# Linear isoperimetric inequality
for normal and integral currents
in compact subanalytic sets

Thierry De Pauw  School of Mathematical Sciences
Shanghai Key Laboratory of PMMP
East China Normal University
500 Dongchuang Road
Shanghai 200062
P.R. of China
and NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai
3663 Zhongshan Road North
Shanghai 200062
China Université Paris Diderot
Sorbonne Université
CNRS
Institut de Mathématiques de Jussieu – Paris Rive Gauche, IMJ-PRG
F-75013, Paris
France thdepauw<EMAIL_ADDRESS>and  Robert M.
Hardt  Department of Mathematics
Rice University
P. O. Box 1892
Houston, Texas 77251<EMAIL_ADDRESS>
###### Abstract.

The isoperimetric inequality for a smooth compact Riemannian manifold $A$
provides a positive ${\bf c}(A)$, so that for any $k+1$ dimensional integral
current $S_{0}$ in $A$ there exists an integral current $S$ in $A$ with
$\partial S=\partial S_{0}$ and $\
{\mathbf{M}}(S)\leqslant{\mathbf{c}}(A){\mathbf{M}}(\partial S)^{(k+1)/k}$.
Although such an inequality still holds for any compact Lipschitz neighborhood
retract $A$, it may fail in case $A$ contains a single polynomial singularity.
Here, replacing $(k+1)/k$ by $1$, we find that a linear inequality
${\mathbf{M}}(S)\leqslant{\mathbf{c}}(A){\mathbf{M}}(\partial S)$ is valid for
any compact algebraic, semi-algebraic, or even subanalytic set $A$. In such a
set, this linear inequality holds not only for integral currents, which have
$\mathbb{Z}$ coefficients, but also for normal currents having $\mathbb{R}$
coefficients and generally for normal flat chains with coefficients in any
complete normed abelian group. A relative version for a subanalytic pair
$B\subseteq A$ is also true, and th ere are applications to variational and
metric properties of subanalytic sets.

###### Key words and phrases:

Isoperimetric inequality, normal and integral currents, subanalytic geometry,
Plateau problem

###### 2010 Mathematics Subject Classification:

Primary 49Q15,49Q20,32B20,14P10; Secondary 52A40,49J45

The first author was partially supported by the Science and Technology
Commission of Shanghai (No. 18dz2271000). The second author was partially
supported by National Science Foundation Grant DMS-1207702.

###### Contents

  1. 1 Introduction
  2. 2 Bilipschitz Equivalence
  3. 3 Two Properties of Subanalytic Sets
  4. 4 Proof of the Theorem
    1. 4.1 Partition into contractible regions
    2. 4.2 Proof of the case $k=0$
    3. 4.3 Proof of the case $k\geqslant 1$
  5. 5 Applications

##  1\. Introduction

Assume that $A$ is a smooth compact Riemannian manifold and that $k$ is a
positive integer. The following hold.

  1. (A)

The singular homology group $H_{k}(A;\mathbb{Z})$ of $A$ with integer
coefficients and the homology group $\mathbf{H}_{k}(A;\mathbb{Z})$ of $A$
defined by means of integral currents are isomorphic.

  2. (B)

If an integral current $T\in\mathbf{I}_{k}(A)$ equals $\partial S_{0}$ for
some $S_{0}\in\mathbf{I}_{k+1}(A)$, then there exists an
$S\in\mathbf{I}_{k+1}(A)$ such that $\partial S=T$ and
$\mathbf{M}(S)\leqslant\mathbf{c}(A)\mathbf{M}(T)^{(k+1)/k}$.

  3. (C)

Each homology class in $\mathbf{H}_{k}(A;\mathbb{Z})$ admits a mass minimizing
representative, i.e. given $T_{0}\in\mathbf{I}_{k}(A)$ with $\partial
T_{0}=0$, the following variational problem admits a minimizer:

$(\mathscr{P})\begin{cases}\text{minimize }\mathbf{M}(T)\\\ \text{among
}T\in\mathbf{I}_{k}(A)\text{ with }T-T_{0}=\partial S\text{ for some
}S\in\mathbf{I}_{k+1}(A).\end{cases}$

These have been established by H. Federer and W.H. Fleming, [10, 6.3]. Note
that a smooth isometric embedding $A\subseteq\mathbb{R}^{n}$ of the manifold
into some Euclidean space exhibits $A$ as a Lipschitz neighborhood retract.
This means there exists an open neighborhood $U$ of $A$ in $\mathbb{R}^{n}$
and a Lipschitzian retraction $f:U\to A$ onto $A$ (one can choose $f$ to have
the same class of smoothness as $A$). It ensues that
$\mathbf{H}_{k}(A;\mathbb{Z})$ and $\mathbf{H}_{k}(U;\mathbb{Z})$ are
isomorphic, and conclusion (A) now follows from the the deformation theorem,
[9, 4.4.2] which shows that each integral cycle $T\in\mathbf{I}_{k}(U)$ is
homologous to some polyhedral cycle $P\in\mathbf{I}_{k}(U)$, of comparable
mass, coming from a fixed complex. Conclusion (B) also follows from a careful
application of this deformation theorem. In order to establish (C), one
considers a mass minimizing sequence $\langle T_{j}\rangle_{j}$ for
$(\mathscr{P})$. According to (B) there are $S_{j}\in\mathbf{I}_{k+1}(A)$ such
that $\partial S_{j}=T_{j}-T_{0}$ and

$\mathbf{M}(S_{j})\leqslant\mathbf{c}(n)\mathbf{M}(T_{j}-T_{0})^{(k+1)/k}\leqslant\mathbf{c}\left(n,\mathbf{M}(T_{0})\right)$

if $j$ is large enough. Referring to the compactness theorem of integral
currents, corresponding subsequences of $\langle T_{j}\rangle_{j}$ and
$\langle S_{j}\rangle_{j}$ converge in flat norm to, respectively
$T\in\mathbf{I}_{k}(A)$ and $S\in\mathbf{I}_{k+1}(A)$ such that $\partial
S=T-T_{0}$ , thus $T$ and $T_{0}$ are homologous in $A$. As $\mathbf{M}$ is
lower semicontinuous with respect to convergence in the flat norm, $T$
minimizes mass in its homology class.

In this paper we study these questions with $A$ being a compact subanalytic
subset of $\mathbb{R}^{n}$, and
$\mathbf{I}_{k}(A)=\mathbf{I}_{k}(\mathbb{R}^{n})\cap\\{T:\operatorname{\mathrm{spt}}T\subseteq
A\\}$. Whereas semi-algebraic sets [2] are defined by finitely many
polynomials, the larger class of subanalytic sets [1] includes sets defined
locally by real analytic functions as well as their images under proper real
analytic maps.

Such sets may fail to be Lipschitz neighborhood retracts, the methods of [10]
do not apply, and the isoperimetric inequality of (B) may in fact fail:

For $N=2,3,\ldots$, we define the semi-algebraic set

$A_{N}=\mathbb{R}^{3}\cap\left\\{(x,y,z):z^{2N}=x^{2}+y^{2}\text{ and
}0\leqslant z\leqslant 1\right\\},$

i.e. $A_{N}$ is obtained from the rotation around the $z$ axis of the graph of
$z=x^{1/N}$, $0\leqslant x\leqslant 1$; it has a cusp at the origin. Given
$0<h<1$ we consider $T_{h}\in\mathbf{I}_{1}(A_{N})$ an oriented circle of
multiplicity 1 on $A_{N}$, at height $h$. It is not hard to show that there
exists a unique $S_{h}\in\mathbf{I}_{2}(A_{N})$ such that $\partial
S_{h}=T_{h}$, and

$\mathbf{M}(S_{h})=2\pi\int_{0}^{h}z^{N}\sqrt{1+(Nz^{N-1})^{2}}dz\,.$

Since $\mathbf{M}(T_{h})=2\pi r=2\pi h^{N}$, where $r$ is the radius of the
circle $A_{N}\cap\\{z=h\\}$, we infer the following. For each $q>1$, choosing
$N$ so large that $q=\frac{N+1}{N}+\varepsilon$ for some $\varepsilon>0$,

$\lim_{h\to 0^{+}}\frac{\mathbf{M}(S_{h})}{\mathbf{M}(T_{h})^{q}}=\lim_{h\to
0^{+}}\frac{2\pi\left(\frac{h^{N+1}}{N+1}\right)}{\left(2\pi
h^{N}\right)^{q}}=\lim_{h\to
0^{+}}\frac{(2\pi)^{1-q}}{N+1}\left(\frac{1}{h}\right)^{N\varepsilon}=\infty\,.$

This shows that in conclusion (B) above, we cannot hope for an inequality
$\mathbf{M}(S)\leqslant\mathbf{c}(A)\mathbf{M}(T)^{q}$ with an exponent $q>1$
depending only on the dimension of $T$ and the constant $\mathbf{c}(A)$
depending only on the semialgebraic set $A$. In fact if $q$ is allowed to
depend only on the dimension of $T$, then $q=1$ is the only possible choice,
as illustrated by these simple calculations. Incidentally the computations
also show that $A_{N}$ is not a Lipschitz neighborhood retract: If there were
a Lipschitzian retraction $f:U\to A_{N}$, considering
$D_{h}\in\mathbf{I}_{2}(\mathbb{R}^{3})$ the unique flat disk in
$\mathbb{R}^{3}$ with $\partial D_{h}=T_{h}$, and $h$ small enough for
$\operatorname{\mathrm{spt}}D_{h}\subseteq U$, we would have
$f_{\\#}D_{h}=S_{h}$ and in turn

$\frac{2\pi
h^{N+1}}{N+1}\leqslant\mathbf{M}(S_{h})=\mathbf{M}(f_{\\#}D_{h})\leqslant\pi(\operatorname{\mathrm{Lip}}f)^{2}h^{2N},$

a contradiction as $h\to 0^{+}$.

Our main result is as follows.

######  Theorem. —

Let $A\subseteq\mathbb{R}^{n}$ be a compact subanalytic set. There exists
$\mathbf{c}(A)>0$ with the following property. For every $k=0,1,2,\ldots$ and
every $S_{0}\in\mathbf{I}_{k+1}(A)$, there exists $S\in\mathbf{I}_{k+1}(A)$
such that $\ \partial S=\partial S_{0}\ $ and $\
\mathbf{M}(S)\leqslant\mathbf{c}(A)\mathbf{M}(\partial S)$.

This is known as a linear isoperimetric inequality because of the absence of
an exponent in the righthand term. Moreover, the linear inequality holds, via
the same proof, with $\mathbb{Z}$ replaced by any normed, complete Abelian
group $G$ of coefficients. When $G=\mathbb{R}$, with the absolute value norm,
the fact that both $\mathbf{M}(\lambda S)=\lambda\mathbf{M}(S)$ and
$\mathbf{M}(\lambda\partial S)=\lambda\mathbf{M}(\partial S)$ for arbitrarily
small positive $\lambda$ shows that both sides of a valid isoperimetric
inequality must be homogeneous of the same degree, which can therefore be set
equal to 1.

Our proof is based on two facts of metric nature, regarding a compact
subanalytic set $A$. A basic topological property is that $A$ is triangulable
[13], [14], i.e. there exists a simplicial complex ${\mathscr{K}}$ and a
subanalytic homeomorphism $\phi:|{\mathscr{K}}|\to A$. For $x\in A$, there
exists $r(x)>0$ such that the intersections with open balls
$|{\mathscr{K}}|\cap\mathbf{U}(\phi^{-1}(x),r)$ are contractible for
$0<r\leqslant r(x)$, and it ensues that $A$ is locally contractible. Here we
will use a strengthening due to G. Valette, [16] stating that the local
homotopy $h:[0,1]\times(A\cap U)\to A\cap U$ from the identity to a constant
can be chosen to be Lipschitz. This already implies $A$ is “locally acyclic”
with respect to the chain complex of integral currents and in turn, that
$H_{k}(A;\mathbb{Z})$ and $\mathbf{H}_{k}(A;\mathbb{Z})$ are isomorphic, see
[5], which is analogous to (A) above. See also [12]. A second consequence of
triangulabi lity is that if $x\in A$ and $0<s\leqslant s(x)$ is small enough,
then, for $s/2<t<t^{\prime}<s$, the spherical links
$A\cap\operatorname{\mathrm{Bdry}}\mathbf{U}(x,t)$ and
$A\cap\operatorname{\mathrm{Bdry}}\mathbf{U}(x,t^{\prime})$ are
subanalytically homeomorphic. Here we use the work of A. Parusiński, [15] on
locally Lipschitz stratification of subanalytic sets to find, away from the
origin, intervals of radii for links which are uniformly bilipschitz
equivalent.

After our Main Theorem and the analog of (B) above are established, the analog
of (C) follows along the same lines sketched at the beginning of this
introduction. It suffices to observe that for proving (B), the particular
power $(k+1)/k$ plays no significant role. An existence theorem for the
Plateau problem in semialgebraic sets has been obtained recently by Q. Funk
using different methods, [12].

As discussed in §5 , the linear isoperimetric inequality for compact
subanalytic sets is also valid for normal currents with $\mathbb{R}$
coefficients or even normal flat chains with coefficients in any complete
normed abelian group. It plays a role in relating not only various homology
theories but also a duality between a homology based on normal currents and a
cohomology based on normal cochains called charges [7].

A large number of geometric variational problems involving support
constraints, boundary constraints, or free boundaries can be formulated with
various groups of chains, cycles, boundaries, or homologies. Generalizing (C),
we give in §5.9, the existence theory for one problem related to minimizing
mass in a relative homology class of a pair $B\subseteq A$ of compact
subanalytic sets. This uses a relative isoperimetric inequality which bounds
the the part of the mass in $A\setminus B$ of a suitable chain in terms of the
part of the mass in $A\setminus B$ of its boundary. There remain many
interesting regularity questions concerning solutions of this and other
variational problems in semi-algebraic and subanalytic sets.

Interesting inequalities concerning functions defined on singular algebraic
varieties or subanalytic domains are found in works of L. Bos and P. Milman
[3], [4] and A.Valette and G. Valette [17] .

Our notations regarding integral and normal currents, and flat chains are
consistent with that of [9].

##  2\. Bilipschitz Equivalence

######  2.1Linear isoperimetric inequality. —

Let $k=0,1,2,\ldots$ and $A\subseteq\mathbb{R}^{n}$. We say that $A$ satisfies
the linear isoperimetric inequality of dimension $k$ whenever the following
holds. There exists $\mathbf{c}(A,k)>0$ such that for each
$S_{0}\in\mathbf{I}_{k+1}(A)$ there exists $S\in\mathbf{I}_{k+1}(A)$ with
$\partial S=\partial S_{0}$ and
$\mathbf{M}(S)\leqslant\mathbf{c}(A,k)\mathbf{M}(\partial S)$.

######  2.2Bilipschitz equivalence. —

Let $X,Y\subseteq\mathbb{R}^{n}$ and $\mathbb{L}>0$. We say that a bijective
map $\phi:X\to Y$ is $\mathbb{L}$-bilipschitz if $\phi$ is Lipschitz,
$\phi^{-1}$ is Lipschitz and
$\max\\{\operatorname{\mathrm{Lip}}\phi,\operatorname{\mathrm{Lip}}\phi^{-1}\\}\leqslant\mathbb{L}$.
In case such map $\phi$ exists, we say that $X$ and $Y$ are bilipschitz
equivalent.

######  2.3 Proposition. —

Let $k=0,1,2,\ldots$, and $X,Y\subseteq\mathbb{R}^{n}$. Assume that $X$ and
$Y$ are bilipschitz equivalent. It follows that $X$ satisfies the linear
isoperimetric inequality of dimension $k$ if and only if $Y$ does.

