimensional skeleton of a fixed cubical complex, where the vertex of the cone is chosen using an averaging argument. Pushing out in this way repeatedly one eventually constructs an isoperimetric extension of $f$ whose image outside of a neighbourhood of $Y$ of controlled size is contained in the $(m-1)$-skeleton of the cubical complex. For controlling Hausdorff content of cycles this construction appeared in [Gu13, Proposition 7.1] and [Y, Lemma 2.5]. Our proof follows the same strategy as the proofs in [Gu13] and [Y]. The difference is that instead of constructing an $m$-chain filling an $(m-1)$-dimensional cycle we construct an extension of an arbitrary continuous map $f$ from a subset $Y$ of a metric space $X$.

Let $Q(r)$ denote the cubical complex in $\mathbb{R}^{n}$ whose $n$-dimensional faces are $\{C(a_{1},...,a_{n})=\{[a_{1}r,(a_{1}+1)r]\times...\times[a_{n}r,(a_{n}+1)r];a_{i}\in\mathbb{Z}\}$, and let $Q(r)^{(k)}$ denote the $k$-skeleton of $Q(r)$, $0\leq k\leq n$.

Let $\lceil m\rceil$ denote the smallest integer that is greater than or equal to $m$.

Lemma 2.4** (Federer-Fleming type inequality).**

Let $X$ be a metric space, $Y\subset X$ and $f:Y\longrightarrow\mathbb{R}^{n}$ continuous map with $f(Y)$ compact. Then for every $m\in[2,n]$ there exists a constant $c_{1}(n)>0$, such that

[TABLE]

Moreover, there exists a constant $c_{2}(n)$, such that for every $\delta>0$ and $R=c_{2}(n)\operatorname{HC}_{m-1}(f(Y))^{\frac{1}{m-1}}+\delta$ we have

[TABLE]

Proof.

As in the proof of Lemma 2.1 we may assume, without any loss of generality, that $Y\subset X\subset\mathbb{R}^{n}$, $Y$ is compact and $f$ is the inclusion map. Fix small $\delta>0$. Let $Q(R)$ denote a cubical complex as defined above. We will show that for some constant $c_{2}(n)$ and $R=c_{2}(n)\operatorname{HC}_{m-1}(f(Y))^{\frac{1}{m-1}}+\delta$ the statement of the Lemma holds.

We can define an open set $Y^{\prime}$ with smooth boundary, a very small $\tau>0$ and $Z=\overline{N_{\tau}(Y^{\prime})\setminus Y^{\prime}}\cong\partial Y^{\prime}\times[0,1]$, such that $Y\subset Y^{\prime}$ and $\operatorname{HC}_{m-1}(N_{\tau}(Y^{\prime}))<\operatorname{HC}_{m-1}(Y)+\delta$. Indeed, this follows by slightly thickening an almost optimal covering of $Y$ by balls.

We will define a map $\Psi:\overline{N_{\tau}(Y^{\prime})}\rightarrow\mathbb{R}^{n}$, such that the following holds:

• (a)

$\Psi(x)=x$ for $x\in Y^{\prime}$;

• (b)

$\Psi(\partial N_{\tau}(Y^{\prime}))\subset Q(R)^{(\lceil m\rceil-2)}$;

• (c)

$dist(x,\Psi(x))\leq const(n)R$;

• (d)

$\operatorname{HC}_{m}(\Psi(\overline{N_{\tau}(Y^{\prime})}))\leq const(n)R(\operatorname{HC}_{m-1}(Y)+\delta)\leq c_{1}(n)\operatorname{HC}_{m-1}(f(Y))^{\frac{m}{m-1}}+\delta$.

Since \Psi\big{(}\partial(\mathbb{R}^{n}\setminus N_{\tau}(Y^{\prime})\big{)}\subset Q(R)^{(\lceil m\rceil-2)}, we can extend $\Psi$ to a map from $\mathbb{R}^{n}$ to $Q(R)^{(\lceil m\rceil-1)}\cup\Psi(N_{\tau}(Y^{\prime}))$. Restricting $\Psi$ to $X\subset\mathbb{R}^{n}$ gives us the desired extension of $f$.

To define map $\Psi$ on the set $Z\cong\partial Y^{\prime}\times[0,1]$ we will define a sequence of homotopies of $\partial Y^{\prime}$ into successively smaller dimensional skeleta $Q(R)^{(k)}$, $k=n,...,\lceil m\rceil-2$.

Let $C$ be a $k$-face in $Q(R)^{(k)}$, $p\in C$ and let $P_{C,p}:C\setminus\{p\}\rightarrow\partial C$ denote the radial projection map. Suppose $k\geq\lceil m\rceil-1$ and $V\subset C$. By the “pushout from an average point” argument [Gu13, Lemma 7.2] there exists a positive constant $c_{0}(k)$, such that if

[TABLE]

then there exists a point $p\in C\setminus V$, such that $\operatorname{HC}_{m-1}(P_{C,p}(V))\leq const(k)\operatorname{HC}_{m-1}(V)$. Define a homotopy $\Phi_{p,V}:[0,1]\times V\rightarrow C$ given by

[TABLE]

By cone inequality we then have

[TABLE]

Note that the homotopy $\Phi_{p,V}$ is constant on $V\cap\partial C$.

We can define homotopies $\Phi_{k}:[0,1]\times V_{k}\rightarrow Q(R)^{(k)}$, $k=n,...,\lceil m\rceil-1$ as follows. We set $V_{n}=\partial Y^{\prime}$ and $V_{k}=\Phi_{k+1}(1,V_{k+1})\subset Q(R)^{(k)}$ for $k<n$. On each $k$-face $C$ of $Q(R)^{(k)}$ define $\Phi_{k}=\Phi_{p(V_{k}\cap C),V_{k}\cap C}$, where the point $p(V_{k}\cap C)\in C$ is chosen as described above. We claim that $V_{k}\cap C$ satisfies (2.1), so $\Phi_{k}$ is well-defined. Note that $\Phi_{k}$ is continuous on all of $V_{k}$ since the homotopy is constant on the boundary of $C$.

To prove the claim we observe the following. Let $\{\beta(p_{j},r_{j})\}$ denote a nearly optimal covering of $V_{k}$, that is $\sum_{j}r_{j}^{m-1}\leq\operatorname{HC}_{m-1}(V_{k})+\delta$. If $\operatorname{HC}_{m-1}(V_{k})\leq R^{m-1}$, then each ball $\beta(p_{j},r_{j})$ intersects at most $const(n)$ faces of $Q(R)^{(k)}$. Hence, if $\{C_{m}\}$ are the $k$-faces of $Q(R)^{(k)}$, then

[TABLE]

Choosing $c_{2}(n)$ in the definition of $R$ sufficiently large we can ensure that inequality (2.1) is satisfied if we replace $V$ with $V_{k}\cap C$ for each $k=n,...,\lceil m\rceil-1$ and each face $C$ of $Q(R)^{(k)}$. With this choice of $c_{2}(n)$ we have that homotopies $\Phi_{k}$ are well-defined for $k=n,...,\lceil m\rceil-1$.