###### Proof.

Let $\phi:X\to Y$ be an $\mathbb{L}$-bilipschitz homeomorphism. One infers
from Kirszbraun’s Theorem [9, 2.10.43] that $\phi,\phi^{-1}$ admit extensions
$f,g:\mathbb{R}^{n}\to\mathbb{R}^{n}$ with
$\max\\{\operatorname{\mathrm{Lip}}f,\operatorname{\mathrm{Lip}}g\\}\leqslant\mathbb{L}$.
Assume $X$ satisfies the linear isoperimetric inequality of dimension $k$ with
constant $\mathbf{c}>0$. Let $S_{0}\in\mathbf{I}_{k+1}(Y)$. Define
$S_{0}^{\prime}=g_{\\#}S_{0}\in\mathbf{I}_{k+1}(\mathbb{R}^{n})$. Notice that
$\operatorname{\mathrm{spt}}(S_{0}^{\prime})\subseteq X$, [9, 4.1.14 p. 371].
Thus there exists $S^{\prime}\in\mathbf{I}_{k+1}(X)$ such that $\partial
S^{\prime}=\partial S^{\prime}_{0}$ and
$\mathbf{M}(S^{\prime})\leqslant\mathbf{c}\mathbf{M}(\partial S^{\prime})$.
Define $S=f_{\\#}S^{\prime}$ and notice as before that
$S\in\mathbf{I}_{k+1}(Y)$. Further note that $f_{\\#}g_{\\#}S=(f\circ
g)_{\\#}S=S$ where the last equality follows from [9, 4.1.15 p. 372] and the
fact that $f\circ g=\operatorname{\mathrm{id}}_{\mathbb{R}^{n}}$ on
$\operatorname{\mathrm{spt}}S$. Finally,

$\mathbf{M}(S)=\mathbf{M}(f_{\\#}g_{\\#}S)\leqslant(\operatorname{\mathrm{Lip}}f)^{k+1}\mathbf{M}(S^{\prime})\leqslant(\operatorname{\mathrm{Lip}}f)^{k+1}\mathbf{c}\mathbf{M}(\partial
S^{\prime})\\\
=(\operatorname{\mathrm{Lip}}f)^{k+1}\mathbf{c}\mathbf{M}(g_{\\#}\partial
S)\leqslant(\operatorname{\mathrm{Lip}}f)^{k+1}(\operatorname{\mathrm{Lip}}g)^{k}\mathbf{c}\mathbf{M}(\partial
S)\leqslant\mathbf{c}{\mathbb{L}}^{2k+1}\mathbf{M}(\partial S).$

∎

##  3\. Two Properties of Subanalytic Sets

A finitely triangulated compact space is trivially locally contractible at
each point. For a subanalytic set $A$, this local contraction may be chosen to
be Lipschitz according to the following result of G. Valette, [16, Theorem
2.3.1] even though $A$ itself may fail to be a Lipschitz neighborhood retract.

######  3.1 Theorem. —

Any point $a$ in a closed subanalytic subset $A$ of $\mathbb{R}^{n}$ has a
compact subanalytic neighborhood $K\subseteq A$ and a Lipschitz deformation
contraction $h:[0,1]\times K\to K$ so that $h(0,x)=x$ and $h(1,x)=a$ for $x\in
K$ and $h(t,a)=a$ for $t\in[0,1]$.

From this we obtain at once a local version of the linear isoperimetric
inequality.

######  3.2 Corollary. —

Suppose that $J\in{\mathbf{I}}_{k}(K)$ and that $\partial J=0$ in case $k>0$.
Then the chain
$H:=-h_{\\#}\left(\boldsymbol{[}\\!\boldsymbol{[}0,1\boldsymbol{]}\\!\boldsymbol{]}\times
J\right)\ $ belongs to ${\mathbf{I}}_{k+1}(K)\ $ and satisfies

$\partial H\ =\ \begin{cases}J&{\rm if}\ \ $k\ >\ 0$\\\
J-J(1)\boldsymbol{[}\\!\boldsymbol{[}a\boldsymbol{]}\\!\boldsymbol{]}&{\rm
if}\ \ k=0\end{cases}\quad\quad{\rm
and}\quad\quad{\mathbf{M}}(H)\leqslant(\operatorname{\mathrm{Lip}}h)^{k+1}{\mathbf{M}}(J)\
.$

###### Proof.

Clearly $H\in{\mathcal{R}}_{k+1}(K)$ because $h$ is Lipschitz, and the
homotopy formula [9, 4.1.9] shows that, for $k>0$,

$\displaystyle-\partial H\ =\
h_{\\#}\partial(\boldsymbol{[}\\!\boldsymbol{[}0,1\boldsymbol{]}\\!\boldsymbol{]}\times
J)\ $
$\displaystyle=h_{\\#}(\boldsymbol{[}\\!\boldsymbol{[}1\boldsymbol{]}\\!\boldsymbol{]}\times
J)-h_{\\#}(\boldsymbol{[}\\!\boldsymbol{[}0\boldsymbol{]}\\!\boldsymbol{]}\times
J)-h_{\\#}(\boldsymbol{[}\\!\boldsymbol{[}0,1\boldsymbol{]}\\!\boldsymbol{]}\times\partial
J)$ $\displaystyle=\quad\quad\;0\quad\quad-\quad\quad J\quad\quad\,-\quad
0\quad\in{\mathcal{R}}_{k}(K)\ ,$

Here $h(0,\ \cdot\ )={\rm id}$ and
$h_{\\#}(\boldsymbol{[}\\!\boldsymbol{[}1\boldsymbol{]}\\!\boldsymbol{]}\times
J)=0\ $, by [9, 4.1.20], because
$\operatorname{\mathrm{spt}}h_{\\#}(\boldsymbol{[}\\!\boldsymbol{[}1\boldsymbol{]}\\!\boldsymbol{]}\times
J)\subseteq\\{a\\}$ and
$h_{\\#}(\boldsymbol{[}\\!\boldsymbol{[}1\boldsymbol{]}\\!\boldsymbol{]}\times
J)\in\mathbf{I}_{k}(\mathbb{R}^{n})$ with $k\geqslant 1$. In case $k=0$, [9,
4.1.9] shows that

$-\partial H\ =\
h_{\\#}(\boldsymbol{[}\\!\boldsymbol{[}1\boldsymbol{]}\\!\boldsymbol{]}\times
J)-h_{\\#}(\boldsymbol{[}\\!\boldsymbol{[}0\boldsymbol{]}\\!\boldsymbol{]}\times
J)\ =\ J(1)\boldsymbol{[}\\!\boldsymbol{[}a\boldsymbol{]}\\!\boldsymbol{]}\ -\
J$

because.
$h_{\\#}(\boldsymbol{[}\\!\boldsymbol{[}1\boldsymbol{]}\\!\boldsymbol{]}\times
J)(f)=J(f(a))=f(a)J(1)$ for $f\in{\mathscr{C}}^{\infty}_{c}(\mathbb{R}^{n})$.
In either case, we easily estimate

$\mathbf{M}(H)\
\leqslant(\operatorname{\mathrm{Lip}}h)^{k+1}{\mathbf{M}}\left(\boldsymbol{[}\\!\boldsymbol{[}0,1\boldsymbol{]}\\!\boldsymbol{]}\times
J\right)\ =\ (\operatorname{\mathrm{Lip}}h)^{k+1}{\mathbf{M}}(J)\ .$

∎

To use this Corollary in the proof of the Main Theorem via local modifications
of the given current $S$, one needs to partition $S$ into finitely many small
pieces each contained in a Lipschitz contractible piece of $A$. The choice of
the partition necessarily depends on the current $S$. Since one still needs
the mass bounds in all constructions to be independent of this choice, we
first verify some bilipschitz equivalences.

######  3.3Finite unions of links. —

Any $a\in A$ and $r>0$ determine a spherical link
$L_{r}^{a}:=A\cap\operatorname{\mathrm{Bdry}}\mathbf{U}(a,r)$. Also for
$\ell$-tuples ${\vec{a}}=(a_{1},\dots,a_{\ell})\in A^{\ell}\ $ and
${\vec{r}}=(r_{1},\dots,r_{\ell})\in(\mathbb{R}_{+}^{*})^{\ell}$ we may denote
the corresponding union of links
$L_{\vec{r}}^{\vec{a}}:=\cup_{i=1}^{\ell}L_{r_{i}}^{a_{i}}$.

The next result follows from work of A. Parusiński, [15] on locally Lipschitz
trivial stratification of subanalytic sets. To indicate individual rescalings
of the $r_{i}$, we use, for
${\vec{\lambda}}=(\lambda_{1},\dots,\lambda_{\ell})$ and
$\vec{r}=(r_{1},\dots,r_{\ell})$, the notation
$\vec{\lambda}\vec{r}=(\lambda_{1}r_{1},\dots,\lambda_{\ell}r_{\ell})$.

######  3.4 Theorem (Link Bilipschitz Equivalence). —

Associated to each pair $\vec{a},\ \vec{r}$ of such $\ell$-tuples are numbers
$\frac{1}{2}<\lambda_{\vec{r},i}^{\vec{a}}<\mu_{\vec{r},i}^{\vec{a}}<1$ so
that, for any
$\lambda_{i},\mu_{i}\in[\lambda_{\vec{r},i}^{\vec{a}},\mu_{\vec{r},i}^{\vec{a}}]$,
the two corresponding link unions ${L}_{{\vec{\lambda}}{\vec{r}}}^{\vec{a}}$
and ${L}_{{\vec{\mu}}{\vec{r}}}^{\vec{a}}$ are uniformly bilipschitz
equivalent by a map sending ${L}_{\lambda_{i}r_{i}}^{a_{i}}$ to
${L}_{\mu_{i}r_{i}}^{a_{i}}$ for $i=1,\dots,\ell$.

###### Proof.

We will work in $\mathbb{R}^{\ell}\times\mathbb{R}^{n}$. Here each set

$L_{i}\ :=\
\bigcup_{\vec{\lambda}\in[\frac{1}{2},1]^{\ell}}\\{\vec{\lambda}\\}\times
L_{\lambda_{i}r_{i}}^{a_{i}}\ =\ \left\\{(\vec{\lambda},x)\ :\
\vec{\lambda}\in[\frac{1}{2},1]^{\ell},\ x\in A,\
|x-a_{i}|^{2}=\lambda_{i}^{2}r_{i}^{2}\ \right\\}$

is clearly subanalytic. Let
$L=\cup_{i=1}^{\ell}L_{i}=\cup_{\vec{\lambda}\in[\frac{1}{2},1]^{\ell}}\\{\vec{\lambda}\\}\times
L_{\vec{\lambda}{\vec{r}}}^{\vec{a}}$. By [15, Theorem 1.4], there exists a
locally Lipschitz trivial stratification ${\mathscr{S}}$ of $L$ having
subanalytic strata and being compatible with every $L_{i}$. This means that
each $S\in{\mathscr{S}}$ which intersects any $L_{i}$ is contained in $L_{i}$.
Thus, the subfamily $\\{S\in{\mathscr{S}}\,:\,S\cap L_{i}\neq\emptyset\\}$
forms a stratification of $L_{i}$ . Let

$p(\vec{\lambda},x)\ =\ \vec{\lambda}\quad{\rm and}\quad
p_{i}(\vec{\lambda},x)\ =\ \lambda_{i}\quad{\rm for\ each}\quad
i\in\\{1,\dots,\ell\\}\ .$

For each $S\in{\mathscr{S}}$, the critical set of $p|S$,
$Z_{S}:=S\cap\\{x:{\rm dim}p({\rm Tan}(S,x))<\ell\\}$, is subanalytic. Hence,
$E={\rm Bdry}[0,1]^{\ell}\cup\cup_{S\in{\mathscr{S}}}p(Z_{S})$ is a compact
subanalytic set of dimension $\leqslant\ell-1$. Similarly the critical set of
$p_{i}|S$, $Z_{S,i}:=S\cap\\{x:p_{i}({\rm Tan}(S,x))=0\\}$, is subanalytic.
And $E_{i}=\\{0\\}\cup\\{1\\}\cup p_{i}(Z_{S,i})$, being compact and
subanalytic of dimension $0$, is finite.

To prove the theorem we choose, for each $i\in\\{1,\dots,\ell\\}$, a
nontrivial closed subinterval $[\lambda_{\vec{r},i}^{\vec{a}},\
\mu_{\vec{r},i}^{\vec{a}}]$ of one of the components $I_{i}$ of
$[\frac{1}{2},1]\setminus E_{i}$ so that the corresponding rectangle

$R_{\vec{r}}^{\vec{a}}\ :=\ \prod_{i=1}^{\ell}[\lambda_{\vec{r},i}^{\vec{a}},\
\mu_{\vec{r},i}^{\vec{a}}]\quad\subseteq\quad\big{[}\,\frac{1}{2}\,,1\big{]}^{\ell}\setminus
E\ .$

To obtain uniformly bilipschitz maps between $L\cap p^{-1}\\{\vec{\lambda}\\}$
and $L\cap p^{-1}\\{\vec{\mu}\\}$ for $\vec{\lambda},\vec{\mu}\in
R_{\vec{r}}^{\vec{a}}$, we repeat arguments of [15] on lifting Lipschitz
vectorfields to locally Lipschitz stratifications. Specifically, as in the
proof of [15, Theorem 1.6, Lemma 1.7], the constant unit vectorfield
${\mathbf{e}}_{i}$ on $\prod_{j=1}^{\ell}I_{j}$ can be lifted via $p$ to a
locally Lipschitz vectorfield $X_{i}$ on $L\cap p_{i}^{-1}(I_{i})$. Moreover,
each of the integral curves of $X_{i}$ stays in the stratum
$S\in{\mathscr{S}}$ where it begins.

To go between two points $\vec{\lambda},\vec{\mu}$ in $R_{\vec{r}}^{\vec{a}}$,
we use intervals $J_{1},\dots,J_{\ell}$ going between the intermediary points

$\vec{\nu}_{0}=\vec{\mu}\
,\quad\vec{\nu}_{i}=(\mu_{1},\dots,\mu_{i},\lambda_{i+1},\dots,\lambda_{\ell})\quad{\rm
for}\ i=1,\dots,\ell-1,\quad\vec{\nu}_{\ell}=\vec{\lambda}\ .$

Thus on the interval $J_{i}$ which goes from $\vec{\nu}_{i-1}$ to
$\vec{\nu}_{i}$ only the $i$th coordinate changes and the tangent is
${\mathbf{e}}_{i}$. Matching the endpoints of the integral curves of the lift
of ${\mathbf{e}}_{i}$ lying above $J_{i}$ gives a bilipschitz equivalence
between $\\{{\vec{\nu}_{i-1}}\\}\times{L}_{\vec{\nu}_{i-1}\vec{r}}^{\vec{a}}$
and $\\{\vec{\nu}_{i}\\}\times{L}_{\vec{\nu}_{i}\vec{r}}^{\vec{a}}$. The
bilipschitz constants are bounded independent of the initial point
$\vec{\nu}_{i-1}\in R_{\vec{r}}^{\vec{a}}$ as in [15, Theorem 1.6, Lemma 1.7].
Taking the composition of these $\ell$ bilipschitz equivalences finally gives
the desired uniformly bilipschitz equivlence between $L\cap
p^{-1}\\{\vec{\lambda}\\}$ and $L\cap p^{-1}\\{\vec{\mu}\\}$. Moreover, for
each individual $i$,
$\\{\lambda\\}\times{L}_{\lambda_{i}r_{i}}^{a_{i}}\to\\{\mu\\}\times{L}_{\mu_{i}r_{i}}^{a_{i}}$
because each integral curve stays completely in only one stratum
$S\in{\mathscr{S}}$, and each such stratum lies in at most one $L_{i}$. ∎

##  4\. Proof of the Theorem

In this section we will use $\mathbf{c}$ (rather than $\mathbf{c}(A)$) to
denote a constant depending only on $A$. Its value may increase in the course
of the proof, even in a single chain of inequalities. Since dividing $A$ into
its finitely many path components decomposes both $S_{0}$ and $\partial
S_{0}$, we may assume that $A$ itself is path-connected.

###  4.1. Partition into contractible regions

By compactness of $A$ and Theorem 3.1, we may choose a finite family of open
balls

$\\{\ \mathbf{U}(a_{i}\,,\,r_{i}/2)\ :\ i=1,2,\dots\,,\ell\ \\}$

covering $A$ where the $a_{i}$ are distinct points of $A$ and each
$\mathbf{U}(a_{i},r_{i})\cap A$ lies in a compact neighborhood $K_{i}\subseteq
A$ of $a_{i}$ having a Lipschitz deformation contraction $h_{i}$ to
$\\{a_{i}\\}$,

We would like to make, for some fixed $\vec{\lambda}\in[1/2,1]^{\ell}$,
separate adjustments of $S_{0}$ in each of the open balls $\
U_{i}^{\lambda_{i}}:=\mathbf{U}(a_{i},\lambda_{i}r_{i})\ .$ Since the
$U_{i}^{\lambda_{i}}$ will likely overlap, we will work with the corresponding
disjoint open sets

$W_{1}^{\vec{\lambda}}:=U_{1}^{\lambda_{1}},\quad
W_{2}^{\vec{\lambda}}:=U_{2}^{\lambda_{2}}\setminus\operatorname{\mathrm{Clos}}U_{1}^{\lambda_{1}},\quad\ldots\
,\quad
W_{\ell}^{\vec{\lambda}}:=U_{\ell}^{\lambda_{\ell}}\setminus\bigcup_{h=1}^{\ell-1}\operatorname{\mathrm{Clos}}U_{i}^{\lambda_{h}}\
,$

noting that
$A\subseteq\cup_{i=1}^{\ell}\operatorname{\mathrm{Clos}}W_{i}^{\vec{\lambda}}$
and
$\operatorname{\mathrm{Bdry}}W_{i}^{\vec{\lambda}}\subseteq\cup_{h=1}^{i}\operatorname{\mathrm{Bdry}}U_{h}^{\lambda_{h}}$,
a finite union of $n-1$ spheres.

We will eventually choose the radii
$\lambda_{1},\lambda_{2},\dots,\lambda_{\ell}$ in order which determine the
sets
$W_{1}^{\vec{\lambda}},W_{2}^{\vec{\lambda}},\dots,W_{\ell}^{\vec{\lambda}}$
(because $W_{i}^{\vec{\lambda}}$ only depends on
$\lambda_{1},\dots,\lambda_{i}$ ).

In choosing the radii $\lambda_{i}$, there are some Lebesgue null sets of
$\lambda$ that one must avoid. To describe one, first, note that, for any
finite Radon measure $\alpha$ on $\mathbb{R}^{n}$, the set

$\Lambda_{\alpha}:=\\{\lambda>0\ :\ \alpha(L_{\lambda r_{i}}^{a_{i}})>0\ {\rm
for\ some}\ i\ \\}\ $

is at most countable because $\\{\lambda:\ \alpha(L_{\lambda
r_{i}}^{a_{i}})>1/j\\}$ is finite for each $j\in{\mathbb{N}}$. One useful
consequences, for any positive
$\lambda_{1},\dots,\lambda_{i}\not\in\Lambda_{\|S_{0}\|+\|\partial S_{0}\|}$,
is that

$(\|S_{0}\|+\|\partial S_{0}\|)\left({\rm
Bdry}(W_{i}^{\vec{\lambda}})\right)=0\ ,\quad S_{0}\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,\overline{W}_{i}^{\vec{\lambda}}=S_{0}\hskip
2.5pt{\vrule height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,W_{i}^{\vec{\lambda}}\ ,\quad(\partial
S_{0})\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,\overline{W}_{i}^{\vec{\lambda}}=(\partial
S_{0})\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,W_{i}^{\vec{\lambda}},$

for $i=1,\dots,\ell$. The disjoint open sets $W_{i}^{\vec{\lambda}}$ now give
the two decompositions

$S_{0}\ =\ \sum_{i=1}^{\ell}S_{0}\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,W_{i}^{\vec{\lambda}}\quad{\rm
and}\quad(\partial S_{0})\ =\ \sum_{i=1}^{\ell}(\partial S_{0})\hskip
2.5pt{\vrule height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,W_{i}^{\vec{\lambda}}\ ,$ (1)

obtained by ignoring the boundaries of the $W_{i}^{\vec{\lambda}}$.

###  4.2. Proof of the case $k=0$

###### Proof.

We will make essentially two different applications of Corollary 3.2 with
$k=0$. First, for $\vec{\lambda}=(\lambda_{1},\dots,\lambda_{\ell})$ as above
and $i\in\\{1,2,\dots,\ell\\}$, we apply Corollary 3.2 with $J=(\partial
S_{0})\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,W_{i}^{\vec{\lambda}}\ $ to find that

$E_{i}\ :=\
-h_{i\\#}\left(\boldsymbol{[}\\!\boldsymbol{[}0,1\boldsymbol{]}\\!\boldsymbol{]}\times[(\partial
S_{0})\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,W_{i}^{\vec{\lambda}}]\right)\in\mathbf{I}_{1}(A)$

has

$\partial E_{i}=(\partial S_{0})\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,W_{i}^{\vec{\lambda}}\ -\
g_{i}\boldsymbol{[}\\!\boldsymbol{[}a_{i}\boldsymbol{]}\\!\boldsymbol{]}\quad{\rm
and}\quad\mathbf{M}(E_{i})\leqslant\operatorname{\mathrm{Lip}}(h_{i})\mathbf{M}[(\partial
S_{0})\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,W_{i}^{\vec{\lambda}}]\ ,$

where $g_{i}=(\partial
S_{0})\left(\mathbbm{1}_{W_{i}^{\vec{\lambda}}}\right)\in\mathbb{Z}$ is the
total multiplicity of $\partial S_{0}$ in $W_{i}^{\vec{\lambda}}$. Observe
that by (1),

$\sum_{i=1}^{\ell}g_{i}\ =\ \sum_{i=1}^{\ell}[(\partial S_{0})\hskip
2.5pt{\vrule height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,W_{i}^{\vec{\lambda}})](1)\
=\left(\sum_{i=1}^{\ell}(\partial S_{0})\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,W_{i}^{\vec{\lambda}}\right)(1)\ =\
(\partial S_{0})(1)\ =\ 0\ .$ (2)

Second, we can use Corollary 3.2 to bound the intrinsic diameter of $A$.
Consider the simple situation in Corollary 3.2 when $J$ is a single point with
multiplicity, say
$J=g\boldsymbol{[}\\!\boldsymbol{[}b,\boldsymbol{]}\\!\boldsymbol{]}$ where
$g\in\mathbb{Z}$ and $b\in K$. The resulting
$H=-h_{\\#}\left(\boldsymbol{[}\\!\boldsymbol{[}0,1\boldsymbol{]}\\!\boldsymbol{]}\times
g\boldsymbol{[}\\!\boldsymbol{[}b\boldsymbol{]}\\!\boldsymbol{]}\right)$ is
$-g$ times the path $t\mapsto h(t,b)$ from $b$ to $a$ and

$\partial H\ =\
g\boldsymbol{[}\\!\boldsymbol{[}b\boldsymbol{]}\\!\boldsymbol{]}-g\boldsymbol{[}\\!\boldsymbol{[}a\boldsymbol{]}\\!\boldsymbol{]}\quad{\rm
and}\quad\mathbf{M}(H)\ =\ |g|{\rm length}\left(h(\cdot,b)\right)\ \leqslant\
|g|\operatorname{\mathrm{Lip}}h\ .$

Thus the intrinsic diameter of $K$ is $\leqslant 2\times{\rm
length}\left(h(\cdot,b)\right)\leqslant 2\operatorname{\mathrm{Lip}}h$.
Inasmuch as the intrinsic diameter of the union of two intersecting sets is at
most the sum of their intrinsic diameters, we readily deduce that the
intrinsic diameter of $A$ is at most
$D(A):=2\sum_{i=1}^{\ell}\operatorname{\mathrm{Lip}}(h_{i})$.

To complete the proof of the $k=0$ case, we choose, for each $i=1,\dots,\ell$,
a curve $\gamma_{i}:[0,1]\to A\ $ from $a_{1}$ to $a_{i}$ with ${\rm
length}(\gamma_{i})\leqslant D(A)$. We now define

$S\ :=\
\sum_{i=1}^{\ell}E_{i}+g_{i}\gamma_{i\\#}\boldsymbol{[}\\!\boldsymbol{[}0,1\boldsymbol{]}\\!\boldsymbol{]}$

and verify, by (2), that

$\displaystyle\partial S\ $ $\displaystyle=\ \sum_{i=1}^{\ell}\left((\partial
S_{0})\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,W_{i}^{\vec{\lambda}}\ -\
g_{i}\boldsymbol{[}\\!\boldsymbol{[}a_{i}\boldsymbol{]}\\!\boldsymbol{]}\right)\
+\
\left(g_{i}\boldsymbol{[}\\!\boldsymbol{[}a_{i}\boldsymbol{]}\\!\boldsymbol{]}-g_{i}\boldsymbol{[}\\!\boldsymbol{[}a_{1}\boldsymbol{]}\\!\boldsymbol{]}\right)$
$\displaystyle=\ \partial S_{0}\ -\
\left(\sum_{i=1}^{\ell}g_{i}\right)\cdot\boldsymbol{[}\\!\boldsymbol{[}a_{1}\boldsymbol{]}\\!\boldsymbol{]}\
=\ \partial S_{0}\ -\ 0\ ,$

and

$\displaystyle\mathbf{M}(S)\ $ $\displaystyle\leqslant\
\sum_{i=1}^{\ell}\mathbf{M}(E_{i})+\sum_{i=1}^{\ell}|g_{i}|D(A)$
$\displaystyle\leqslant\
\sum_{i=1}^{\ell}\operatorname{\mathrm{Lip}}(h_{i})\mathbf{M}[(\partial
S_{0})\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,W_{i}^{\vec{\lambda}}]\ +\
D(A)\mathbf{M}[(\partial S_{0})\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,W_{i}^{\vec{\lambda}}]\
\leqslant\mathbf{c}\mathbf{M}(\partial S_{0})\ .$

∎

###  4.3. Proof of the case $k\geqslant 1$

###### Proof.

We use induction on $\operatorname{\mathrm{dim}}A$. For
$\operatorname{\mathrm{dim}}A=0$, the current $S_{0}\in\mathbf{I}_{k+1}(A)$
necessarily vanishes, and the theorem is trivially true. Also for
$\operatorname{\mathrm{dim}}A=1$, $S_{0}\in\mathbf{I}_{k+1}(A)$ is
nonvanishing only for $k=0$, in which case the theorem was established in the
previous section §4.2. So we now assume that $k\geqslant 1$, that
$\operatorname{\mathrm{dim}}A\geqslant 2$ and inductively, that the theorem is
true for any compact subanalytic set of dimension less than
$\operatorname{\mathrm{dim}}A$.

For this inductive step, note that the links
$L_{\lambda_{i}r_{i}}^{a_{i}}=A\cap\operatorname{\mathrm{Bdry}}U_{i}^{\lambda_{i}}$,
as well as all the sets
$A\cap\operatorname{\mathrm{Bdry}}W_{i}^{\vec{\lambda}}$, are compact and
subanalytic of dimension $<\dim A$. We need some more discussion about the
choice of the radii $\lambda_{i}$ , the corresponding balls
$U_{i}^{\lambda_{i}}$, disjoint open sets $W_{i}^{\vec{\lambda}}$ and their
boundaries. First the distinctness of the centers of the $U_{i}^{\lambda_{i}}$
already guarantees the pairwise transversality of their spherical boundaries.

We describe a particular subdivision of
$\bigcup_{i=1}^{\ell}\operatorname{\mathrm{Bdry}}W_{i}^{\vec{\lambda}}$ by
using the sequence of closed spherical regions

$\Gamma_{i}^{\vec{\lambda}}:=(\operatorname{\mathrm{Bdry}}U_{i}^{\lambda_{i}})\cap(\operatorname{\mathrm{Bdry}}W_{i}^{\vec{\lambda}})\quad=\quad(\operatorname{\mathrm{Bdry}}U_{i}^{\lambda_{i}})\setminus\bigcup_{h=1}^{i-1}U_{h}^{\lambda_{h}}\
,$

whose interiors $\ \Gamma_{i}^{\vec{\lambda}\,\circ}\ $are disjoint. While
each $\Gamma_{i}^{\vec{\lambda}}$ clearly does not overlap any
$W_{h}^{\vec{\lambda}}$ for $h\leqslant i$, we will also be interested in the
cover of $\ \Gamma_{i}^{\vec{\lambda}}\ $ by the closed $n-1$ dimensional
spherical regions

$\Gamma_{i,j}^{\vec{\lambda}}:=\Gamma_{i}^{\vec{\lambda}}\cap\operatorname{\mathrm{Clos}}W_{j}^{\vec{\lambda}}\quad{\rm
for}\quad j=i+1,\dots,\ell\ ,$

whose interiors $\Gamma_{i,j}^{{\vec{\lambda}}\,\circ}$ are disjoint and whose
boundaries are contained in $n-2$ dimensional spheres. Note that

$A\cap\operatorname{\mathrm{Bdry}}W_{i}^{\vec{\lambda}}\quad=\quad
A\cap\left(\Gamma_{i}^{\vec{\lambda}}\ \cup\
\bigcup_{h=1}^{i-1}\Gamma_{h,i}^{\vec{\lambda}}\right)\quad=\quad
A\cap\left(\bigcup_{h=1}^{i-1}\Gamma_{h,i}^{\vec{\lambda}}\ \cup\
\bigcup_{j=i+1}^{\ell}\Gamma_{i,j}^{\vec{\lambda}}\right)\ .$

Again all the corresponding interior spherical regions,
$\Gamma_{h,i}^{{\vec{\lambda}}\,\circ}$ or
$\Gamma_{i,j}^{{\vec{\lambda}}\,\circ}$, are disjoint for each fixed $i$, with
$h$ ranging in $\\{1,\dots,i-1\\}$ and $j$ ranging in $\\{i+1,\dots,\ell\\}$.

Many of the partitioning domains $W_{i}^{\vec{\lambda}}$ or many of their
boundary regions $\Gamma_{i,j}^{{\vec{\lambda}}\,\circ}$ may be empty. In any
case, for each nonempty $W_{i}^{\vec{\lambda}}$, the boundary
$\operatorname{\mathrm{Bdry}}W_{i}^{\vec{\lambda}}$ has a single nonempty
“outward-pointing” region
$\Gamma_{i}^{\vec{\lambda}}\subseteq\operatorname{\mathrm{Bdry}}U_{i}^{\lambda_{i}}$,
which may decompose into various $\Gamma_{i,j}^{\vec{\lambda}}$ for some later
$j>i$. And any remainder
$\operatorname{\mathrm{Bdry}}W_{i}^{\vec{\lambda}}\setminus\Gamma_{i}^{\vec{\lambda}}$
then consists of “inward-pointing” regions $\Gamma_{h,i}^{\vec{\lambda}}$
coming from distinct spheres for some earlier $h<i$.

Each set $A\cap\Gamma_{i}^{\vec{\lambda}}\ $ is contained in the link union
$L_{\vec{\lambda}\vec{r}}^{\vec{a}}$. Moreover,
$A\cap\Gamma_{i}^{\vec{\lambda}}$ is mapped to $A\cap\Gamma_{i}^{\vec{\mu}}$
under the bilipschitz equivalence that maps
$L_{\vec{\lambda}\vec{r}}^{\vec{a}}$ to $L_{\vec{\mu}\vec{r}}^{\vec{a}}$,
obtained in Theorem 3.4. In fact, since each individual link
$L_{\lambda_{i}r_{i}}^{a_{i}}=A\cap\operatorname{\mathrm{Bdry}}U_{i}^{\lambda_{i}}$
is mapped to the corresponding link
$L_{\mu_{i}r_{i}}^{a_{i}}=A\cap\operatorname{\mathrm{Bdry}}U_{i}^{\mu_{i}}$, a
similar $\vec{\lambda}\to\vec{\mu}$ transfer property holds for corresponding
finite intersections, set differences, connected components, or disjoint
unions of such links. We easily see that each region
$\Gamma^{\vec{\lambda}}_{i}$ partitions into finitely many pieces, each given
as a connected component of a difference of finite intersections of spheres
$\operatorname{\mathrm{Bdry}}U_{j}^{\lambda_{j}}$. Thus sets obtained by
intersecting these with $A$ all inherit the desired
$\vec{\lambda}\to\vec{\mu}$ transfer property.

Since each set $A\cap\Gamma_{i}^{\vec{\lambda}}$ is compact and subanalytic of
dimension $<\dim A$, it has, by our induction on $\dim A$, its own linear
isoperimetric inequality. Moreover, by the bilipschitz equivalence from
Theorem 3.4 and Proposition 2.3, we may now assume

(*) The linear isoperimetric inequality is true with the same constant $\
\mathbf{c}\ $ for

$\ A\cap\Gamma_{1}^{\vec{\lambda}}\ ,\ \ldots\ ,\
A\cap\Gamma_{\ell}^{\vec{\lambda}}\quad{\bf for\ all}\quad\vec{\lambda}\in
R_{\vec{r}}^{\vec{a}}\ .$

The uniformity of this estimate will allow us to choose and fix an $\ell$
tuple of scaling factors $\vec{\lambda}=(\lambda_{1},\dots,\lambda_{\ell})\in
R_{\vec{r}}^{\vec{a}}a\ $ depending on the given chain
$S_{0}\in{\mathbf{I}}_{k+1}(A)$, but still have mass estimates independent of
$S_{0}$.