Performing homotopies $\Phi_{k}$ we obtain a homotopy of $\partial Y^{\prime}$ into $Q(R)^{(\lceil m\rceil-2)}$. This defines map $\Psi$ on $Z$ and define $\Psi$ to be the identity on $Y^{\prime}$. It follows from the construction that condition $(a)-(d)$ are satisfied. ∎

Observe that in the course of the proof of the previous proposition we also established (cf. [Gu13, Proposition 7.1]) that:

Lemma 2.5**.**

Let $Y\subset\mathbb{R}^{n}$ be compact, $m\in[1,n-1]$. There exists a compact $(\lceil m\rceil-1)$-simplicial complex $Q\subset\mathbb{R}^{n}$ and a map $\varrho:Y\longrightarrow Q$ such that for each $y\in Y$ $dist(y,\varrho(y))\leq const(n)\operatorname{HC}_{m}(Y)^{1\over m}$.

Here $m$ corresponds to $m-1$ form the previous lemma; $Q$ is a bounded subcomplex of $Q(R)^{(\lceil m\rceil-1)}$.

Also, for every $n$-dimensional normed linear space $L$ there exists a linear isomorphism $T:L\longrightarrow\mathbb{R}^{n}$ such that $\|T\|\|T^{-1}\|\leq\sqrt{n}$ (F. John’s theorem). Therefore, two previous lemmae can be imeediately generalized to an arbitrary $n$-dimensional ambient normed linear space instead of $\mathbb{R}^{n}$.

3. Isoperimetric extension inequality

In this section we prove the isoperimetric inequality Theorem 1.1. Recall that given $Y\subset X$ and a map $f:Y\rightarrow U$ we denote by $\mathcal{F}_{m}(f,X,U)$ the infimum of $\operatorname{HC}_{m}(F(X))$ over all extensions $F:X\rightarrow U$ of the map $f$ and by $\mathcal{F}_{m}(f,X,U;W)$ the analogous infimum of $\operatorname{HC}_{m}(F(X);W)$, where Hausdorff content is measured with respect to balls whose center lies in $W$. The following result can be thought of as a version of an isoperimetric and filling radius inequalities in Banach spaces (compare with Theorem A*′′′* in Appendix 2 of [Gr]).

Recall, that a CW complex $P$ is called $k$-connected if $P$ is connected and $\pi_{j}(P)=0$ for $j=1,...,k$.

Theorem 3.1**.**

Let $m\in(1,\infty)$, $S$ be a Banach space, $U\subset S$ a closed ball, $X$ metric space and $Y\subset X$ closed subspace. Suppose $f:Y\longrightarrow U$ is a continuous map with $f(Y)$ compact. There exists constant $I_{1}(m)>0$, such that

[TABLE]

Moreover, we have the following estimate for the filling radius. There exists a constant $I_{2}(m)>0$ with the following property. For every $\delta>0$ let $R=I_{2}(m)\operatorname{HC}_{m-1}(f(Y))^{1\over m-1}+\delta$ and let $\varepsilon>0$ be arbitrary. There exists an extension $F:X\longrightarrow U$ of $f$, such that

[TABLE]

and an $(\lceil m\rceil-1)$-dimensional CW complex $W\subset U$, such that $F(X)\setminus N_{R}(f(Y))\subset W$. If $N_{R}(f(Y))\subset P\subset U$, where $P$ is an $(\lceil m\rceil-2)$-connected open set, then we can ensure that $W\subset P$, and, therefore,

[TABLE]

Our proof yields explicit values for constants $I_{1}(m)$ and $I_{2}(m)$. One can take $I_{1}(m)=(100m)^{m}$ and $I_{2}(m)=(1500m)^{m}$. Here $S$ may be infinite dimensional and $m$ non-integer. Observe, that constant $\varepsilon$ and $\delta$ are required in this theorem only for the case, when $\operatorname{HC}_{m-1}(Y)=0$. If $\operatorname{HC}_{m-1}(Y)\not=0$, then $\varepsilon$ and $\delta$ can be omitted from the formulae in the text of the theorem.

Proof of Theorem 3.1. First observe that the last sentence of Theorem 3.1 follows from the second last sentence. Indeed, if $W$ is $(\lceil m\rceil-1)$-dimensional CW complex in $U$ and $P$ is $(\lceil m\rceil-2)$-connected, then there exists a Lipschitz map $F^{\prime}:W\longrightarrow P$ that is equal to the identity on $P\cap W$. The statement follows by composing $F^{\prime}$ with $F$.

For an arbitrarily small $\varepsilon>0$ consider a covering of $Y$ by finitely many balls $\{\beta(q_{i},\varrho_{i})\}$, such that $\sum\varrho_{i}^{m-1}\leq\operatorname{HC}_{m-1}(Y)+\varepsilon$. Let $S^{\prime}\subset S$ be a finite-dimensional Banach space that contains $\bigcup q_{i}$ and the center of ball $U$.

We fix $S^{\prime}$ for the rest of the proof. The reason why we need $S^{\prime}$ is the following. Assume we proved the result for $\operatorname{HC}_{l}$ with $l\leq m$ and we want to prove it for $l=m+1$. Similarly to the argument of Wenger, using repeated coning in a certain carefully chosen collection of balls $\{B_{j}\}$ and inductive assumption applied to $\partial B_{j}\cap Y$ we will reduce the extension problem to the case when $Y$ has arbitrarily small Hausdorff content. In a similar situation Wenger [W] also applies coning to finish the proof. In our case coning will not give us the desired bound for the filling radius. Instead, we use projection onto $S^{\prime}$. By the Kadec-Snobar theorem this projection will increase the Hausdorff content at most by a factor of $const(dimension(S^{\prime}))$. We then apply Federer-Fleming extension Lemma 2.4 (see subsection 3.3). In order for this argument to work we need to fix finite dimensional subspace $S^{\prime}$ in the beginning of the proof.

Observe that to prove Theorem 3.1 it is now enough to construct an extension $F$ of the map $f$, such that $F(X)\subset(N_{R}(Y)\cup W)\cap U$, where $W$ is some $(\lceil m\rceil-1)$-dimensional CW complex, for $R\leq I_{2}(m)\operatorname{HC}_{m-1}(Y;S^{\prime})^{\frac{1}{m-1}}+\delta$ and $\operatorname{HC}_{m}(F(X))\leq I_{1}(m)\operatorname{HC}_{m-1}(f(Y);S^{\prime})^{m\over m-1}+\varepsilon$. In particular, we can state the following Proposition that implies Theorem 3.1.

Proposition 3.2**.**

Let $m\in(1,\infty)$, $S$ be a Banach space, $U\subset S$ a closed ball, $S^{\prime}\subset S$ a finite-dimensional linear subspace, $X$ metric space, $Y\subset X$ closed subset and $f:Y\longrightarrow U$ continuous map with $f(Y)$ compact.