As we did in §4.1, we again consider some Lebesgue null sets of $\lambda_{i}$
to avoid to guarantee some desired properties relative to the given chain.

For any fixed $S\in\mathbf{I}_{k+1}({\mathbb{R}^{n}})$ and positive
$\lambda_{1},\dots,\lambda_{i}\not\in\Lambda_{\|S\|+\|\partial S\|}$, we see
as before that

$(\|S\|+\|\partial S\|)\left({\rm Bdry}(W_{h}^{\vec{\lambda}})\right)=0\
,\quad S\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,\overline{W}_{h}^{\vec{\lambda}}=S\hskip
2.5pt{\vrule height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,W_{h}^{\vec{\lambda}}\ ,\quad(\partial
S)\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,\overline{W}_{h}^{\vec{\lambda}}=(\partial
S)\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,W_{h}^{\vec{\lambda}},$

for $h=1,\dots,i$. Now we find, in $\cup_{h=1}^{i}U_{h}^{\lambda_{h}}$, the
two decompositions

$S\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,\left(\cup_{h=1}^{i}U_{h}^{\lambda_{h}}\right)\
=\ \sum_{h=1}^{i}S\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,W_{h}^{\vec{\lambda}}\quad{\rm
and}\quad(\partial S)\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,\left(\cup_{h=1}^{i}U_{h}^{\lambda_{h}}\right)=\
\sum_{h=1}^{i}(\partial S)\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,W_{h}^{\vec{\lambda}}\ ,$ (3)

obtained by ignoring the boundaries of the $W_{h}^{\vec{\lambda}}$.

Using the Lipschitz map $u_{i}(x)=r_{i}^{-1}|x-a_{i}|$, we have that
$U_{i}^{\lambda}\ =\ \\{x\,:\,u_{i}(x)<\lambda\\}$, and there is another
exceptional null set $\Lambda^{S}_{i}\subseteq[0,1]$, on the complement of
which one obtains integral chain slices satisfying the formulas

$\begin{split}\langle S,u_{i},\lambda\rangle&=\partial(S\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U_{i}^{\lambda})-(\partial S)\hskip
2.5pt{\vrule height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U_{i}^{\lambda}\quad\in\mathbf{I}_{k}(A\cap\operatorname{\mathrm{Bdry}}U_{i}^{\lambda})\
,\\\ \partial\langle S,u_{i},\lambda\rangle\ &=\ -\langle\partial
S,u_{i},\lambda\rangle=-\partial[(\partial S)\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U_{i}^{\lambda}]\quad\in\mathbf{I}_{k-1}(A\cap\operatorname{\mathrm{Bdry}}U_{i}^{\lambda})\
.\end{split}$ (4)

and the corresponding measures satisfy

$\|\partial(S\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U_{i}^{\lambda})\|\ =\ \|\partial
S\|\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U_{i}^{\lambda}\ +\ \|\langle
S,u_{i},\lambda\rangle\|\ ,\quad\ \|\partial[(\partial S)\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U_{i}^{\lambda}]\,\|\ =\
\|\langle\partial S,u_{i},\lambda\rangle\|\ ,$

with $\|\partial S\|\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U_{i}^{\lambda}$ and $\|\langle
S,u_{i},\lambda\rangle\|$ being mutually singular.

Moreover, since $\operatorname{\mathrm{Lip}}u_{i}=r_{i}^{-1}$, we have the
integral slice mass inequality

$\int_{\lambda_{\vec{r},i}^{\vec{a}}}^{\mu_{\vec{r},i}^{\vec{a}}}\mathbf{M}\langle\partial
S,u_{i},\lambda\rangle\,d\lambda\leqslant r_{i}^{-1}\mathbf{M}(\partial S)\ .$
(5)

We will also need a similar discussion for a fixed
$Q\in\mathbf{I}_{k}(A\cap\Gamma_{i}^{\vec{\lambda}})$. If
$j\in\\{i+1,\dots,\ell\\}$ and $\lambda_{j}\not\in\Lambda_{\|Q\|}\ $, then $\
\|Q\|\left({\operatorname{\mathrm{Bdry}}}(W_{j}^{\vec{\lambda}})\right)=0\ $.
In case this is true for every such $j\in\\{i+1,\dots,\kappa\\}$, one gets the
decomposition

$Q\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,(U_{1}^{\lambda_{1}}\cup\dots\cup
U_{\kappa}^{\lambda_{\kappa}})\ =\ \sum_{j=i+1}^{\kappa}Q\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,\Gamma_{i,j}^{\vec{\lambda}}\ .$ (6)

There is also another exceptional null set $\Lambda^{Q}_{i}\subseteq[0,1]$ so
that each $\lambda\in[0,1]\setminus\Lambda^{Q}_{i}$ gives an integral chain
slices $\langle Q,u_{i},\lambda\rangle$ satisfying

$\langle Q,u_{i},\lambda\rangle=\partial(Q\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U_{i}^{\lambda})-(\partial Q)\hskip
2.5pt{\vrule height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U_{i}^{\lambda}\quad\in\mathbf{I}_{k-1}(A\cap\Gamma_{i}^{\vec{\lambda}})\
,$ (7)

with the orthogonal decomposition of measures

$\|\partial(Q\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U_{i}^{\lambda})\|\ =\ \|\partial
Q\|\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U_{i}^{\lambda}\ +\ \|\langle
Q,u_{i},\lambda\rangle\|\ .$

We shall now choose below $\lambda_{1},\lambda_{2},\dots,\lambda_{\ell}$ in
order, and use these radii to construct chains

$H_{1},\dots H_{\ell}\in\bf I_{k+1}(A)$ so that

$\partial(H_{1}+\cdots+H_{\ell})\ =\ \partial S_{0}\quad{\rm
and}\quad\mathbf{M}(H_{1})\ ,\dots,\ \mathbf{M}(H_{\ell})\
\leqslant\mathbf{c}\mathbf{M}(\partial S_{0})\ ,$

with $\mathbf{c}$ depending only on $A$, and independent of $S_{0}$. This will
complete the proof by letting $S=H_{1}+\cdots+H_{\ell}$.

For our first choice, we apply (4) and the second inequality of (5) with
$i=1$, $S=S_{0}$, to find a “good” radius

$\lambda_{1}\in[\lambda_{\vec{r},1}^{\vec{a}},\mu_{\vec{r},1}^{\vec{a}}]\
\setminus\ \left(\Lambda_{\|S_{0}\|+\|\partial
S_{0}\|}\cup\Lambda^{S_{0}}_{1}\right)$

so that the chain

$R_{1}\ :=\ \partial\langle S_{0},u_{1},\lambda_{1}\rangle\ =\
-\langle\partial S_{0},u_{1},\lambda_{1}\rangle\ \ =-\partial[(\partial
S_{0})\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U_{1}^{\lambda_{1}}]\
\in\mathbf{I}_{k-1}(A\cap\Gamma_{1}^{\vec{\lambda}})$

has mass satisfying

$\mathbf{M}(R_{1})\ \leqslant\
[(\mu_{\vec{r},1}^{\vec{a}}-\lambda_{\vec{r},1}^{\vec{a}})r_{i}]^{-1}\mathbf{M}(\partial
S_{0})\ =\ \mathbf{c}\mathbf{M}(\partial S_{0})\ .$

Since the chain $P_{1}:=\langle
S_{0},u_{1},\lambda_{1}\rangle\in\mathbf{I}_{k}(A\cap\Gamma_{1}^{\vec{\lambda}})$
has $\partial P_{1}=R_{1}$, we may apply our dimension induction (*), with $A$
replaced by the lower dimensional set $A\cap\Gamma_{1}^{\vec{\lambda}}$ and
$S_{0}$ replaced by $P_{1}$, to obtain a chain
${Q_{1}}\in\mathbf{I}_{k}(A\cap\Gamma_{1}^{\vec{\lambda}})$ with $\partial
Q_{1}=\partial P_{1}=R_{1}$ and

$\mathbf{M}(Q_{1})\ \leqslant\mathbf{c}\mathbf{M}(\partial Q_{1})\ =\
\mathbf{c}\mathbf{M}(R_{1})\ \leqslant\ \mathbf{c}\mathbf{M}(\partial S_{0})\
.$

Note that $\quad J_{1}\ :=\ (\partial S_{0})\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,W_{1}^{\vec{\lambda}}\ +\
Q_{1}\quad\in\mathbf{I}_{k}(A\cap\operatorname{\mathrm{Clos}}U_{1}^{\lambda_{1}})$
has

$\partial J_{1}\ =\ \partial[(\partial S_{0})\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,W_{1}^{\vec{\lambda}}]+\ \partial
P_{1}\ =\ -R_{1}+R_{1}\ =\ 0\ ,$

and

$\mathbf{M}(J_{1})\ \leqslant\ \mathbf{M}(\partial S_{0})+\mathbf{M}(Q_{1})\
\leqslant\ \mathbf{c}\mathbf{M}(\partial S_{0})\ .$

Inasmuch as the cycle $J_{1}$ has support in $\ A\cap\mathbf{U}(a_{1},r_{1})\
$, we infer from Corollary 3.2 that the contraction

$H_{1}\ :=\
-h_{1\\#}\left(\boldsymbol{[}\\!\boldsymbol{[}0,1\boldsymbol{]}\\!\boldsymbol{]}\times
J_{1}\right)\ \in\mathbf{I}_{k+1}(A)\ $

has $\quad\partial H_{1}\ =\ J_{1}\quad$ and

$\mathbf{M}(H_{1})+\mathbf{M}(\partial H_{1})\ \leqslant\
\left((\operatorname{\mathrm{Lip}}h_{1})^{k+1}+1\right)\mathbf{M}(J_{1})\
\leqslant\ \mathbf{c}\mathbf{M}(\partial S_{0})\ .$

Letting $S_{1}:=S_{0}-H_{1}$, we have that

$\mathbf{M}(\partial S_{1})\ \leqslant\mathbf{M}(\partial
S_{0})+\mathbf{M}(\partial H_{1})\ \leqslant\mathbf{c}\mathbf{M}(\partial
S_{0})\ ,$

and we easily see that one may replace $S_{0}$ by $S_{1}$ to prove the
theorem. The advantage in passing from $S_{0}$ to $S_{1}$ is that subtractng
$H_{1}$ essentially “moves the boundary” out of the ball
$U_{1}^{\lambda_{1}}$. In fact, the chain

$\partial S_{1}\ =\ \partial S_{0}-\partial H_{1}\ =\ \partial S_{0}-(\partial
S_{0})\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,W_{1}^{\vec{\lambda}}-Q_{1}\ =\
(\partial S_{0})\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,(A\setminus
W_{1}^{\vec{\lambda}})-Q_{1}\ $

has

$\operatorname{\mathrm{spt}}(\partial S_{1})\ \subseteq\ (A\setminus
W_{1}^{\vec{\lambda}})\cup\Gamma_{1}^{\vec{\lambda}}\ =\ A\setminus
U_{1}^{\lambda_{1}}\ .$

We wish to continue moving the boundary in steps out of the remaining balls
$U_{i}^{\lambda_{i}}$. However, to find $H_{2}$ for the next modification $\
S_{2}=S_{1}-H_{2}\ $, or generally $H_{i}$ for $\ S_{i}=S_{i-1}-H_{i}$, is
somewhat more involved. For example, to get the boundary of $S_{2}$ out of
both balls

$U_{1}^{\lambda_{1}}\cup U_{2}^{\lambda_{2}}\ =\
W_{1}^{\vec{\lambda}}\cup\Gamma_{1,2}^{\vec{\lambda}\ o}\cup
W_{2}^{\vec{\lambda}}\ ,$

we will below need to choose $H_{2}$ to attach to $H_{1}$ along the interface
$\Gamma_{1,2}^{\vec{\lambda}\ o}$. For the reader’s convenience, we will first
go through this second step, involving the choice of $H_{2}$, carefully before
describing the general $i$th step.

For the second step, we again start with the choice of radius $\lambda_{2}$.
By applying (4), (5), (6), (7) with $i=2$, $S=S_{1}$, and $Q=Q_{1}$. We obtain

$\lambda_{2}\in[\lambda_{\vec{r},2}^{\vec{a}},\mu_{\vec{r},2}^{\vec{a}}]\
\setminus\ \left(\Lambda_{\|S_{0}\|+\|\partial S_{0}\|+\|S_{1}\|+\|\partial
S_{1}\|+\|Q_{1}\|}\cup\Lambda^{S_{0}}_{1}\cup\Lambda^{S_{1}}_{1}\cup\Lambda^{Q_{1}}_{1}\right)$

so that the chain

$R_{2}\ :=\ \partial\langle S_{1},u_{2},\lambda_{2}\rangle\ =\
-\langle\partial S_{1},u_{2},\lambda_{2}\rangle\ =-\partial[(\partial
S_{1})\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U_{2}^{\lambda_{2}}]$

has mass satisfying

$\mathbf{M}(R_{2})\ \leqslant\
[(\mu_{\vec{r},2}^{\vec{a}}-\lambda_{\vec{r},2}^{\vec{a}})r_{2}]^{-1}\mathbf{M}(\partial
S_{1})\ \leqslant\ \mathbf{c}\mathbf{M}(\partial S_{1})\ \leqslant\
\mathbf{c}\mathbf{M}(\partial S_{0})\ .$

Since the chain $P_{2}:=\langle
S_{1},u_{2},\lambda_{2}\rangle\in\mathbf{I}_{k}(A\cap\Gamma_{2}^{\vec{\lambda}})$
has $\partial P_{2}=R_{2}$, we may again apply our dimension induction (*),
this time with $A$ replaced by the lower dimensional set
$A\cap\Gamma_{2}^{\vec{\lambda}}$ and $S_{0}$ replaced by $P_{2}$, to obtain a
chain ${Q_{2}}\in\mathbf{I}_{k}(A\cap\Gamma_{2}^{\vec{\lambda}})$ with
$\partial Q_{2}=\partial P_{2}=R_{2}$ and

$\mathbf{M}(Q_{2})\ \leqslant\mathbf{c}\mathbf{M}(\partial Q_{2})\ =\
\mathbf{c}\mathbf{M}(R_{2})\ \leqslant\ \mathbf{c}\mathbf{M}(\partial S_{0})\
.$

The chain $Q_{1,2}:=Q_{1}\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,\Gamma_{1,2}^{\vec{\lambda}}=Q_{1}\hskip
2.5pt{\vrule height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,\ \overline{W}_{2}^{\vec{\lambda}}$
shows up in the boundary calculation

$\begin{split}\partial(S_{1}\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U_{2}^{\lambda_{2}})\ &=\ \langle
S_{1},u_{2},\lambda_{2}\rangle\ +\ (\partial S_{1})\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U_{2}^{\lambda_{2}}\\\ &=\ P_{2}\ +\
(\partial S_{1})\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,\overline{W}_{2}^{\vec{\lambda}}\\\ &=\
P_{2}\ +\ (\partial S_{0})\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,\overline{W}_{2}^{\vec{\lambda}}\ -\
(\partial H_{1})\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,\overline{W}_{2}^{\vec{\lambda}}\\\ &=\
P_{2}\ +\ (\partial S_{0})\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,\overline{W}_{2}^{\vec{\lambda}}\ -\
(\partial S_{0})\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,\
(W_{1}^{\vec{\lambda}}\cap\overline{W}_{2}^{\vec{\lambda}})\ -\ Q_{1}\hskip
2.5pt{\vrule height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,\overline{W}_{2}^{\vec{\lambda}}\\\ &=\
P_{2}\ +\ (\partial S_{0})\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,\ W_{2}^{\vec{\lambda}}\ +0\ -\
Q_{1,2}\ .\end{split}$

Thus the chain $\quad J_{2}\ :=\ (\partial S_{0})\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,\
W_{2}^{\vec{\lambda}}+Q_{2}-Q_{1,2}\quad$ has

$\partial J_{2}\ =\ \partial[(\partial S_{0})\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,\ W_{2}^{\vec{\lambda}}+P_{2}-Q_{1,2}]\
=\ \partial^{2}(S_{1}\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U_{2}^{\vec{\lambda}})\ =\ 0\ ,$

and

$\mathbf{M}(J_{2})\leqslant\ \mathbf{M}(\partial
S_{0})+\mathbf{M}(Q_{2})+\mathbf{M}(Q_{1})\ \leqslant\
\mathbf{c}\mathbf{M}(\partial S_{0})\ .$

Since $J_{2}$ also has support in $\ A\cap\mathbf{U}(a_{2},r_{2})\ $, we find
from Corollary 3.2 that the contraction

$H_{2}\ :=\
-h_{2\\#}\left(\boldsymbol{[}\\!\boldsymbol{[}0,1\boldsymbol{]}\\!\boldsymbol{]}\times
J_{2}\ \right)\ \in\mathbf{I}_{k+1}(A)\ $

has $\quad\partial H_{2}\ =\ J_{2}\quad$ and

$\mathbf{M}(H_{2})+\mathbf{M}(\partial H_{2})\ \leqslant\
\left((\operatorname{\mathrm{Lip}}h_{2})^{k+1}+1\right)\mathbf{M}(J_{2})\
\leqslant\ \mathbf{c}\mathbf{M}(\partial S_{0})\ .$

Letting $S_{2}:=S_{1}-H_{2}=S_{0}-H_{1}-H_{2}$ , we have

$\mathbf{M}(\partial S_{2})\ \leqslant\mathbf{c}[\mathbf{M}(\partial
S_{1})+\mathbf{M}(\partial H_{2})]\ \leqslant\mathbf{c}\mathbf{M}(\partial
S_{0})\ ,$

and

$\begin{split}\partial S_{2}\ &=\partial S_{1}-\partial H_{2}\ =\ (\partial
S_{0})\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,(A\setminus
W_{1}^{\vec{\lambda}})-Q_{1}\ -Q_{2}+Q_{1,2}-(\partial S_{0})\hskip
2.5pt{\vrule height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,W_{2}^{\vec{\lambda}}\\\ &=\ (\partial
S_{0})\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,[A\setminus(W_{1}^{\vec{\lambda}}\cup
W_{2}^{\vec{\lambda}})]\ -\ Q_{1}+Q_{1,2}-\ Q_{2}\\\ &=\ (\partial
S_{0})\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,[A\setminus(U_{1}^{\lambda_{1}}\cup
U_{2}^{\lambda_{2}})]\ -\ Q_{1}+Q_{1,2}-\ Q_{2}\ .\end{split}$

because $\|\partial
S_{0})\|(\operatorname{\mathrm{Bdry}}U_{1}^{\lambda_{1}})=0$. Inasmuch as

$Q_{1}\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U_{1}^{\lambda_{1}}=0\ ,\ \ Q_{1}\hskip
2.5pt{\vrule height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U_{2}^{\lambda_{2}}=Q_{1,2}\ ,\ \
Q_{1,2}\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U_{1}^{\lambda_{1}}=0\ ,\ \ Q_{2}\hskip
2.5pt{\vrule height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U_{1}^{\lambda_{1}}=0\ ,\ \ Q_{2}\hskip
2.5pt{\vrule height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U_{2}^{\lambda_{2}}=0\ ,$

we see that

$\begin{split}(\partial S_{2})\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,(U_{1}^{\lambda_{1}}\cup
U_{2}^{\lambda_{2}})\ &=\ 0\ -\ (Q_{1}-Q_{1,2})\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,(U_{1}^{\lambda_{1}}\cup
U_{2}^{\lambda_{2}})\ -\ Q_{2}\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,(U_{1}^{\lambda_{1}}\cup
U_{2}^{\lambda_{2}})\\\ &=\ 0\ -\ (Q_{1,2}-Q_{1,2})-\ 0\ =\ 0\ ,\end{split}$

and conclude that

$\operatorname{\mathrm{spt}}(\partial S_{2})\ \subseteq\ A\setminus\
(U_{1}^{\lambda_{1}}\cup U_{2}^{\lambda_{2}})\ ,$

which completes the second step.

Modifying the above, we now describe how to obtain the $i$th step from the
$(i-1)$st. We assume that $i\leqslant\ell-1$ and that we have already chosen,
for each $h\in\\{1,\ \dots,i-1\ \\}$,

$\lambda_{h}>0\ ,\quad
Q_{h}\in\mathbf{I}_{k}(A\cap\Gamma_{h}^{\vec{\lambda}})\ ,\quad
H_{h}\in\mathbf{I}_{k+1}(A)\ ,\quad
S_{h}:=S_{0}-\sum_{\eta=1}^{i-1}H_{\eta}\in\mathbf{I}_{k+1}(A)\ ,$

to satisfy the three conditions: (I)h $\quad\partial H_{h}\ =\ (\partial
S_{0})\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,\ W_{h}^{\vec{\lambda}}\ +\ Q_{h}-\
\sum_{\eta=1}^{h-1}Q_{\eta,h}\ $ where $\ Q_{\eta,h}:=Q_{\eta}\hskip
2.5pt{\vrule height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,\Gamma_{\eta,h}^{\vec{\lambda}}$ ,
(II)h $\quad\mathbf{M}(Q_{h})\ +\ \mathbf{M}(H_{h})\ +\ \mathbf{M}(\partial
H_{h})\ +\ \mathbf{M}(\partial S_{h})\ \leqslant\
\mathbf{c}\mathbf{M}(\partial S_{0})$ . (III)h
$\quad\quad\quad\operatorname{\mathrm{spt}}(\partial S_{h})\ \subseteq\
A\setminus\ (U_{1}^{\lambda_{1}}\cup\ \cdots\ \cup U_{h}^{\lambda_{h}}\ )$ ,
(Here the formula for $\partial H_{1}$ is correct with the convention
$\sum_{\eta=1}^{0}Q_{\eta,h}=0$.)

We first choose $\lambda_{i}$ by applying (4), (5), (6), and (7), this time
with $i=i$, $S=S_{i-1}$, and $Q=Q_{1},\dots,Q_{i-1}$ . We obtain

$\lambda_{i}\in[\lambda_{\vec{r},i}^{\vec{a}},\mu_{\vec{r},i}^{\vec{a}}]\
\setminus\ \Lambda_{\sum_{h=0}^{i-1}\|S_{h}\|+\|\partial
S_{h}\|+\|Q_{h}\|}\setminus\cup_{h=0}^{i-1}\left(\Lambda^{S_{h}}_{1}\cup\Lambda^{Q_{h}}_{1}\right)$

so that the chain

$R_{i}\ :=\ \partial\langle S_{i-1},u_{i},\lambda_{i}\rangle\ =\
-\langle\partial S_{i-1},u_{i},\lambda_{i}\rangle\ =-\partial[(\partial
S_{i-1})\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U_{i}^{\lambda_{i}}]\in\mathbf{I}_{k-1}(A\cap\Gamma_{i}^{\vec{\lambda}})$

has mass satisfying

$\mathbf{M}(R_{i})\ \leqslant\
[(\mu_{\vec{r},i}^{\vec{a}}-\lambda_{\vec{r},i}^{\vec{a}})r_{i}]^{-1}\mathbf{M}(\partial
S_{i-1})\ \leqslant\ \mathbf{c}\mathbf{M}(\partial S_{i-1})\
\leqslant\mathbf{c}\mathbf{M}(\partial S_{0})\ .$

Since the chain $P_{i}:=\langle
S_{i-1},u_{i},\lambda_{i}\rangle\in\mathbf{I}_{k-1}(A\cap\Gamma_{i}^{\vec{\lambda}})$
has $\partial P_{i}=R_{i}$, we may again apply our dimension induction (*),
this time with $A$ replaced by the lower dimensional set
$A\cap\Gamma_{i}^{\vec{\lambda}}$ and with $S_{0}$ replaced by $P_{i}$, to
obtain a chain ${Q_{i}}\in\mathbf{I}_{k}(A\cap\Gamma_{i}^{\vec{\lambda}})$
such that $\ \partial Q_{i}=\partial P_{i}=R_{i}\ $ and

$\mathbf{M}(Q_{i})\ \leqslant\ \mathbf{c}\mathbf{M}(\partial Q_{i})\ =\
\mathbf{c}\mathbf{M}(R_{i})\ \leqslant\ \mathbf{c}\mathbf{M}(\partial S_{0})\
.$

The chains $Q_{h,i}:=Q_{h}\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,\Gamma_{h,i}^{\vec{\lambda}}=Q_{h}\hskip
2.5pt{\vrule height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,\overline{W}_{i}^{\vec{\lambda}}$ for
$h=1,\dots,i-1$, show up in the following boundary calculation using (4),
(III)i-1, and (II)h,

$\begin{split}\partial(S_{i-1}&\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U_{i}^{\lambda_{i}})\ =\ \langle
S_{i-1},u_{i},\lambda_{i}\rangle\ +\ (\partial S_{i-1})\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U_{i}^{\lambda_{i}}\\\ &=\ \langle
S_{i-1},u_{i},\lambda_{i}\rangle\ +\ (\partial S_{i-1})\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,\overline{W}_{i}^{\vec{\lambda}}\\\ &=\
P_{i}\ +\ (\partial S_{0})\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,\overline{W}_{i}^{\vec{\lambda}}\ -\
\sum_{h=1}^{i-1}(\partial H_{h})\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,\overline{W}_{i}^{\vec{\lambda}}\\\ &=\
P_{i}\ +\ (\partial S_{0})\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,\overline{W}_{i}^{\vec{\lambda}}\ -\
\sum_{h=1}^{i-1}\left((\partial S_{0})\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,\
(W_{h}^{\vec{\lambda}}\cap\overline{W}_{i}^{\vec{\lambda}})\ +\ Q_{h}\hskip
2.5pt{\vrule height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,\overline{W}_{i}^{\vec{\lambda}}\ -\
\sum_{\eta=1}^{h}Q_{\eta,h}\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,\overline{W}_{i}^{\vec{\lambda}}\right)\\\
&=\ P_{i}\ +\ (\partial S_{0})\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,\ W_{i}^{\vec{\lambda}}\ -\
\sum_{h=1}^{i-1}\ (\ 0\ +\ Q_{h,i}\ -\ 0\ )\ .\end{split}$

Then the chain $\quad J_{i}\ =\ (\partial S_{0})\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,W_{i}^{\vec{\lambda}}\ +\ Q_{i}\ -\
\sum_{h=1}^{i-1}Q_{h,i}\in\mathbf{I}_{k}(A\cap\operatorname{\mathrm{Clos}}U_{i}^{\lambda_{i}})\quad$
satisfies

$\partial J_{i}\ =\ \partial\left(\ (\partial S_{0})\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,\ W_{i}^{\vec{\lambda}}\ +\ P_{i}\ -\
\sum_{h=1}^{i-1}Q_{h,i}\right)\ =\ \partial^{2}(S_{i-1}\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U_{i}^{\lambda_{i}})\ =\ 0\ $

and, by (II)h,

$\mathbf{M}(J_{i})\ \leqslant\ \mathbf{M}(\partial S_{0})\ +\
\mathbf{M}(Q_{i})+\sum_{h=1}^{i-1}\mathbf{M}(Q_{h})\ \leqslant\
\mathbf{c}\mathbf{M}(\partial S_{0})\ .$

Since this cycle has support in $A\cap\mathbf{U}(a_{i},r_{i})$, we see from
Corollary 3.2 that the contraction

$H_{i}\ :=\
-h_{i\\#}\left[\boldsymbol{[}\\!\boldsymbol{[}0,1\boldsymbol{]}\\!\boldsymbol{]}\times
J_{i}\ \right]\ \in\mathbf{I}_{k+1}(A)\ $

has $\quad\partial H_{i}\ =\ J_{i}\quad$ and that

$\mathbf{M}(H_{i})+\mathbf{M}(\partial H_{i})\ \leqslant\
\left((\operatorname{\mathrm{Lip}}h_{i})^{k+1}+1\right)\mathbf{M}(J_{i})\leqslant\
\mathbf{c}\mathbf{M}(\partial S_{0})\ .$

Letting $\ S_{i}:=\ S_{i-1}-H_{i}\ =\ S_{0}-\sum_{h=1}^{i}H_{h}$ , we have
from (II)i-1 that

$\mathbf{M}(\partial S_{i})\ \leqslant\ \mathbf{c}[\mathbf{M}(\partial
S_{i-1})+\mathbf{M}(\partial H_{i})]\ \leqslant\ \mathbf{c}\mathbf{M}(\partial
S_{0})\ ,$

and from (I)h that

$\begin{split}\partial S_{i}\ &=\ \partial S_{0}\ -\ \sum_{h=1}^{i}\partial
H_{h}\\\ &=\ \partial S_{0}\ -\ \sum_{h=1}^{i}\left(\ (\partial S_{0})\hskip
2.5pt{\vrule height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,W_{h}^{\vec{\lambda}})\ +\
Q_{h}-\sum_{\eta=1}^{h-1}Q_{\eta,h}\ \right)\\\ &=\ (\partial S_{0})\hskip
2.5pt{\vrule height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,(A\setminus\cup_{h=1}^{i}W_{h}^{\vec{\lambda}})\
-\ \sum_{h=1}^{i}\left(Q_{h}-\sum_{\eta=1}^{h-1}Q_{\eta,h}\ \right)\\\ &=\
(\partial S_{0})\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,(A\setminus\cup_{h=1}^{i}U_{h}^{\lambda_{h}})\
-\ \sum_{h=1}^{i}Q_{h}\ +\
\sum_{h=2}^{i}\sum_{\eta=1}^{h-1}Q_{\eta,h}\end{split}$

because $\ \|\partial
S_{0})\|(\operatorname{\mathrm{Bdry}}U_{h}^{\lambda_{h}})=0$ for $h=1,\dots,i\
$. Abbreviating ${\tilde{U}}_{i}=\cup_{h=1}^{i}U_{h}^{\lambda_{h}}$, we see
that $\ Q_{i}\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,{\tilde{U}}_{i}=0\ $ because
$\Gamma_{i}^{\vec{\lambda}}\cap{\tilde{U}}_{i}=\ \emptyset$, and that we may
decompose $Q_{h}\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,{\tilde{U}}_{i}$ by applying (6) with
$Q$, $i$, $\kappa$ replaced by $Q_{h}$, $h$, $i$. We deduce that

$\begin{split}(\partial S_{i})\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,{\tilde{U}}_{i}\ &=\ (\partial
S_{0})\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,(A\setminus{\tilde{U}}_{i})\hskip
2.5pt{\vrule height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,{\tilde{U}}_{i}\ -\
\sum_{h=1}^{i-1}Q_{h}\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,{\tilde{U}}_{i}\ +\
\sum_{h=2}^{i}\sum_{\eta=1}^{h-1}Q_{\eta,h}\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,{\tilde{U}}_{i}\\\ &=\ 0\ -\
\sum_{h=1}^{i-1}\sum_{j=h+1}^{i}Q_{h,j}\ +\
\sum_{h=2}^{i}\sum_{\eta=1}^{h-1}Q_{\eta,h}\quad=\quad 0\ .\end{split}$

Thus

$\operatorname{\mathrm{spt}}(\partial S_{i})\ \subseteq\ A\setminus\
{\tilde{U}}_{i}\ =\ A\setminus\ (U_{1}^{\lambda_{1}}\cup\ \cdots\ \cup
U_{h}^{\lambda_{h}}\ )\ ,$

and we have now verified (I)i, (II)i, and (III)i and completed the $i$th step
for all $i\leqslant\ell-1$. In particular,

$\operatorname{\mathrm{spt}}(\partial S_{\ell-1})\ \subseteq\
A\setminus\cup_{i=1}^{\ell-1}U_{i}^{\lambda_{i}}\ \subseteq\
A\setminus\cup_{i=1}^{\ell-1}\mathbf{U}(a_{i},r_{i}/2)\ \subseteq
A\cap\mathbf{U}(a_{\ell},r_{\ell}/2)\ .$

Finally by defining $\ H_{\ell}\ :=\
-h_{\ell\\#}\left(\boldsymbol{[}\\!\boldsymbol{[}0,1\boldsymbol{]}\\!\boldsymbol{]}\times\partial
S_{\ell-1}\right)\ $ and $\ S=H_{1}+\cdots+H_{\ell}\ $, we find that

$\partial H_{\ell}\ =\ \partial S_{\ell-1}\ =\ \partial S_{0}\ -\
\sum_{i=1}^{\ell-1}\partial H_{i}\ ;\quad{\rm hence,}\quad\partial S\
=\partial S_{0}\ ,$

and, by using (II)1,…,(II)ℓ-1, that

$\mathbf{M}(S)\ \leqslant\ \mathbf{c}\sum_{i=1}^{\ell-1}\mathbf{M}(\partial
H_{i})+\mathbf{M}(H_{\ell})\ \leqslant\ \mathbf{c}\mathbf{M}(\partial
S_{0})+(\operatorname{\mathrm{Lip}}h_{\ell})^{k}\mathbf{M}(\partial
S_{\ell-1})\ \leqslant\ \mathbf{c}\mathbf{M}(\partial S_{0})\ ,$

which completes the proof. ∎

##  5\. Applications

######  5.1Normal Currents. —

The exact same proof shows that if $A\subseteq\mathbb{R}^{n}$ is compact and
subanalytic, $k=0,1,2,\ldots$ and $S_{0}\in\mathbf{N}_{k+1}A)$ is a normal
current of dimension $k+1$ supported in $A$, then there exists
$S\in\mathbf{N}_{k+1}(A)$ such that $\partial S=\partial S_{0}$ and
$\mathbf{M}(S)\leqslant\mathbf{c}(A)\mathbf{M}(\partial S)$. This seems to be
new even in the case when $A$ is a compact real analytic submanifold of
$\mathbb{R}^{n}$.