There exist constants $I_{1}(m)>0$ and $I_{2}(m)>0$, such that for all $\delta>0$ and $\varepsilon>0$ the following holds. Let $R=I_{2}(m)\operatorname{HC}_{m-1}(f(Y);S^{\prime})^{1\over m-1}+\delta$. There exists an extension $F:X\longrightarrow U$ of $f$, such that

[TABLE]

and an $(\lceil m\rceil-1)$-dimensional CW complex $W\subset U$, such that $F(X)\setminus N_{R}(f(Y))\subset W$.

The rest of the proof will be devoted to demonstrating Proposition 3.2. We will proceed by induction on $m$.

3.1. The base case $m\in(1,2]$.

We can apply a well-known generalization of Tietze extension theorem to locally convex target spaces (see [Du, Theorem 4.1]) to find a map $\Phi:X\longrightarrow U$ that coincides with $f$ on $Y$. As in the proof of Theorem 2.1 we observe that Proposition 3.2 immediately follows from its particular case, when $X\subset U$, as we can just apply this particular case to the sets $Y^{\prime}=f(Y)$ and $X^{\prime}=\Phi(X)$ and the inclusion map $f^{\prime}:f(Y)\hookrightarrow U$ and then compose the resulting map that has small Hausdorff content of the image with $\Phi$. So without any loss of generality we assume in this subsection that $Y\subset X\subset U$ and $f$ is the inclusion.

We start from a general overview of our strategy. We are going to partition $X$ into several pieces and define maps from each piece to $U$ that can be combined into a single continuous map $F$ from $X$ to $U$ and will have the desired properties.

All but one of these “pieces” will be small closed neighbourhoods of some disjoint closed metric balls in $S$ with centers in $S^{\prime}$ providing an almost optimal covering of $Y$ (from the perspective of $\operatorname{HC}_{m-1}$ with centers in $S^{\prime}$). These pieces will be mapped using the coning construction, so that their boundaries will be mapped to points $p_{i}$. Finally, it would remain to map the last “piece”, i.e. the closure of the complement to the (finite) union of all previously described “pieces”. The map on its boundary is already defined; it sends the boundary to a finite collection of points, and one can extend it to a map into an arc connecting all these points using Tietze extension theorem.

Let $\{B_{i}\}_{i=1}^{N}$ be a finite collection of closed metric balls covering $Y$ with centers in $S^{\prime}$ and $\sum r_{i}^{m-1}\leq\operatorname{HC}_{m-1}(Y;S^{\prime})+\delta$. Without any loss of generality we may assume that each $B_{i}$ contains a point of $Y$.

Observe, that, as $m-1\leq 1$, $\sum r_{i}^{m-1}\geq(\sum_{i}r_{i})^{m-1}$. Using this observation, it is straightforward to prove by induction on the number of balls in $N$ in the covering that there exists another covering $\{\beta_{j}\}_{j=1}^{N^{\prime}}$, $N^{\prime}\leq N$ with centers in $S^{\prime}$ such that balls $\beta_{j}$ are pairwise disjoint and, still, $\sum R_{i}^{m-1}\leq\operatorname{HC}_{m-1}(Y;S^{\prime})+\delta$, where $R_{i}$ denotes the radius of $\beta_{i}$. Indeed, if two balls $B(p_{k},r_{k})$, $B(p_{l},r_{l})$ of the covering intersect, consider the ball with the radius $r_{k}+r_{l}$ centered at the point $p$ on the straight line segment $[p_{k}p_{l}]$ at the distance $r_{k}$ from $p_{2}$. It is obvious, that $p\in S^{\prime}$ and $B(p,r_{k}+r_{l})$ contains both balls $B(p_{k},r_{k})$, $B(p_{l},r_{l})$ . On the other hand $(r_{k}+r_{l})^{m-1}\leq r_{k}^{m-1}+r_{l}^{m-1},$ so we can replace both this balls by $B(p,r_{k}+r_{l})$. Now we can repeat this procedure until all balls of the covering will become disjoint.

Now denote centers and radii of $\beta_{j}$ as $p_{j}$ and $R_{j}$. Choose a small positive $\varrho$ such that the slightly larger concentric closed balls $B(p_{j},R_{j}+\varrho)$ still do not pairwise intersect.

Consider a map from $\partial B(p_{j},R_{j}+\varrho)\cup(Y\cap\beta_{j})$ to $\beta_{j}$ that sends $\partial B(p_{j},R_{j}+\varrho)$ to $p_{j}$ and is the identity map on $(Y\cap\beta_{j})$. (Observe that our assumption about $\varrho$ implies that $Y\cap\beta_{j}=Y\cap B(p_{j},R_{j}+\varrho)$.) Cone inequality Lemma 2.1 implies that this map can be continuously extended to the map $B(p_{j},R_{j}+\varrho)\longrightarrow\beta_{j}$. Denote the restriction of this map to $X\cap B(p_{j},R_{j}+\varrho)$ by $F_{j}$. By construction $F_{j}$ sends $X\cap\partial B(p_{j},R_{j}+\varrho)$ to the center $p_{j}$ of $\beta_{j}$. Note that $\operatorname{HC}_{m-1}$ of the image of $F_{j}$ cannot exceed $\operatorname{HC}_{m-1}(\beta_{j})\leq R_{j}^{m-1}$. Further, $\operatorname{HC}_{m-1}(\cup_{j}F_{j}(X\cap B(R_{j}+\varrho))\leq\sum_{j}R_{j}^{m-1}\leq\operatorname{HC}_{m-1}(Y)+\delta$. Also, each point of $\cup_{j}F_{j}(X\cap B(R_{j}+\varrho))$ is $(R_{j}+\varrho)$-close to one of the points $p_{j}$. As $R_{j}\leq\sum_{j}R_{j}\leq(\sum_{j}R_{j}^{m-1})^{1\over m-1}\leq(\operatorname{HC}_{m-1}(Y)+\delta)^{1\over m-1}$, one can choose a sufficiently small $\delta$ such that $\cup_{j=1}^{N^{\prime}}F_{j}(X\cap B(R_{j}+\varrho))$ is $(\operatorname{HC}_{m-1}^{1\over m-1}(Y)+\varepsilon)$-close to the [math]-dimensional complex $\{p_{1},\ldots,p_{N^{\prime}}\}$.

Connect $p_{1}\in S^{\prime}$ to points $p_{2}$, $p_{3}$, etc, by straight line segments so that we obtain a finite contractible 1-complex $W$. The map $\cup_{j}X\cap\partial B(p_{j},R_{j}+\varrho)\longrightarrow\cup_{j}p_{j}$ can be continuously extended to a map $F_{N^{\prime}+1}:X\setminus\cup_{j=1}^{N^{\prime}}int(B(p_{j},R_{j}+\varrho)\longrightarrow W$. (Here $int(B)$ denotes the interior of $B$; recall that all our balls are closed.) Indeed, one can choose a small $\mu$ such that all closed balls $B(p_{j},R_{j}+\varrho+\mu)$ are disjoint. We can map the part of $X$ outside of the union of these balls to $p_{1}$. Each point $x$ of $X$ in the annulus $B(p_{j},R_{j}+\varrho+\mu)\setminus B(p_{j}+R_{j}+\mu)$ will be mapped to $p_{j}+{dist(x,p_{j})-R_{j}-\varrho\over\mu}(p_{1}-p_{j})$.) Obviously, $d$-dimensional Hausdorff content of the image of $F_{N^{\prime}+1}$ is zero for any $d>1$, and, in particular, for $d=m$.