######  5.2Other coefficients groups. —

We observe that the proof further generalizes to the case of a general normed,
complete, Abelian group $G$ of coefficients, [19], [6]. Here
$\mathcal{R}_{k}(A;G)$ and $\mathscr{F}_{k}(A;G)$ denote the groups consisting
of those $k$ dimensional, respectively rectifiable and flat chains with
coefficients in $G$, supported in $A$, and

$\begin{split}\mathcal{I}_{k+1}(A;G)&=\mathcal{R}_{k+1}(A;G)\cap\left\\{S:\partial
S\in\mathcal{R}_{k}(A;G)\right\\}\\\
\mathscr{N}_{k}(A;G)&=\mathscr{F}_{k}(A;G)\cap\left\\{S:\mathbf{M}(S)+\mathbf{M}(\partial
S)<\infty\right\\}\end{split}$

where $\mathbf{M}$ denotes the usual Euclidean Hausdorff mass of a rectifiable
chain in $\mathbb{R}^{n}$, relaxed to the class of flat chains. If
$k=1,2,\ldots$ and $S_{0}\in\mathcal{I}_{k+1}(A;G)$ (resp.
$S_{0}\in\mathscr{N}_{k+1}(A;G)$), then there exists
$S\in\mathcal{I}_{k+1}(A;G)$ (resp. $S\in\mathscr{N}_{k+1}(A;G)$) such that
$\partial S=\partial S_{0}$ and
$\mathbf{M}(S)\leqslant\mathbf{c}(A)\mathbf{M}(\partial S)$. This indeed
encompasses the previous cases since
$\mathcal{I}_{k+1}(A;\mathbb{Z})\cong\mathbf{I}_{k+1}(A)$ and
$\mathscr{N}_{k+1}(A;\mathbb{R})\cong\mathbf{N}_{k+1}(A)$.

We ought to say a word about the case $k=0$ of the proof. Here one replaces
the expression $(\partial
S_{0})\left(\mathbbm{1}_{W_{i}^{\vec{\lambda}}}\right)$ using the total
multiplicity morphism $\chi:\mathscr{F}_{0}(A;G)\to G$, see [6, 4.3.3] to find
that $g_{i}=\chi[(\partial S_{0})\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,W_{i}^{\vec{\lambda}}]$, while
recalling that $\chi$ is finitely additive and that
$|\chi(T)|\leqslant\mathscr{F}(T)\leqslant\mathbf{M}(T)$.

######  5.3Comparing homology groups. —

We use the same notations as in 5.2, and we define the groups of cycles and
boundaries

$\begin{split}\mathbf{Z}_{k}^{\mathcal{I}}(A;G)&=\mathcal{I}_{k}(A;G)\cap\left\\{T:\partial
T=0\right\\}\\\
\mathbf{B}_{k}^{\mathcal{I}}(A;G)&=\mathcal{I}_{k}(A;G)\cap\left\\{T:T=\partial
S\text{ for some }S\in\mathcal{I}_{k+1}(A;G)\right\\}\end{split}$

as well as the corresponding homology group
$\mathbf{H}^{\mathcal{I}}_{k}(A;G)=\mathbf{Z}_{k}^{\mathcal{I}}(A;G)/\mathbf{B}_{k}^{\mathcal{I}}(A;G)$.
One checks that $\mathbf{H}^{\mathcal{I}}_{0}(\\{0\\};G)=G$ and, as in [5,
3.7, 3.9, 3.10, 3.14] one shows that the functors
$\mathbf{H}^{\mathcal{I}}_{k}(\cdot;G)$ and $\check{H}_{k}(\cdot;G)$ (Čech
homology with coefficients in $G$) are naturally equivalent on the category of
$(\mathbf{H}^{\mathcal{I}},k)$ locally connected subsets of Euclidean space
and their Lipschitz maps. According to Theorem 3.1 each compact subanalytic
set $A\subseteq\mathbb{R}^{n}$ is $(\mathbf{H}^{\mathcal{I}},k)$ locally
connected, recall [5, 3.11]. Thus in that case,
$\mathbf{H}^{\mathcal{I}}_{k}(A;G)\cong\check{H}_{k}(A;G)\cong H_{k}(A;G)$
where $H_{k}(A;G)$ denotes singular homology and the last equivalence holds
because $A$ is triangulable.

######  5.4Homology of Normal Currents. —

We can repeat the argument made in 5.3 with the usual normal currents. Letting

$\begin{split}\mathbf{Z}_{k}(A)&=\mathbf{N}_{k}(A)\cap\left\\{T:\partial
T=0\right\\}\\\ \mathbf{B}_{k}(A)&=\mathbf{N}_{k}(A)\cap\left\\{T:T=\partial
S\text{ for some }S\in\mathbf{N}_{k+1}(A)\right\\}\end{split}$

and $\mathbf{H}_{k}(A)=\mathbf{Z}_{k}(A)/\mathbf{B}_{k}(A)$ we note that
$\mathbf{H}_{0}(\\{0\\})=\mathbb{R}$ and, referring to [5, 3.14] again that
$\mathbf{H}_{k}(A)\cong\check{H}_{k}(A;\mathbb{R})\cong H_{k}(A;\mathbb{R})$
in case $A\subseteq\mathbb{R}^{n}$ is compact and subanalytic.

######  5.5Cohomology of Charges. —

A complex of cochains on a compact subset $A\subseteq\mathbb{R}^{n}$ is
defined and studied in [7]. A charge of degree $k$ on $A$ is a linear
functional $\alpha:\mathbf{N}_{k}(A)\to\mathbb{R}$ with the following
property. For every $\varepsilon>0$ there exists $\theta>0$ such that
$|\alpha(T)|\leqslant\theta\mathbf{F}(T)+\varepsilon\mathbf{N}(T)$ where
$\mathbf{F}(T)$ is the flat norm of $T$ and
$\mathbf{N}(T)=\mathbf{M}(T)+\mathbf{M}(\partial T)$. According to the main
result of [8], a linear functional $\alpha:\mathbf{N}_{k}\to\mathbb{R}$ is a
charge of degree $k$ if and only if there exist continuous forms $\omega\in
C(\mathbb{R}^{n},\bigwedge_{k}\mathbb{R}^{n})$ and $\zeta\in
C(\mathbb{R}^{n},\bigwedge_{k-1}\mathbb{R}^{n})$ such that

$\alpha(S)=\int_{\mathbb{R}^{n}}\langle\omega,\vec{S}\rangle
d\|S\|+\int_{\mathbb{R}^{n}}\langle\zeta,\vec{\partial S}\rangle d\|\partial
S\|$

whenever $S\in\mathbf{N}_{k}(A)$, in other words $\alpha=\omega+d\zeta$. Let
$\mathbf{CH}^{k}(A)$ denote the vector space of charges of degree $k$ in $A$.
The notions of cocycle and coboundary for charges in $A$ are readily defined
in terms of their exterior derivatives $d=\partial^{*}$:

$\begin{split}\mathbf{Z}^{k}(A)&=\mathbf{CH}^{k}(A)\cap\left\\{\alpha:d\alpha=0\right\\}\\\
\mathbf{B}^{k}(A)&=\mathbf{CH}^{k}(A)\cap\left\\{d\beta:\beta\in\mathbf{CH}^{k-1}(A)\right\\}\,.\end{split}$

This in turn yields a cohomology space
$\mathbf{H}^{k}(A)=\mathbf{Z}^{k}(A)/\mathbf{B}^{k}(A)$. Furthermore
$\mathbf{CH}^{k}(A)$ is given a structure of Banach space with the norm

$\|\alpha\|=\sup\left\\{\alpha(S):S\in\mathbf{N}_{k}(A)\text{ and
}\mathbf{N}(S)\leqslant 1\right\\}\,.$

The relevance of the linear isoperimetric inequality 5.1 in this context is as
follows. We recall [7, Chapter 14] that the compact set
$A\subseteq\mathbb{R}^{n}$ is called $q$-bounded, $q=0,1,2,\ldots$, whenever
the following holds: There exists $\mathbf{c}(A,q)>0$ such that for every
$T\in\mathbf{B}_{q}(A)$ there exists $S\in\mathbf{N}_{q+1}(A)$ with $\partial
S=T$ and $\mathbf{M}(S)\leqslant\mathbf{c}(A,q)\mathbf{M}(T)$. It follows from
[7, 14.4 and 13.10] that $A$ is $q$-bounded if and only $\mathbf{B}^{q}(A)$ is
closed in $\mathbf{CH}^{q}(A)$. In that case $\mathbf{H}^{q}(A)$ is a Banach
space. Furthermore, according to [7, 14.9] $\mathbf{H}^{q}(A)$ is the strong
dual of $\mathbf{H}_{q}(A)$ equipped with an appropriate locally convex vector
topology, [7, Chapter 12]. From §5.1 it readily follows that:

If $A\subseteq\mathbb{R}^{n}$ is compact and subanalytic, then $A$ is
$q$-bounded for all $q=0,1,2,\ldots$ and
$\mathbf{H}^{q}(A)\cong\mathbf{H}_{q}(A)^{*}\cong H^{q}(A;\mathbb{R})$, the
singular cohomology with real coefficients. The latter may be interpreted as a
de Rham Theorem in this context.

######  5.6Plateau problem in a homology classes. —

Let $G$ be a complete normed Abelian group with norm $|\cdot|$ and assume it
satisfies the following extra two conditions.

  1. (A)

$G\cap\\{g:|g|\leqslant\kappa\\}$ is compact for every $\kappa>0$;

  2. (B)

$G$ is a White group, i.e. $G$ contains no nonconstant curve of finite length.

We also let $A\subseteq\mathbb{R}^{n}$ be a compact subanalytic set, and
$k=1,2,\ldots$. Recall the notations of 5.2. Given
$T_{0}\in\mathcal{R}_{k}(A;G)$ with $\partial T_{0}=0$, the following
variational problem admits a minimizer:

$(\mathscr{P})\begin{cases}\text{minimize }\mathbf{M}(T)\\\ \text{among
}T\in\mathcal{R}_{k}(A;G)\text{ with }T-T_{0}=\partial S\text{ for some
}S\in\mathcal{R}_{k+1}(A;G).\end{cases}$

As we show below this is a consequence of our linear isoperimetric inequality
and of work of B. White.

###### Existence of solution.

Let $\langle T_{j}\rangle_{j}$ be a minimizing sequence of $(\mathscr{P})$.
For each $j$ there exists $S_{j}\in\mathcal{R}_{k+1}(A;G)$ such that
$T_{j}-T_{0}=\partial S_{j}$. According to the linear isoperimetric inequality
5.2, we may choose $\hat{S}_{j}\in\mathcal{I}_{k+1}(A;G)$ so that
$\partial\hat{S}_{j}=\partial S_{j}$ and
$\mathbf{M}(\hat{S}_{j})\leqslant\mathbf{c}(A)\mathbf{M}(\partial\hat{S}_{j})$.
Thus $T_{j}-T_{0}=\partial\hat{S}_{j}$ and

$\ \mathbf{M}(\hat{S}_{j})\ \leqslant\
\mathbf{c}(A)\mathbf{M}(\partial\hat{S}_{j})=\mathbf{c}(A)\mathbf{M}(T_{j}-T_{0})\
\leqslant\mathbf{c}(A)[1+\inf(\mathscr{P})+\mathbf{M}(T_{0})]\ <\ \infty\ .$

for $j$ sufficiently large. Since $A$ is compact, the deformation theorem [18]
together with condition (A) above imply that both sets
$\\{T_{j}:j=1,2,\ldots\\}$ and $\\{\hat{S_{j}}:j=1,2,\ldots\\}$ are totally
bounded in the flat norm $\mathscr{F}$. Consequently there are integers
$j_{1}<j_{2}<\ldots$ and flat $G$ chains $T\in\mathscr{F}_{k}(A;G)$ and
$\hat{S}\in\mathscr{F}_{k+1}(A;G)$ such that $\
\lim_{i}\mathscr{F}(T-T_{j_{i}})=0=\lim_{i}\mathscr{F}(\hat{S}-\hat{S}_{j_{i}})$.
Note that

$T-T_{0}\ =\ \lim_{i\to\infty}(T_{j_{i}}-T_{0})\ =\
\lim_{i\to\infty}\partial\hat{S}_{j_{i}}\ =\ \partial\hat{S}\ .$

Inasmuch as $\mathbf{M}$ is lower semicontinuous with respect to $\mathscr{F}$
convergence, one infers that $\mathbf{M}(T)<\infty$ and
$\mathbf{M}(\hat{S})<\infty$. It therefore follows from [19] and condition (B)
above that $T\in\mathcal{R}_{k}(A;G)$ and $\hat{S}\in\mathcal{R}_{k+1}(A;G)$.
∎

######  5.7A linear relative isoperimetric inequality. —

Here we work with a pair $B\subseteq A$ of compact subanalytic subsets of
$\mathbb{R}^{n}$ and verify an isoperimetric inequality generalizing our Main
Theorem.

Assuming that $U:=\mathbb{R}^{n}\setminus B$ and that $S$ is a chain of finite
mass with support in $A$, we will be interested in the mass of $S$ in $U$,

$\mathbf{M}(S\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U)\ =\ \|S\|(U)\ =\ \|S\|(A\setminus
B)\ =\ \mathbf{M}[S\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,(A\setminus B)\,]\ ,$

rather than the total mass $\mathbf{M}(S)$. For integral currents, our
relative version is:

######  Theorem. —

There is a constant $\mathbf{c}(A,B)>0$ so that, for every
$k\in\\{1,2,\ldots\\}$ and every $S_{0}\in\mathbf{I}_{k}(A)$, there exists an
$S\in\mathbf{I}_{k}(A)$ satisfying

$(\partial S)\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U\ =\ (\partial S_{0})\hskip
2.5pt{\vrule height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U\quad{\rm and}\quad\mathbf{M}(S\hskip
2.5pt{\vrule height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U)\ \leqslant\
\mathbf{c}(A,B)\mathbf{M}[(\partial S)\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U]\ ,$

###### Proof.

Note that this does not follow by simply applying the statement of the Main
Theorem to $S\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U$ because the righthand side of the
inequality is missing the term $\mathbf{M}[\partial(S\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U)]-\mathbf{M}[(\partial S)\hskip
2.5pt{\vrule height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U]$. Nevertheless, we can obtain the
desired $S$ by slightly modifying the arguments in §3 and §4. Of course, the
various constructions and choices, as well as the final chain $S$, will now
depend on $B$ (and $U=\mathbb{R}^{n}\setminus B$) as well as on $A$ and
$S_{0}$.

For the new relative version of Theorem 3.1, we simply require the extra
statement that if $a\in B$, then the Lipschitz contraction $h$ preserves
$B\cap K$, that is,

$h\left([0,1]\times(B\cap K)\right)\ =\ B\cap K\ .$

This extra property has already been essentially treated in $(A_{n})(1)$ of
the proof of [16, Th.2.3.1]. There, the inductive argument, in G. Valette’s
notation, gives, for any finite collection $X_{1},X_{2},\dots,X_{s}$ of
compact subanalytic subsets of $\mathbb{R}^{n}$, a single Lipschitz
neighborhood contraction $r$ of a neighborhood $U_{\varepsilon}$ of $x_{0}$
whose restrictions simultaneously contract the $U_{\varepsilon}\cap X_{j}$ .
So here we are simply using the two sets $X_{1}=A$, $X_{2}=B$ to get the
desired contractions $h=r$ of
$K=X_{1}\cap\operatorname{\mathrm{Clos}}(U_{\varepsilon})$ to $a=x_{0}$ that
preserves $B\cap K$.