By construction, we can combine $F_{j}$, $j=1,\ldots,N^{\prime}+1$ into one continuous map $F$ of $X$ into $S$. For $m\in(1,2]$ we have

[TABLE]

since $F_{N^{\prime}+1}(X\setminus\cup_{j=1}^{N^{\prime}}int(B(p_{j},R_{j}+\varrho)))\subset W$ has [math] $m$-dimensional Hausdorff content. In particular, $\operatorname{HC}_{m}(F(X))\leq(\operatorname{HC}_{m-1}(Y)+\delta)^{m\over m-1}$.

3.2. Inductive step

By induction we assume the conclusions of Proposition 3.2 to be true for all dimensions less than or equal to $m$. We will now prove it for $m+1$. The following Proposition is the analogue of the Proposition from [W].

Proposition 3.3**.**

Let $A(m)>m$ and $Y^{\prime}\subset S$ be a compact subset. For every $\varepsilon>0$ there exists a finite set of disjoint balls $\{B_{j}=B(p_{j},r_{j})\}$ with centers in $S^{\prime}$ and a constant $\alpha\in(\frac{1}{12},1]$, such that the following inequalities hold:

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Proof.

The main difficulty in the construction of the covering is that Hausdorff content is not additive for disjoint sets. This can be circumvented by considering a modification of Hausdorff content of subsets of $Y^{\prime}$ with respect to a fixed nearly optimal covering $Q$ of $Y^{\prime}$. We will find a collection of disjoint balls $\{B_{j}=B(p_{j},r_{j})\}$, such that each $B_{j}$ contains a slightly smaller ball $B(p_{j},r(p_{j}))$ with the property that optimal coverings of $B(p_{j},r(p_{j}))$ and $B(p_{i},r(p_{i}))$, $i\neq j$, by balls chosen from $Q$, are disjoint. This will imply approximate additivity of the modified Hausdorff content for $\{B_{j}\}$.

Fix a finite covering of $Y^{\prime}$ by closed balls $\beta_{k}$, $k=1,\ldots,N$ (for some $N$) of radius $\tilde{r}_{k}$ and with centers in $S^{\prime}$, so that $\sum_{k=1}^{N}\tilde{r}_{k}^{m}\leq\operatorname{HC}_{m}(Y^{\prime};S^{\prime})+\varepsilon$, where $\varepsilon$ can be chosen arbitrarily small. Without any loss of generality we can assume that no ball $\beta_{k}$ (even with a very small radius) is contained in the union of the other balls of the collection. (This requirement can, obviously, be satisfied by inductively removing such “unnecessary” balls from the collection.) We denote this collection of balls $Q$, and the center of $\beta_{k}$ by $q_{k}$. For each subset $W$ of $Y^{\prime}$ define $m$-dimensional Hausdorff content $\widetilde{\operatorname{HC}}_{m}(W)$ with respect to $Q$ as the infimum of $\sum_{k\in J}\tilde{r}_{k}^{m}$ over all subsets $J\subset\{1,\ldots,N\}$ such that $W\subset\bigcup_{k\in J}\beta_{k}$. In other words, we calculate the Hausdorff content with respect to only balls from the collection $Q$. Clearly, for each $W$ we have $\widetilde{\operatorname{HC}}_{m}(W)\geq\operatorname{HC}_{m}(W;S^{\prime})$, so any upper bound for $\widetilde{\operatorname{HC}}_{m}$ will be automatically an upper bound for $\operatorname{HC}_{m}$. On the other hand, it immediately follows from definitions that $\widetilde{\operatorname{HC}}_{m}(Y^{\prime})\leq\operatorname{HC}_{m}(Y^{\prime},S^{\prime})+\varepsilon.$

Fix a point $p\in Y^{\prime}$ and consider quantity $\lambda_{p}(r)={\widetilde{\operatorname{HC}}(B(p,r)\cap Y^{\prime})\over r^{m}}$. (Recall that $B(p,r)$ denotes a closed ball of radius $r$ centred at $p$). Observe that the following properties of $\lambda_{p}(r)$ directly follow from the definition:

1. (1)

$\lambda_{p}(r)$ is piecewise continuous; 2. (2)

$\lim_{x\rightarrow r^{-}}\lambda_{p}(x)=\lambda_{p}(r)\leq\lim_{x\rightarrow r^{+}}\lambda_{p}(x)$; 3. (3)

$\lim_{x\rightarrow 0}\lambda_{p}(x)=\infty$; 4. (4)

$\lim_{x\rightarrow\infty}\lambda_{p}(x)=0$; 5. (5)

For each $k\in\{1,\ldots,N\}$ we have $\lambda_{q_{k}}(\tilde{r}_{k})=1$.

The third equality holds since $\widetilde{\operatorname{HC}}$ of any non-empty subset of $Y^{\prime}$ is bounded from below by the m-th power of the radius of the smallest ball. For each $p$ we define $r(p)=\sup\{r|\lambda_{p}(r)\geq{1\over A(m)^{m}}\}$. Observe that there exists a sequence of radii $r_{l}$ approaching $r(p)$ from below with $\lambda_{p}(r_{l})\geq{1\over A(m)^{m}}$; on the other hand, $\lambda_{p}(r)<{1\over A(m)^{m}}$ for every $r>r(p)$. It follows from the properties of $\lambda_{p}(r)$ that

[TABLE]

Property (5) of $\lambda$ implies that for each center $q_{k}$ we have $r(q_{k})>\tilde{r}(q_{k})$.

Also, for every $\theta\geq 1$ we have

[TABLE]

Now we would like to define concentric balls $B(p,\overline{r}(p))$ for somewhat larger radii $\overline{r}(p)\geq r(p)$. For this purpose consider the annulus $A=B(p,(1+{1\over m})^{2}r(p))\setminus B(p,(1+{1\over m})r(p))$. Applying the coarea inequality (Lemma 2.2) to $A\cap Y^{\prime}$ we see that there exists $\overline{r}(p)\in[(1+{1\over m})r(p),(1+{1\over m})^{2}r(p)]$ such that

[TABLE]

(observe, that our coarea inequality applies to $\widetilde{\operatorname{HC}}$).

Observe that, as $\overline{r}(q_{k})>r(q_{k})>\tilde{r}(q_{k})$, the collection of balls $B(q_{k},\overline{r}(q_{k}))$, $k=1,\ldots,N$, covers $Y^{\prime}$. Therefore, one can use the Vitali covering construction to find a finite set of disjoint balls $B_{j}=B(q_{i_{j}},\overline{r}(q_{i_{j}}))$, $j\in\{1,\ldots,L\}$ from the collection $Q$, so that $\bigcup_{j=1}^{L}B(q_{i_{j}},3\overline{r}(q_{i_{j}}))$ covers all $Y^{\prime}$. We set $p_{j}=q_{i_{j}}$ and $r_{j}=\overline{r}(q_{i_{j}})$. Define the density constant $\alpha$ as

[TABLE]

We claim that $\alpha\in(\frac{1}{12},1]$, and the set of balls $\{B_{j}\}_{j=1}^{L}$, ($L\leq N$), satisfies inequalities (3.1)-(3.5).

It will be convenient to define $\eta_{j}(\theta)=\widetilde{\operatorname{HC}}_{m}(B(p_{j},\theta r(p_{j}))\cap Y^{\prime})$ and $\theta_{j}=\frac{r_{j}}{r(p_{j})}\in[1+\frac{1}{m},(1+\frac{1}{m})^{2}]$. Since $\eta_{j}(\theta)=\theta^{m}r(p_{j})^{m}\lambda_{p_{j}}(\theta)$ we can write inequality (3.7) as follows. For $\theta\geq 1$

[TABLE]

Note that \alpha=\big{(}{\sum_{j=1}^{L}\eta_{j}(1)\over\widetilde{\operatorname{HC}}_{m}(Y^{\prime})}\big{)}^{\frac{1}{m}}.

Proof of inequality (3.1). Using the definition of $r_{j}=\overline{r}(p_{j})$ and (3.9) we have

[TABLE]

Proof of inequality (3.2) and the inequalities $\frac{1}{12}<\alpha\leq 1$. The key observation to prove the rest of the inequalities is that we have additivity of $\widetilde{\operatorname{HC}}_{m}$ for disjoint sets $B(p_{j},r(p_{j}))\cap Y^{\prime}\subset B_{j}\cap Y^{\prime}$. Indeed, let $\{\beta_{k_{l}}\}$ denote the covering of $B(p_{j},r(p_{j}))\cap Y^{\prime}$ realizing its Hausdorff content with respect to $Q$, $\widetilde{\operatorname{HC}}_{m}(B(p_{j},r(p_{j}))\cap Y^{\prime})=\sum_{l}rad(\beta_{k_{l}})^{m}$. By (3.9) we have

[TABLE]

for each $l$. In particular, if $i\not=j$, then none of balls $\beta_{k_{l}}$ can appear as a ball $\beta_{m_{l^{\prime}}}$ in a covering that realizes $\widetilde{\operatorname{HC}}_{m}(B(p_{i},r(p_{i}))\cap Y^{\prime})$. (Indeed, recall that the concentric balls $B(p_{i},\overline{r}(p_{i}))$ and $B(p_{j},\overline{r}(p_{j}))$ are disjoint, and $\overline{r}(p_{i})\geq r(p_{i})+\frac{r(p_{i})}{m}$, $\overline{r}(p_{j})\geq r(p_{j})+\frac{r(p_{j})}{m}$.) On the other hand, every ball in $Q$ is necessary for $Q$ to be a covering of $Y^{\prime}$. Therefore, the $m$th powers of the radii of all balls $\beta_{k_{l}}$ appear in the expression for $\widetilde{H}C_{m}(Y^{\prime})$. Hence, $\sum_{j}\eta_{j}(1)\leq\widetilde{\operatorname{HC}}_{m}(Y^{\prime}),$ or, equivalently, $\alpha\leq 1$.

Also, $\beta_{k_{l}}$ does not intersect $Y^{\prime}\setminus B(p_{j},r_{j})$. It follows that

[TABLE]

which immediately implies (3.2). On the other hand, since $Y^{\prime}\subset\bigcup B(q_{i},3\overline{r}(q_{i}))$ and utilizing (3.9) as well as the inequalities $\overline{r}(p_{j})\leq r(p_{j})(1+\frac{1}{m})^{2}$ for all $j$, we have

[TABLE]

Hence, $\alpha\geq{1\over 3(1+\frac{1}{m})^{2}}>{1\over 12}$ (as $m>1$). We conclude that

[TABLE]

Proof of inequality (3.3). From inequalities (3.8) and (3.9) we obtain

[TABLE]

We use (3.12), (3.9) and the definition of $\alpha$ to bound from above the the left hand side of (3.3).

[TABLE]

Estimate $(2(m+1)(1+\frac{1}{m})^{2m-2})^{\frac{m}{m-1}}(1+\frac{1}{m})^{2}\leq 200m\ 4^{1\over m-1}$ in the third line can be verified by an elementary but tedious calculation using the inequality $m>1$. This finishes the proof of (3.3).

Proof of inequalities (3.4) and (3.5). Using (3.12) and the definition of $\alpha$ we obtain:

[TABLE]

Using (3.9), the definition of $\alpha$, and the inequality $\theta_{j}\leq(1+{1\over m})^{2}$, we get

[TABLE]

∎

Let $I_{1}(m)=(100m)^{m},A(m)=[100m4^{1\over m-1}I_{1}(m)]^{m-1\over m}<(100m)^{m}$, $I_{2}(m)=10m12^{m}A(m)<(1500m)^{m}$.

To finish the proof of Proposition 3.2 we will need the following definition:

Given an open set $H\subset X$ and an extension $F:X\longrightarrow U$ of $f$, we will say that $(H,F)$ is an $(\alpha,\varepsilon)$-improvement pair for $X,Y,f$ if the following conditions hold ($\bar{H}$ below denotes the closure of $H$):

1. (1)

$\operatorname{HC}_{m+1}(F(X\cap\bar{H});S^{\prime})\leq\frac{1}{4}I_{1}(m)\alpha^{m+1}\operatorname{HC}_{m}(f(Y);S^{\prime})^{\frac{m+1}{m}}+\varepsilon$; 2. (2)

$\operatorname{HC}_{m}(F(X\cap\partial H)\cup F(Y\setminus H);S^{\prime})\leq(1-\frac{\alpha^{m}}{2})\operatorname{HC}_{m}(f(Y);S^{\prime})+\varepsilon$; 3. (3)

$F(\bar{H})\subset N_{R}(Y)$ for $R\leq 3A(m)\operatorname{HC}_{m}(f(Y);S^{\prime})^{\frac{1}{m}}+\varepsilon$.

Lemma 3.4**.**

Assume the same set up as in Proposition 3.2. For every $\varepsilon>0$ there exists $\alpha\in[\frac{1}{12},1]$ and an $(\alpha,\varepsilon)$-improvement pair $(H,F)$ for $X,Y,f$.

Proof.

As in the subsection 3.1 we can assume, without any loss of generality, that $Y\subset X\subset S$ and $Y$ is compact. Apply Proposition 3.3 to $Y^{\prime}=Y$ to obtain a set of disjoint closed balls $\{B_{j}\}$ with centers in $S^{\prime}$, and then determine the density constant $\alpha$. We take $H=\cup_{j}int(B_{j})$, where $int(B_{j})$ denotes the open ball with the same center and radius as $B_{j}$.