In the new relative version of Corollary 3.2, the fact that $h$ preserves both
$K$ and $B\cap K$ implies that $h$ preserves $U\cap K=K\setminus B$, that is,
$\ h([0,1)\times(U\cap K))\ =\ U\cap K$. Thus we obtain, by applying $\hskip
2.5pt{\vrule height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U$ in the proof of Corollary 3.2, that

$H\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U\ =\ -h_{\\#}\left([0,1]\times
J\right)\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U\ =\ -h_{\\#}\left([0,1]\times(J\hskip
2.5pt{\vrule height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U)\ \right)\ ,$

which gives the additional conclusion

$\mathbf{M}(H\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U)\leqslant(\operatorname{\mathrm{Lip}}h)^{k+1}{\mathbf{M}}(J\hskip
2.5pt{\vrule height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U)\ ,$ (8)

which is essentially the local case of our linear relative isoperimetric
inequality.

If the point $a\not\in B$ and $K$ is small enough so that $K\cap B=\emptyset$,
we can still use the contraction $h$ as before and inequality (8) remains true
because $J\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U=J$ and $H\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U=H$.

For the new version of Theorem 3.4, we need the extra property that the
resulting bilipschitz equivalence of $A$ links also preserves the $B$ links,
ie. for $\vec{\lambda},\vec{\mu}$ in $R_{\vec{r}}^{\vec{a}}$,

$B\cap\operatorname{\mathrm{Bdry}}\mathbf{U}(a_{i},\lambda r_{i})\ =\ B\cap
L_{\lambda r_{i}}^{a_{i}}\ \to\
B\cap\operatorname{\mathrm{Bdry}}\mathbf{U}(a_{i},\mu r_{i})\ =\ B\cap L_{\mu
r_{i}}^{a_{i}}\ .$

Our present proof will give this property if we simply add the requirement
that our stratification $\mathscr{S}$ also be compatible with the set
$\mathbb{R}\times B$.

We now repeat all the constructions of §4, look at the restrictions to $U$,
and estimate the masses in $U$ in terms of ${\mathbf{M}}[(\partial
S_{0})\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U]$ rather than ${\mathbf{M}}(\partial
S_{0})$. In these estimates we will use the symbol $\mathbf{c}$ to abbreviate
a constant $\mathbf{c}(A,B)$, depending only on $A$ and $B$.

When we first employ compactness to find the finite collection of balls
${\mathbf{U}}(a_{i},r_{i}/2)$ covering $A$, we may choose just from those
balls ${\mathbf{U}}(a,r/2)$ where the center $a\in A$ and either $a\in B$ or
${\mathbf{B}}(a,r)\cap B=\emptyset$. Thus, when we repeat the constructions
with the resulting $U_{i}^{\lambda_{i}}=\mathbf{U}(a_{i},\lambda_{i}r_{i})$,
we can use either the new or old version of Theorem 3.4, and Corollary 3.2
depending on whether $a_{i}\in B$ or $a_{i}\not\in B$.

We again argue by induction on $\operatorname{\mathrm{dim}}A$. As before, the
bilipschitz equivalence discussion gives us the assumption (*) on the
inductive validity of the new relative theorem with the same constant when
$A,B$ is replaced by every pair $A\cap\Gamma_{i}^{\vec{\lambda}},\
B\cap\Gamma_{i}^{\vec{\lambda}}$.

For $\lambda\in\mathbb{R}\setminus\Lambda_{i}^{S}$, the slices of $S$ and
$\partial S$ by $u_{i}$ are given by integration, and so they may be
restricted to the open set $U$. That is, these slices all commute with the
operation $\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U$. In particular, simply applying
$\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U$ to every term in (4), (5), and (7)
results in

$\begin{split}\langle S\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U,u_{i},\lambda\rangle\ =\ \langle
S,u_{i},\lambda\rangle\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U\ &=\ [\partial(S\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U_{i}^{\lambda})]\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U\ -(\partial S)\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,(U_{i}^{\lambda}\cap U)\ ,\\\
\langle(\partial S)\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U,u_{i},\lambda\rangle\ =\
\langle\partial S,u_{i},\lambda\rangle\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U\ &=\ \partial[(\partial S)\hskip
2.5pt{\vrule height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U_{i}^{\lambda}]\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U\ ,\\\
\int_{\lambda_{\vec{r},i}^{\vec{a}}}^{\mu_{\vec{r},i}^{\vec{a}}}\mathbf{M}(\langle
S,u_{i},\lambda\rangle\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U)\,d\lambda\ \ &\leqslant\
r_{i}^{-1}\mathbf{M}(S\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U)\\\
\int_{\lambda_{\vec{r},i}^{\vec{a}}}^{\mu_{\vec{r},i}^{\vec{a}}}\mathbf{M}(\langle\partial
S,u_{i},\lambda\rangle\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U)\,d\lambda\ &\leqslant\
r_{i}^{-1}\mathbf{M}((\partial S)\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U)\ .\\\ \langle Q\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U,u_{i},\lambda\rangle\ =\ \langle
Q,u_{i},\lambda\rangle\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U\ &=\ [\partial(Q\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U_{i}^{\lambda})]\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U\ -(\partial Q)\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,(U_{i}^{\lambda}\cap U)\ ,\end{split}$

In the $i$th step of the new proof, we find $\lambda_{i}$ so that the slice at
$R_{i}\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U\ :=\ \partial\langle
S_{i-1},u_{i},\lambda_{i}\rangle\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U$, has mass

$\mathbf{M}(R_{i}\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U)\ \leqslant\
\mathbf{c}\mathbf{M}[(\partial S_{i-1})\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U]\ \leqslant\
\mathbf{c}\mathbf{M}[(\partial S_{0})\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U]\ .$

Now the induction on dimension allows us to to replace $P_{i}=\langle
S_{i-1},u_{i},\lambda_{i}\rangle$ by a chain
${Q_{i}}\in\mathbf{I}_{k-1}(A\cap\Gamma_{i}^{\vec{\lambda}})$ such that
$\partial Q_{i}=\partial P_{i}=R_{i}$ and

$\mathbf{M}(Q_{i}\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U)\ \leqslant\
\mathbf{c}\mathbf{M}[(\partial Q_{i})\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U]\ =\ \mathbf{c}\mathbf{M}(R_{i}\hskip
2.5pt{\vrule height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U)\ \leqslant\
\mathbf{c}\mathbf{M}[(\partial S_{0})\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U]\ .$

By defining $Q_{\eta,h}$ and $H_{i}$ exactly as before, based on
$Q_{1},\dots,Q_{i}$, $S_{0}$, and contractions $h_{i}$, we now find mass-
in-$U$ estimates

$\mathbf{M}(H_{i}\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U)+\mathbf{M}[(\partial H_{i})\hskip
2.5pt{\vrule height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U]\ \leqslant\
\mathbf{c}\left(\mathbf{M}[(\partial S_{0})\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U]\ +\
\sum_{h=1}^{i}\mathbf{M}(Q_{h}\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U)\right)\ \leqslant\
\mathbf{c}\mathbf{M}[(\partial S_{0})\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U]\ ,$

since the $h_{i}$ preserves $U\cap K_{i}$. With the last modification
$H_{\ell}$ also defined exactly as before, we similarly verify
$\mathbf{M}(H_{i}\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U)\leqslant\mathbf{c}\mathbf{M}[(\partial
S_{0})\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U]\ $. It follows, as before, that
$S=H_{1}+\cdots+H_{\ell}$ satisfies the relative theorem. ∎

######  5.8Remark. —

Again both the statement and proof of Theorem 5.7 carry over to chains with
coefficients in a complete normed abelian group $G$. The application below in
§5.9 generalizes §5.6 and so involves the rectifiability and the group
assumptions of §5.6(A)(B). But we will no longer be assuming that all chains
and their boundaries have finite mass or are rectifiable everywhere. Noting
that the statement and proof of Theorem 5.7 involve the behavior of the chains
and their masses only in $U$, we see that we can further generalize Theorem
5.7 to flat chains $T$ where both $T$ and $\partial T$ are rectifiable with
finite mass in $U$. Here a chain $T\in\mathscr{F}_{k}(A;G)$ is rectifiable
with finite mass in $U$ provided that

$T\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U_{i}\in\mathcal{R}_{k}(A;G)\quad{\rm
and}\quad\sup_{i}\mathbf{M}(T\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U_{i})<\infty\ ,$

for some open sets $U_{1}\subseteq U_{2}\subseteq\cdots$ with
$\cup_{i=1}^{\infty}U_{i}=U$. In this case $\langle T\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U_{i}\rangle_{i}$ is $\mathbf{M}$
Cauchy, and we let $T\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U$ denote the $\mathbf{M}$ limit. This
limit is well-defined independently of the choice of open sets, is
rectifiable, and coincides with the usual definition of $T\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U$ in case $\mathbf{M}(T)<\infty$. Also
one checks that $\operatorname{\mathrm{spt}}(T-T\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U)\subseteq B$ [6, §5.5].

######  5.9A relative homology Plateau problem. —

First recall that, for rectifiable chains, the groups of relative cycles,
relative boundaries, and relative homology can be defined:

$\begin{split}\mathbf{Z}_{k}^{\mathcal{R}}(A,B;G)&=\mathcal{R}_{k}(A;G)\cap\left\\{T:{\operatorname{\mathrm{spt}}}(\partial
T)\subseteq B\right\\}\\\
\mathbf{B}_{k}^{\mathcal{R}}(A,B;G)&=\mathcal{R}_{k}(A;G)\cap\left\\{T:{\operatorname{\mathrm{spt}}}(T-\partial
S)\subseteq B\text{ for some }S\in\mathcal{R}_{k+1}(A;G)\right\\}\\\
\mathbf{H}^{\mathcal{R}}_{k}(A,B;G)&=\mathbf{Z}_{k}^{\mathcal{R}}(A,B;G)/\mathbf{B}_{k}^{\mathcal{R}}(A,B;G)\
.\end{split}$

Here we discuss how the relative isoperimetric inequality of §5.7 is useful
for the following relative homology Plateau problem that generalizes §5.6.

Given $T_{0}\in\mathcal{R}_{k}(A;G)$ where $B\subseteq A$ are compact
subanalytic subsets of $\mathbb{R}^{n}$ and $G$ satisfies §5.6(A)(B), consider
the problem:

$(\mathscr{P}_{B})\begin{cases}\text{minimize }\mathbf{M}(T)\\\ \text{among
}T\in\mathcal{R}_{k}(A;G)\text{ with }\rm{spt}(T-T_{0}-\partial S)\subseteq
B\text{ for some }S\in\mathcal{R}_{k+1}(A;G).\end{cases}$

Note that in case $\operatorname{\mathrm{spt}}(\partial T_{0})\subseteq B$,
i.e. $T_{0}\in\mathbf{Z}_{k}^{\mathcal{R}}(A,B;G)$, one is minimizing mass in
the relative homology class

$[T_{0}]\ :=\ \mathbf{Z}_{k}^{\mathcal{R}}(A,B;G)\cap\left\\{T\ :\
T-T_{0}\in\mathbf{B}_{k}^{\mathcal{R}}(A,B;G)\right\\}\ \in\
\mathbf{H}^{\mathcal{R}}_{k}(A,B;G)\ .$

###### Existence of solution.

Since $T_{0}$ is admissible, one easily obtains a mass minimizing sequence
$\langle T_{j}\rangle_{j}$ in $\mathcal{R}_{k}(A;G)$ and $\langle
S_{j}\rangle_{j}$ in $\mathcal{R}_{k+1}(A;G)$ with
$\operatorname{\mathrm{spt}}(T_{j}-T_{0}-\partial S_{j})\subseteq B$. The
sequence $\mathbf{M}(T_{j})$ has a finite upper bound which we may assume is
$\mathbf{M}(T_{0})$. The chain $T\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U$ is admissible whenever $T$ is
because $\operatorname{\mathrm{spt}}(T-T\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U)\subseteq B$. Also we may assume
$T_{0}=T_{0}\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U$ because $T_{0}\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U$ gives the same admissible class as
$T_{0}$ does .

Even though the sequence $\mathbf{M}(\partial T_{j})$ is not obviously bounded
above, the equation $(\partial T_{j})\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U=(\partial T_{0})\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U$ gives a bound on the mass in $U$ of
$\partial T_{j}$. While there is no bound for the mass in $U$ of $S_{j}$, the
equation $(\partial S_{j})\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U=T_{j}\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U\ -\ T_{0}\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U$ shows that $\partial S_{j}$ has
bounded mass in $U$ and is rectifiable in $U$. Just like in §5.6, we can now
use Theorem 5.7, with $S_{0}=S_{j}$, and Remark 5.8 to replace $S_{j}$ by
another chain $\hat{S}_{j}$, with the same boundary in $U$, to assure that the
sequence $\hat{S}_{j}$ has bounded mass in $U$. Specifically,

$\mathbf{M}(\hat{S}_{j}\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U)\leqslant\mathbf{c}\mathbf{M}((\partial
S_{j})\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U)=\mathbf{c}\mathbf{M}[(T_{j}-T_{0})\hskip
2.5pt{\vrule height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U]\leqslant
2\mathbf{c}\mathbf{M}(T_{0})\ .$

To construct the desired rectifiable chains $S\in\mathcal{R}_{k+1}(A;G)$ and
$T\in\mathcal{R}_{k}(A;G)$ so that $\rm{spt}(T-T_{0}-\partial S)\subseteq B$
and $T$ is a mass minimizer for $(\mathscr{P}_{B})$, we will take limits
inside of $U$ away from $B$ and then use a diagonal argument. Accordingly, we
define, for $\delta>0$, $U_{\delta}:=\ \\{x\in U\ :\ u(x)>\delta\\}$ where
$u(x)=\rm{dist}(x,B)$. Inasmuch as

$\begin{split}\int_{0}^{\infty}\liminf_{j\to\infty}&[\mathbf{M}\langle{\hat{S}}_{j},u,t\rangle+\mathbf{M}\langle
T_{j},u,t\rangle]\,dt\ \leqslant\
\liminf_{j\to\infty}\int_{0}^{\infty}[\mathbf{M}\langle{\hat{S}}_{j},u,t\rangle+\mathbf{M}\langle
T_{j},u,t\rangle]\,dt\\\ &\leqslant\ \sup_{j}\mathbf{M}({\hat{S}}_{j}\hskip
2.5pt{\vrule height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U)+\sup_{j}\mathbf{M}(T_{j}\hskip
2.5pt{\vrule height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U)\ \leqslant\
(2\mathbf{c}+1)\mathbf{M}(T_{0})\ <\ \infty\ ,\end{split}$

we can choose a sequence $t_{i}\downarrow 0$ so that, for all $i$,

$\liminf_{j\to\infty}[\mathbf{M}\langle{\hat{S}}_{j},u,t_{i}\rangle+\mathbf{M}\langle
T_{j},u,t_{i}\rangle]\ <\ \infty\
,\quad\mathbf{M}\langle{T}_{0},u,t_{i}\rangle\ <\
\infty,\quad\mathbf{M}\langle\partial T_{0},u,t_{i}\rangle\ <\ \infty\ .$

We can also insist that, for all $i$ and $j$,

$\begin{split}\langle{\hat{S}}_{j},u,t_{i}\rangle\in\mathcal{R}_{k}(A;G)\
,&\quad\langle T_{j},u,t_{i}\rangle\in\mathcal{R}_{k-1}(A;G)\ ,\\\
\partial({\hat{S}}_{j}\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U_{t_{i}})=(\partial{\hat{S}}_{j})\hskip
2.5pt{\vrule height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U_{t_{i}}+\langle{\hat{S}}_{j},u,t_{i}\rangle\
,&\quad\partial({T}_{j}\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U_{t_{i}})=(\partial{T}_{j})\hskip
2.5pt{\vrule height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U_{t_{i}}+\langle{T}_{j},u,t_{i}\rangle\
.\end{split}$

Inasmuch as $(\partial{\hat{S}}_{j})\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U=(T_{j}-T_{0})\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U$ and $(\partial T_{j})\hskip
2.5pt{\vrule height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U=(\partial T_{0})\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U$, we also have, for every
$i=1,2,\dots$, that