Now, first, we define $F$ on $\bigcup_{j}X\cap\partial B_{j}$. This is accomplished by applying (using our inductive assumption) Proposition 3.2 one dimension lower with $X$, $Y_{j}=Y\cap\partial B_{j}$ playing the role of $Y$, and $B_{j}$ playing the role of ball $U$. We obtain a map $\tau_{j}:X\rightarrow B_{j}$ satisfying

[TABLE]

Now we would like to restrict $\tau_{j}$ to $S_{j}=X\cap\partial B_{j}$, and use these mappings to define $F$ on $\bigcup_{j}X\cap B_{j}$. Let $Z_{j}=(Y\cap B_{j})\cup\tau_{j}(S_{j})$. We apply Lemma 2.1 to define a cone over $Z_{j}$ map $F_{j}:X\cap B_{j}\longrightarrow B_{j}$, such that $F_{j}=\tau_{j}$ on $S_{j}$ and $F_{j}$ is the identity on $Y\cap B_{j}$. This map satisfies the inequality $\operatorname{HC}_{m+1}(F_{j}(X\cap B_{j});S^{\prime})\leq emr_{j}\operatorname{HC}_{m}(Z_{j};S^{\prime})$.

Using (3.14) and (3.3), (3.5) from Proposition 3.3 we estimate

[TABLE]

We estimate the Hausdorff content of $Y_{1}=(\bigcup_{j}\tau_{j}(S_{j}))\cup(Y\setminus\bigcup_{j}B_{j})$. Combining inequalities (3.14) and (3.2), (3.4) from Proposition 3.3 we get

[TABLE]

by our choice of $A(m)$. (Recall that $\alpha\geq{1\over 12}$.) We also have that by (3.1) for all $j$ the image of $F_{j}(X_{j})$ is contained in $N_{R}(P)$ for $R\leq\max_{j}r_{j}\leq 3A(m)\operatorname{HC}_{m}(Y;S^{\prime})^{1\over m}+\varepsilon$. ∎

Remark. To prove Theorem 1.5 we will need the following consequence of the above construction. Given a compact $Y\subset U\subset S$, let $\tilde{Y}=(Y\setminus\bigcup_{j}B_{j})\cup(\cup_{j}\tau_{j}(Y\cap B_{j}))$, where $B_{j}$ and $\tau_{j}$ were defined in the proof of the previous lemma. Now we can define a continuous map $\theta:Y\longrightarrow\tilde{Y}$ defined as (the restriction of) $\tau_{j}$ on $Y\cap B_{j}$ for each $j$, and the identity map outside of $\bigcup_{j}B_{j}$. For each point $y\in Y$ $dist(y,\theta(y))$ does not exceed $\max_{j}r_{j}\leq 3A(m)HC_{m}(Y;S^{\prime})^{1\over m}+\varepsilon.$ Also, using the fact that $\tau_{j}(Y\cap B_{j})$ is contained in $\tau_{j}(X)$, inequality (3.14), and, finally, the same argument as in inequality (3.16), we see that $\operatorname{HC}_{m}(\tilde{Y};S^{\prime})\leq(1-{1\over 2*12^{m}})\operatorname{HC}_{m}(Y;S^{\prime})+\varepsilon$. We can record our observation as the following lemma:

Lemma 3.5**.**

Given a Banach space $S$, its finite-dimensional subset $S^{\prime}$, a closed ball $U$ in $S$, a compact subset $Y$ of $U$, and an arbitrarily small positive $\varepsilon$, there exists a compact $\tilde{Y}\subset S$ and a continuous map $\theta:Y\longrightarrow\tilde{Y}$ such that

1. (1)

$\operatorname{HC}_{m}(\tilde{Y};S^{\prime})\leq(1-{1\over 2*12^{m}})\operatorname{HC}_{m}(Y;S^{\prime})+\varepsilon$; and 2. (2)

For each $y\in Y$ $\|y-\theta(y)\|\leq 3A(m)\operatorname{HC}_{m}(Y;S^{\prime})^{1\over m}+\varepsilon,$ where $A(m)<(100m)^{m}$.

We continue our proof of Proposition 3.2. We now define the desired extension $F$ by repeatedly applying Lemma 3.4.

Fix $\varepsilon>0$. We set $X_{1}=X$, $f_{1}=f$ and $Y_{1}=Y$. Inductively, let $(H_{k},F_{k})$ be an $(\alpha_{k},\varepsilon_{k})$-improvement pair defined in Lemma 3.4 for $X_{k},Y_{k},f_{k}$, where $\alpha_{k}\in[\frac{1}{12},1]$ and $\varepsilon_{k}$ is a sequence of positive numbers rapidly converging to [math], so that $\sum_{k=1}^{\infty}\varepsilon_{k}$ can be made arbitrarily small in comparison with $\varepsilon$. More specifically, we can choose $\varepsilon_{k}=\frac{\varepsilon}{3m10^{m}A(m)2^{k}}$. (Note, that the notation $F_{k}$ has a new meaning now that differs from its meaning in the proof of Lemma 3.4.)

We let $X_{k+1}=X_{k}\setminus H_{k}$, $Y_{k+1}=\partial H_{k}\cup(Y_{k}\setminus H_{k})$ and $f_{k}$ be the restriction of $F_{k}$ to $Y_{k}$. For a large $K>1$ and for each $k\leq K$ we define map $F:\bigcup_{k=1}^{K}H_{k}\longrightarrow S$ by setting $F(x)=F_{k}(x)$. Note that this map is well-defined and continuous on $\bigcup_{k=1}^{K}H_{k}$. Indeed, if $x\in H_{k_{1}}\cap H_{k_{2}}$ for $k_{1}<k_{2}$, then $x\in\partial H_{k}$ for all $k_{1}\leq k\leq k_{2}$ and $F_{k_{2}}(x)=F_{k_{1}}(x)$. We also have that $F(x)=f(x)$ for all $x\in Y\cap\bigcup H_{k}$.

By property 1) of $(\alpha,\varepsilon_{k})$-improvement pairs we have that for an arbitrarily large $K$, the following inequality holds:

[TABLE]

(The last inequality can be easily derived once one notices that the generalized binomial theorem implies that the denominator $1-(1-\frac{\alpha^{m}}{2})^{\frac{m}{m+1}}>\frac{m}{m+1}\frac{\alpha^{m}}{2}$.)

To estimate the filling radius observe that $F(\bigcup_{k=1}^{K}H_{k})\subset N_{R}(Y)$, for $R$ satisfying

[TABLE]

Last inequality follows by our choice of $I_{2}(m)$.

Now we are going to digress and observe that we have obtained the following lemma:

Lemma 3.6**.**

Let $Y$ be a compact subset of a closed ball in Banach space $S$, $S^{\prime}$ a finite-dimensional subspace of $S$. There exists a sequence of compact subsets $Y^{(i)},$ $i=1,2,\ldots$ with $Y^{(1)}=Y$ and maps $\theta_{i}:Y^{(i)}\longrightarrow Y^{(i+1)}$ for all $i$ so that for all $k$

(1) $\operatorname{HC}_{m}(Y^{(k)};S^{\prime})\leq(1-{1\over 2*12^{m}})^{k-1}\operatorname{HC}_{m}(Y;S^{\prime})+\varepsilon;$

(2) Let $\Theta_{k}$ denote $\theta_{k}\circ\theta_{k-1}\circ\ldots\theta_{1}$. For each $y\in Y$ $dist(y,\Theta_{k}(y))\leq I_{2}(m)\operatorname{HC}_{m}(Y;S^{\prime})+\varepsilon$.

Proof.

We repeatedly apply Lemma 3.5. Inductively $\theta_{i}$ is the map $\theta$ from the remark for $Y=Y^{(i)}$, and $Y^{(i+1)}=\tilde{Y}$. The property (1) immediately follows from the remark. Property (2) is proven exactly as (3.18). ∎

3.3. The case of $\operatorname{HC}_{m}(Y_{K+1})<\varepsilon_{0}$

Let $X^{\prime}=X\setminus\bigcup_{k=1}^{K}H_{k}$. To finish the proof we need to define an extension $F:X^{\prime}\longrightarrow S$ of map $f_{K+1}:Y_{K+1}\longrightarrow S$ with $\operatorname{HC}_{m}(Y_{K+1})<\varepsilon_{0}$, where we can make $\varepsilon_{0}$ arbitrarily small (in particular, much smaller than $\varepsilon$) by picking $K$ sufficiently large.

By Kadec-Snobar estimate [Wo, Theorem III.B.10] there exists a projection map $\Pi_{S^{\prime}}:S\longrightarrow S^{\prime}$, such that $||\Pi(v)||\leq\sqrt{N}||v||$, where $N$ is the dimension of $S^{\prime}$. Let $\Pi_{U}:S^{\prime}\longrightarrow U\cap S^{\prime}$ denote the radial retraction onto the ball $U\cap S^{\prime}$. We define map $\Pi=\Pi_{U}\circ\Pi_{S^{\prime}}$. We have that the Lipschitz constant of $\Pi$ satisfies $Lip(\Pi)\leq const(N)$. Let $\{B(p_{i},r_{i})\}$, $p_{i}\in S^{\prime}$, be a finite collection of open balls covering $Y_{K+1}$ with $\sum r_{i}^{m}\leq 2\varepsilon_{0}$. Let $V$ be an open set with $\bigcup_{i}B(p_{i},r_{i})\subset V\subset\bigcup_{i}B(p_{i},2r_{i})$ and define a map $\Phi:\bigcup B(p_{i},2r_{i})\longrightarrow S$ setting $\Phi(x)=(1-\phi(x))x+\phi(x)\Pi(x)$, where $\phi:S\longrightarrow[0,1]$ is a continuous function, such that $\phi(x)=0$ for $x\in\bigcup B(p_{i},r_{i})$ and $\phi(x)=1$ for $x\in\partial V$.

Observe that we have $\operatorname{HC}_{m}(\Phi(F_{K}(X)\cap V))\leq const(N)\varepsilon_{0}$. Also, $\Phi(F_{K}(X)\cap V)$ is $const(N)R^{\prime}$-close to $V$, and, therefore, to $Y_{K}$, where $R^{\prime}=const(N)\max_{i}r_{i}\leq const(N)\varepsilon_{0}^{1\over m-1}$. Finally, we apply Federer-Fleming deformation Lemma 2.4 to define map $F$ that extends $\Phi\circ F_{K}$ restricted to $F_{K}^{-1}(\partial V\cap X)$ to a map on $X\setminus F_{K}^{-1}(V)$. By choosing $\varepsilon_{0}(N)$ sufficiently small we can guarantee that $\operatorname{HC}_{m+1}(F(X\setminus V))<\varepsilon/2$ and $F(X\setminus V)\setminus N_{R}(Y_{K})$ is contained in an $(\lceil m\rceil-1)$-dimensional CW complex $W$ for $R<\varepsilon/2$.

This finishes the proof of Proposition 3.2 and, therefore, Theorem 3.1.

Similarly, we can project $Y_{K+1}$ to finite-dimensional $S^{\prime}$ using $\Pi_{S^{\prime}}$ and then use Lemma 2.5 and the remark after it to map $Y_{K+1}$ to a subset of the $(m-1)$-dimensional skeleton of the cubic complex $Q$ in $S^{\prime}$. Note that $\Pi_{S^{\prime}}$ maps each point $y\in Y^{K+1}$ to $\sqrt{N}r_{i}$-close point in $S^{\prime}$, where $r_{i}\leq const(m)\varepsilon_{0}^{1\over m}$ is the radius of the ball $B(x_{i},r_{i})$ that contains $y$. Composing the resulting map to the $(m-1)$-simplicial complex with $\Theta_{K}$ from Lemma 3.6 we obtain a continuous map of $Y$ to a $(m-1)$-dimensional simplicial complex such that the distance between each point $y$ and its image does not exceed $Const(m)\operatorname{HC}_{m}(Y)^{1\over m}$.

To prove Theorem 1.5 change the notation $X$ to $Y$ and observe that one can assume that $Y$ is a subset of Banach space $L^{\infty}(Y)$, where the inclusion sends each point $y\in Y$ to the distance function $d_{y}(x)$ on $Y$ defined as $d_{y}(x)=dist(x,y)$ (compare [Gr]). Now the assertion at the end of the previous paragraph immediately yields Theorem 1.5.

Observe that our proof of Theorem 1.5 was almost a “subset” of the proof of Theorem 1.1. This is not surprising. Indeed, the general theory of absolute extensors for metric spaces implies that absolute extensors are precisely absolute retracts, i.e. contractible ANR. In fact, if we are interested only in extensions of maps from some fixed metric space $Y$, the requirement of contractibility can be replaced by the weaker condition that the image of $Y$ is contractible in the considered space. Correspondingly, our strategy of proving Theorem 3.1 was to embed $Y$ in an open set in $const(m)\operatorname{HC}_{m}(Y)^{1\over m}$-neighbourhood of $Y$, so that we have the desired upper bound for $\operatorname{HC}_{m+1}$ of this set. Moreover, this set is contractible to a bounded subcomplex of $(m-1)$-skeleton of a subdivision of $n$-dimensional linear space into cubes. (At this stage we already have Theorem 1.5.) Adding the cone over this subset (or embedding it into the $m$-skeleton of the same cubic complex as in Lemma 2.4) we obtain an absolute extensor (or an absolute extensor for $Y$). We constructed explicit mappings from arbitrary $X$ to the constructed space, but could have just used the relevant results from Borsuk’s theory of retracts.

3.4. Proof of Proposition 1.3

Now we are going to prove Proposition 1.3. First, we are going to establish the isoperimetric inequality stated in the proposition.