$\begin{split}&\liminf_{j\to\infty}\left[\mathbf{M}({\hat{S}}_{j}\hskip
2.5pt{\vrule height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U_{t_{i}})+\mathbf{M}\partial({\hat{S}}_{j}\hskip
2.5pt{\vrule height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U_{t_{i}})+\mathbf{M}(T_{j}\hskip
2.5pt{\vrule height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U_{t_{i}})+\mathbf{M}\partial(T_{j}\hskip
2.5pt{\vrule height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U_{t_{i}})\right]\\\
&\leqslant\sup_{j}\left(\mathbf{M}({\hat{S}}_{j}\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U)+\mathbf{M}[(\partial{\hat{S}}_{j})\hskip
2.5pt{\vrule height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U]+\mathbf{M}(T_{j}\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U)+\mathbf{M}[(\partial T_{j})\hskip
2.5pt{\vrule height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U]\right)\\\ &\hskip 144.54pt+\
\liminf_{j\to\infty}\left[\mathbf{M}\langle{\hat{S}}_{j},u,t_{i}\rangle+\mathbf{M}\langle
T_{j},u,t_{i}\rangle\right]\\\
&\leqslant(2\mathbf{c}+2+1)\mathbf{M}(T_{0})+\mathbf{M}[(\partial T_{0})\hskip
2.5pt{\vrule height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U]+\
\liminf_{j\to\infty}\left[\mathbf{M}\langle{\hat{S}}_{j},u,t_{i}\rangle+\mathbf{M}\langle
T_{j},u,t_{i}\rangle\right]\ <\ \infty\ .\end{split}$

We find a subsequence $\ \langle i^{(1)}_{j}\rangle_{j}\ $ of $\ \langle
j\rangle_{j}$ giving, as $j\to\infty$, flat convergences

${\hat{S}}_{i^{(1)}_{j}}\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U_{t_{1}}\to{\hat{S}}^{(1)}\in\mathcal{R}_{k+1}(A;G)\quad{\rm
and}\quad T_{i^{(1)}_{j}}\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U_{t_{1}}\to
T^{(1)}\in\mathcal{R}_{k}(A;G)\ .$

For $m=2,3,\dots$, we similarly inductively find subsequences $\ \langle
i^{(m)}_{j}\rangle_{j}\ $ of $\ \langle i^{(m-1)}_{j}\rangle_{j}\ $ and,

as $j\to\infty$, flat convergences

${\hat{S}}_{i^{(m)}_{j}}\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U_{t_{m}}\to{\hat{S}}^{(m)}\in\mathcal{R}_{k+1}(A;G)\quad{\rm
and}\quad\ T_{i^{(m)}_{j}}\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U_{t_{m}}\to
T^{(m)}\in\mathcal{R}_{k}(U;G)\ .$

The lower semicontinuity of $\mathbf{M}$ implies that
$\mathbf{M}({\hat{S}}^{(m)})\leqslant 2\mathbf{c}\mathbf{M}(T_{0})$ and
$\mathbf{M}({\hat{T}}^{(m)})\leqslant\mathbf{M}(T_{0})$.

For $\ell<m$, $U_{t_{\ell}}\subseteq U_{t_{m}}$, and one has
${\hat{S}}^{(\ell)}={\hat{S}}^{(m)}\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U_{t_{\ell}}$ and
${\hat{T}}^{(\ell)}={\hat{T}}^{(m)}\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U_{t_{\ell}}$. It follows that the
sequences ${\hat{S}}^{(m)}$ and ${\hat{T}}^{(m)}$ are $\mathbf{M}$-Cauchy and
$\mathbf{M}$-convergent to chains $S\in\mathcal{R}_{k+1}(A;G)$ and
$T\in\mathcal{R}_{k}(A;G)$, respectively, characterized by having $S\hskip
2.5pt{\vrule height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U_{t_{m}}={\hat{S}}^{(m)}$ and $T\hskip
2.5pt{\vrule height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U_{t_{m}}=T^{(m)}$ for every
$m\in\\{1,2,\dots\\}$. Taking the diagonal subsequence $\ \langle
j^{\prime}\rangle_{j}\ =\ \langle i^{(j)}_{j}\rangle_{j}\ $, one now has, for
all $m$, the flat convergences

$\lim_{j\to\infty}{\hat{S}}_{j^{\prime}}\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U_{t_{m}}={\hat{S}}^{(m)}=S\hskip
2.5pt{\vrule height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U_{t_{m}}\quad{\rm
and}\quad\lim_{j\to\infty}T_{j^{\prime}}\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U_{t_{m}}={\hat{T}}^{(m)}=T\hskip
2.5pt{\vrule height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U_{t_{m}}\ .$

To verify the boundary relation that $R:=T-T_{0}-\partial S$ has support in
$B$, it suffices to show that $R\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U_{t}=0$ for a.e. $t>0$. For each
$m\in\\{1,2,\dots\\}$, a.e. $t\in[t_{m+1},t_{m}]$, and $j^{\prime}>m$, we have
that

$\begin{split}R\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U_{t}\ &=\ (T-T_{0}-\partial S)\hskip
2.5pt{\vrule height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U_{t}\\\ &=\
(T_{j^{\prime}}-T_{0}-\partial{\hat{S}}_{j^{\prime}})\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U_{t}+(T-T_{j^{\prime}})\hskip
2.5pt{\vrule height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U_{t}-(\partial
S-\partial{\hat{S}}_{j^{\prime}})\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U_{t}\ .\end{split}$

Taking the flat norm, integrating, and applying [6][Thm.5.2.3(2)] gives

$\begin{split}\int_{t_{m+1}}^{t_{m}}\mathscr{F}(R\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U_{t})\,dt\ &=\
\int_{t_{m+1}}^{t_{m}}\mathscr{F}[(T-T_{0}-\partial S)\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U_{t}]\,dt\\\ &\leqslant\
\int_{t_{m+1}}^{t_{m}}\left(0+\mathscr{F}[(T-T_{j^{\prime}})\hskip
2.5pt{\vrule height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U_{t}]+\mathscr{F}[(\partial
S-\partial{\hat{S}}_{j^{\prime}})\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U_{t})]\right)dt\\\ &\leqslant\
(t_{m}-t_{m+1}+1)\left[\mathscr{F}(T-T_{j^{\prime}})+\mathscr{F}\partial(S-{\hat{S}}_{j^{\prime}})\right]\\\
&\leqslant\
(t_{m}-t_{m+1}+1)\left[\mathscr{F}(T-T_{j^{\prime}})+\mathscr{F}(S-{\hat{S}}_{j^{\prime}})\right]\
\to\ 0\ {\rm as}\ j\to\infty\ .\end{split}$

We conclude that $(T-T_{0}-\partial S)\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U_{t}=0$ for a.a. positive $t$, and
$\operatorname{\mathrm{spt}}(T-T_{0}-\partial S)\subseteq B$.

Since $T$ is thus admissible for $(\mathscr{P}_{B})$ and $\langle
T_{j}\rangle_{j}$ is a mass minimizing sequence,

$\mathbf{M}(T)\geqslant{\mathscr{M}}:=\lim_{j\to\infty}\mathbf{M}(T_{j})$ . To
get that the reverse inequality, we may for any $\varepsilon>0$, choose an $m$
sufficiently large so that $\mathbf{M}(T^{(m)})>\mathbf{M}(T)-\varepsilon$. We
may combine this with the mass lower semicontinuity, under the flat
convergence of $T_{i^{(m)}_{j}}\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U_{t_{m}}$ to $T^{(m)}$, to deduce that

$\mathbf{M}(T)-\varepsilon\ <\ \mathbf{M}(T^{(m)})\ \leqslant\
\liminf_{j\to\infty}\mathbf{M}(T_{i^{(m)}_{j}}\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,U_{t_{m}})\ \leqslant\
\liminf_{j\to\infty}\mathbf{M}(T_{i^{(m)}_{j}})\ =\ \mathscr{M}\ ,$

Letting $\varepsilon\downarrow 0$ gives $\ \mathbf{M}(T)\leqslant\mathscr{M}$,
showing that $T$ is the desired mass minimizer for $(\mathscr{P}_{B})$. ∎

######  5.10A Poincaré inequality. —

Suppose the set $M$ of regular points of a compact subanalytic subset $A$ of
dimension $k$ of $\mathbb{R}^{n}$ is connected and orientable. There is a
constant $\mathbf{c}(A)$ so that for any $f\in{\bf BV}(M)$ and any
$m\in\mathbb{R}$ satisfying $\ \mathscr{H}^{k}\\{x\,:\,f(x)<m\\}=\
\mathscr{H}^{k}\\{x\,:\,f(x)>m\\}\ $, (i.e. $m$ is a median of $f$ ), one has
the inequalities

$\displaystyle(1)\quad\int_{M}|f-m|\,d{\mathscr{H}}^{k}\ \leqslant\
\mathbf{c}(A)\int_{M}\|Df\|\ \quad{\rm and}$
$\displaystyle(2)\quad\int_{M}|f-\overline{f}|\,d{\mathscr{H}}^{k}\ \leqslant\
\mathbf{c}(A)\int_{M}\|Df\|\quad{\rm where}\quad\overline{f}\ =\
{\mathscr{H}}^{k}(A)^{-1}\int_{M}f\,d{\mathscr{H}}^{k}\ .$

###### Proof.

Since $A\setminus M$ is subanalytic of dimension $<k$, we may assume
$\overline{M}=A$. For (1) we may subtract the constant function $m$ to assume
that $m=0$. Thus the two sets $M_{-}=\\{x\in M\,:\,f(x)<0\\}$ and
$M_{+}=\\{x\in M\,:\,f(x)>0\\}$ have the same ${\mathscr{H}}^{k}$ measure,
which is finite because ${\mathscr{H}}^{k}(M)={\mathscr{H}}^{k}(A)<\infty$.
Fix an orientation for $M$, and let
$\boldsymbol{[}\\!\boldsymbol{[}M\boldsymbol{]}\\!\boldsymbol{]}$ denote the
corresponding $k$ dimensional rectifiable current. Here, the flat chain
$\partial\boldsymbol{[}\\!\boldsymbol{[}M\boldsymbol{]}\\!\boldsymbol{]}$ is
also rectifiable because
$\operatorname{\mathrm{spt}}\partial\boldsymbol{[}\\!\boldsymbol{[}M\boldsymbol{]}\\!\boldsymbol{]}\subseteq
B:=\overline{M}\setminus M$ and the constancy theorem [9, 4.1.31] may be
applied to the $k-1$ dimensional strata. From [9, 4.5.9(12)], we see that for
almost all $s>0$, the chain
$\boldsymbol{[}\\!\boldsymbol{[}M\boldsymbol{]}\\!\boldsymbol{]}_{s}:=\boldsymbol{[}\\!\boldsymbol{[}M\boldsymbol{]}\\!\boldsymbol{]}\hskip
2.5pt{\vrule height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,\\{x\in M\,:\,f(x)>s\\}$ is rectifiable
and of finite mass and finite boundary mass in $U:=\mathbb{R}^{n}\setminus B$.

Applying the linear relative isoperimetric inequality, Theorem 5.7, with
$S_{0}=\boldsymbol{[}\\!\boldsymbol{[}M\boldsymbol{]}\\!\boldsymbol{]}_{s}$ we
find an $S_{s}\in\mathbf{I}_{k}(A)$ satisfying

$(\partial S_{s})\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,M\ =\
(\partial\boldsymbol{[}\\!\boldsymbol{[}M\boldsymbol{]}\\!\boldsymbol{]}_{s})\hskip
2.5pt{\vrule height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,M\quad{\rm
and}\quad\mathbf{M}(S_{s}\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,M)\ \leqslant\
\mathbf{c}(A)\mathbf{M}[(\partial\boldsymbol{[}\\!\boldsymbol{[}M\boldsymbol{]}\\!\boldsymbol{]}_{s}\hskip
2.5pt{\vrule height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,M]\ ,$

Inasmuch as
$\partial(S_{s}-\boldsymbol{[}\\!\boldsymbol{[}M\boldsymbol{]}\\!\boldsymbol{]}_{s})\hskip
2.5pt{\vrule height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,M=0$, the constancy theorem gives an
integer $j$ so that

$(S_{s}-\boldsymbol{[}\\!\boldsymbol{[}M\boldsymbol{]}\\!\boldsymbol{]}_{s})]\hskip
2.5pt{\vrule height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,M\
=j\boldsymbol{[}\\!\boldsymbol{[}M\boldsymbol{]}\\!\boldsymbol{]}\quad{\rm
and}\quad S_{s}\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,M=(j-1)\left(\boldsymbol{[}\\!\boldsymbol{[}M\boldsymbol{]}\\!\boldsymbol{]}\
-\boldsymbol{[}\\!\boldsymbol{[}M\boldsymbol{]}\\!\boldsymbol{]}_{s}\right)+j\boldsymbol{[}\\!\boldsymbol{[}M\boldsymbol{]}\\!\boldsymbol{]}_{s}\
.$

Since
$\mathbf{M}(\boldsymbol{[}\\!\boldsymbol{[}M\boldsymbol{]}\\!\boldsymbol{]}_{s})\leqslant{\mathscr{H}}^{k}(M_{+})=\frac{1}{2}{\mathscr{H}}^{k}(M)=\frac{1}{2}\mathbf{M}(\boldsymbol{[}\\!\boldsymbol{[}M\boldsymbol{]}\\!\boldsymbol{]})$,
we deduce that

$\mathbf{M}(\boldsymbol{[}\\!\boldsymbol{[}M\boldsymbol{]}\\!\boldsymbol{]}_{s})\
\leqslant\ \mathbf{M}(S_{s}\hskip 2.5pt{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,M)\ \leqslant\
\mathbf{c}(A)\mathbf{M}\left(\partial\boldsymbol{[}\\!\boldsymbol{[}M\boldsymbol{]}\\!\boldsymbol{]}_{s}\hskip
2.5pt{\vrule height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,M\right)\ .$

We now use the BV coarea formula as in [9, 4.5.9(13)] to see that

$\displaystyle\int_{M_{+}}|f|\,d{\mathscr{H}}^{k}\ $ $\displaystyle=\
\int_{0}^{\infty}{\mathscr{H}}^{k}\\{f>s\\}\,ds\ =\
\int_{0}^{\infty}\mathbf{M}(\boldsymbol{[}\\!\boldsymbol{[}M\boldsymbol{]}\\!\boldsymbol{]}_{s})\,ds$
$\displaystyle\leqslant\
\mathbf{c}(A)\int_{0}^{\infty}\mathbf{M}\left(\partial\boldsymbol{[}\\!\boldsymbol{[}M\boldsymbol{]}\\!\boldsymbol{]}_{s})\hskip
2.5pt{\vrule height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,M\right)\ \,ds\ =\
\mathbf{c}(A)\int_{M_{+}}\|Df\|\ .$

The same argument applied to $-f$ gives

$\int_{M_{-}}|f|\,d{\mathscr{H}}^{k}\ \leqslant\
\mathbf{c}(A)\int_{M_{-}}\|Df\|\ ,$

and the conclusion of (1)

$\displaystyle\int_{M}|f|\,d{\mathscr{H}}^{k}\ $
$\displaystyle=\int_{M_{-}}|f|\,d{\mathscr{H}}^{k}+\
\int_{M_{+}}|f|\,d{\mathscr{H}}^{k}$ $\displaystyle\leqslant\
\mathbf{c}(A)\int_{M_{-}}\|Df\|\ +\mathbf{c}(A)\int_{M_{+}}\|Df\|\
\leqslant\mathbf{c}(A)\int_{M}\|Df\|\ .$

Conclusion (2) easily follows from (1) because

$\int_{M}|f-\overline{f}|\,d{\mathscr{H}}^{k}\ \leqslant\
\int_{M}|f-m|\,d{\mathscr{H}}^{k}\ +\
\int_{M}|m-\overline{f}|\,d{\mathscr{H}}^{k}\ ,$

and

$|m-\overline{f}|\ =\ {\mathscr{H}}^{k}(A)^{-1}\big{|}\int_{M}(m\ -\
f)\,d{\mathscr{H}}^{k}\big{|}\ \leqslant\
{\mathscr{H}}^{k}(A)^{-1}\int_{M}|f-m|\,d{\mathscr{H}}^{k}\ ,$

∎

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