The proof is almost identical to that of Loomis-Whitney inequality [LW]. Consider a covering of $\partial\Omega$ by cubes (nearly) realizing the Hausdorff content. We can approximate this covering arbitrarily well using cubes $\{C_{i}\}$ in a sufficiently fine lattice, with cubes of side length $2r$, where $r$ can be arbitrarily small. (Here we are using the assumption that $n$ is the dimension of the cubes. For a cube with side length $R$, $R^{n}$ is its volume, and, therefore, $R^{n}$ will be only somewhat smaller than the sum of volumes of smaller cubes, providing that the union of smaller cubes includes the original cube, and is only “slightly” larger than the ambient cube.) Let $\pi_{j}$ denote the projection onto the $j$th coordinate $(n-1)$-plane and let $U_{j}=\pi_{j}(\bigcup C_{i})$. Note that $\Omega\subset\bigcap\pi_{j}^{-1}(U_{j})$. Let $N$ denote the number of $r$-cubes in $\bigcap\pi_{j}^{-1}(U_{j})$, and $N_{j}$ be the number of $r$-cubes in $U_{j}$. By choosing the size $2r$ of the grid sufficiently small, we may assume that, for an arbitrarily small $\varepsilon>0$, $\operatorname{HC}_{n-1}(\partial\Omega)\geq\operatorname{HC}_{n-1}(\pi_{j}(\partial\Omega))\geq N_{j}r^{n-1}-\varepsilon$. By Theorem 2 in [LW]

[TABLE]

We conclude that $HC_{n}(\Omega)\leq Nr^{n}\leq(\prod_{j=1}^{n}N_{j}r^{n(n-1)})^{1\over n-1}\leq(HC_{n-1}(\partial\Omega)+\varepsilon)^{n\over n-1}$.

Now it remains to check what is going on in the case of coordinate cubes. If $\Omega$ is a coordinate cube with side length $R$, then $\operatorname{HC}_{n}(\Omega)=({R\over 2})^{n}$ (Lemma 1.11). Also, $\operatorname{HC}_{n-1}(\partial\Omega)$ is not less than $\operatorname{HC}_{n-1}$ of any of its $(n-1)$-dimensional faces, that are $(n-1)$-dimensional coordinate cubes with side length $R$. Therefore, $\operatorname{HC}_{n-1}(\partial\Omega)\geq({R\over 2})^{n-1}$. On the other hand, the closure of $\Omega$, including $\partial\Omega$, can be covered by just one closed metric ball (i.e. the coordinate cube) of radius ${R\over 2}$. Hence, $\operatorname{HC}_{n-1}(\partial\Omega)^{n\over n-1}=({R\over 2})^{n}=\operatorname{HC}_{n}(\Omega)$.

4. The proof of systolic inequalities.

Here we prove the systolic inequalities stated in subsection 1.3. We will be using the upper bounds for $UW_{m-1}$ provided by Theorems 1.5, 1.6.

The proof is modelled on the argument from [Gr] used there to deduce the inequality $sys_{1}(M^{n})\leq c(n)vol^{\frac{1}{n}}(M^{n}).$ First, observe that according to [Gr], Appendix 1, Proposition (D) on p. 128, the Kuratowski embedding $f:X\longrightarrow L^{\infty}(X)$ is at the distance ${1\over 2}UW_{m-1}(X)$ from some $(m-1)$-degenerate map $g:X\longrightarrow L^{\infty}(X)$, that is, a map $g$ which is a composition of a map $g_{1}$ of $X$ into a $(m-1)$-dimensional polyhedron $K$, and a map $g_{2}$ of $K$ into $L^{\infty}(X)$. Let $W$ denote the mapping cylinder of $g_{1}$: the quotient space of the cylinder $X\times[0,1]$ by the quotient map $g_{1}:X\times\{1\}\longrightarrow K$. Define $F:W\longrightarrow L^{\infty}(X)$ as $f$ on the “bottom” $X\times\{0\}$ of $W$, $g$ (or, equivalently, $g_{2}$) on the “top”, and as straight line segments connecting $f(x)$ and $g(x)$ in $L^{\infty}(X)$ on all “vertical” segments of $W$ “above” $x\in X\times\{0\}$.

Exactly as in the proof of Lemma 1.2.B from [Gr] one can prove that if $sys_{1}(X)>3UW_{m-1}(X)$, then the classifying map $Q:X\longrightarrow K(\pi_{1}(X),1)$ can be extended to $W$ by first mapping it as above to $L^{\infty}(X)$ and then extending the classifying map defined on $X\times\{0\}\subset W\subset L^{\infty}(X)$. As in the proof of Lemma 1.2.B from [Gr] one considers a very fine triangulation of $W$ and performs the extension to [math]-dimensional, then $1$-dimensional, then $2$-dimensional skeleta of the chosen triangulation of $W$. All new vertices of the triangulation are being mapped first to the nearest points of $f(X)$, and then to $K(\pi_{1}(X),1)$ via the classifying map $Q$. All $1$-dimensional simplices are first mapped to minimal geodesics between the images of their endpoints in $f(X)$, and then to $K(\pi_{1}(X),1)$ using $Q$. Observe that the triangle inequality implies that their images in $f(X)$ have length $\leq UW_{m-1}(X)+\varepsilon$, where $\varepsilon$ can be made arbitrarily small by choosing a sufficiently fine initial triangulation of $W$. We observe that an easy compactness argument implies that there exists a positive $\delta$ such that each closed curve of length $\leq 3UW_{m-1}(X)+\delta$ is still contractible. We choose $\varepsilon$ above as ${\delta\over 3}$. Now the boundary of each new $2$-simplex in $W$ has been already mapped to a closed curve of length $\leq 3(UW_{m-1}(X)+\varepsilon)$ in $X$ that is contractible in $X$. So, we can map the corresponding $2$-simplex in $W$ by, first, contracting the image of its boundary in $f(X)$ to a point, and then mapping the resulting $2$-disc in $f(X)$ to $K(\pi_{1}(X),1)$ using the classifying map $Q$.

Finally, one argues that the extension to the skeleta of all higher dimensions is always possible as $K(\pi_{1}(X),1)$ is aspherical, as the corresponding obstructions live in homology groups of the pair $(W,f(X))$ with coefficients in trivial (higher) homotopy groups of the target space $K(\pi_{1}(X),1)$.

It remains to notice that this extension is impossible as the inclusion $X\times\{0\}\longrightarrow W$ is homotopic to $g=g_{1}\circ g_{2}$, and, therefore, induces trivial homomorphisms of all homology groups in dimensions $\geq m$. Therefore, the existence of such an inclusion would contradict the assumption that $X$ is $m$-essential.

Acknowledgements. A part of this work was done during the authors’ visit to the Research Institute for Mathematical Studies, Kyoto University, in July, 2018. Another part of this work was done while three of the authors (Yevgeny Liokumovuch, Alexander Nabutovsky and Regina Rotman) were members of the Institute for Advanced Study in January-April, 2019. The authors would like to thank both the RIMS (Kyoto University) and the IAS for their kind hospitality.

The authors would like to thank Gregory Chambers, Larry Guth, Stephane Sabourau, Christina Sormani and Robert Young for stimulating discussions.

The research of Yevgeny Liokumovich was partially supported by NSF Grant DMS-1711053 and NSERC Discovery grant RGPAS-2019-00085. The research of Boris Lishak was supported by the Australian Research Council’s Discovery funding scheme (project number DP160104502). The research of Alexander Nabutovsky was partially supported by his NSERC Discovery Grant RGPIN-2017-06068. The research of Regina Rotman was partially supported by her NSERC Discovery Grant RGPIN-2018-04523.

The authors are grateful to the anonymous referees for numerous corrections and helpful suggestions.