# Poincaré inequality and energy of separating sets

Emanuele Caputo  Mathematics Institute, Zeeman Building, University of
Warwick, Coventry, CV4 7AL, United Kingdom<EMAIL_ADDRESS>and
Nicola Cavallucci  Institute of Mathematics, EPFL, Station 8, 1015 Lausanne,
Switzerland<EMAIL_ADDRESS>
###### Abstract.

We study geometric characterizations of the Poincaré inequality in doubling
metric measure spaces in terms of properties of separating sets. Given a
couple of points and a set separating them, such properties are formulated in
terms of several possible notions of energy of the boundary, involving for
instance the perimeter, codimension type Hausdorff measures, capacity,
Minkowski content and approximate modulus of suitable families of curves. We
prove the equivalence within each of these conditions and the $1$-Poincaré
inequality.

###### Key words and phrases:

Poincaré inequality, Separating sets, Perimeter, Minkowski content, metric
measure spaces

###### 2020 Mathematics Subject Classification:

30L15, 53C23, 49J52

###### Contents

  1. 1 Introduction
    1. 1.1 Structure of the paper
    2. 1.2 Acknowledgments
  2. 2 Preliminaries
    1. 2.1 Curves in metric spaces
    2. 2.2 Modulus and approximate modulus of a family of curves
    3. 2.3 The Poincaré inequality and the Riesz potential
  3. 3 Several energies to measure boundaries
  4. 4 Relations between the energies under Poincaré inequality
  5. 5 Relations between the different energies weighted with the Riesz potential
  6. 6 Proof of the main theorem

##  1\. Introduction

Since the work of Heinonen-Koskela [HK98] it has been clear the importance of
the class of doubling metric measure spaces satisfying a Poincaré inequality,
called for short PI-spaces. On such spaces many analytical properties that
were classically studied for Euclidean spaces or Riemannian manifolds hold
(see for instance the monographs [BB11] and [HK00]). However, despite its
definition being analytical, it was clear from the beginning of the theory
that a Poincaré inequality has strong geometric consequences. One of the most
notable result in such a direction is Cheeger’s theorem ([Che99]) saying that
PI spaces are Lipschitz differentiability spaces. Moreover, as noticed by
Semmes, every such space must be $L$-quasiconvex, namely there exists a
constant $L\geq 1$, such that each couple of points can be connected by a
$L$-quasigeodesic, i.e. a curve whose length is bounded by $L$ times the
distance of the points (see for instance [Che99, Thm. 17.1]). The relation
between the validity of a Poincaré inequality and the presence of ‘many’
curves in the space was highlighted in [Kei03] by Keith. He proved that, given
a doubling metric measure space $({\rm X},{\sf d},\mathfrak{m})$, a
$p$-Poincaré inequality is equivalent to a uniform lower bound on the
$p$-modulus of curves joining two points $x,y\in{\rm X}$, but computed with
respect to the weighted measure
$\mathfrak{m}_{x,y}^{L}=R_{x,y}^{L}\mathfrak{m}$, where

$R_{x,y}^{L}(z):=\chi_{B_{x,y}^{L}}(z)\left(\frac{{\sf
d}(x,z)}{\mathfrak{m}(B_{{\sf d}(x,z)}(x))}+\frac{{\sf
d}(y,z)}{\mathfrak{m}(B_{{\sf d}(y,z)}(y))}\right).$

Here $B_{x,y}^{L}:=B_{2L{\sf d}(x,y)}(x)\cup B_{2L{\sf d}(x,y)}(y)$. We call
$R_{x,y}^{L}$ the _$L$ -truncated Riesz potential with poles at $x,y$_. The
Riesz potential, together with the Hardy-Littlewood maximal function, has been
a central tool in the theory of Poincaré inequalities from the beginning
([Hei01], [HK00]).
Semmes conjectured that the $1$-Poincaré inequality is equivalent to having
the existence of a $1$-pencil of curves for every couple of points, that is a
measure on quasigeodesics connecting them with bounded overlap controlled in
terms of the Riesz potential (see Proposition 2.2 for the precise definition).
The conjecture is true as proven independently by [FO19] and [DCEBKS21]. The
proof of the latter one can be extended for $p>1$ as we notice in [CC24].
There we also clarify the roles of the different constants in the definitions.
Another remarkable condition that encodes the Poincaré inequality for $p>1$ in
geometric terms is defined through an obstacle avoiding condition. The
condition is informally saying that for each couple of points and for each
Borel set that is not too dense around these two points (in a sense quantified
with the maximal function) there exists a quasigeodesic connecting the two
points that does not spend too much time inside the set, see [EB19] and
[EBG21]. The same condition for $p=1$ is studied in detail in [CC24], where we
clarify its relation with the $1$-Poincaré inequality.

All the conditions above turned out to be useful in order to prove that
relevant class of spaces appearing in metric measure geometry and analysis on
fractals satisfy or do not satisfy a Poincaré inequality. All the proofs of
these facts use a characterization which is relevant for the geometry under
study. For instance, using the obstacle-avoidance approach [EBG21] studied the
Poincaré inequality on non-self similar Sierpinski carpets and more general
fractals (see [MTW13] for previous results in this direction). Let us comment
with an application of the $1$-pencil of curves. Ohta [Oht07] and Sturm
[Stu06] introduced a notion of metric measure spaces satisfying a weak lower
bound on the Ricci curvature by $K$ and an upper bound on the dimension by $N$
in a synthetic sense, the so called ${\rm MCP}(K,N)$ spaces. A straightforward
consequence of the definition of ${\rm MCP}(0,N)$ spaces implies the existence
of $1$-pencils of curves and so a $1$-Poincaré inequality, under the
additional assumption of $N$-Ahlfors regularity.

The goal of our paper is to go in this direction and provide a new
characterizations of the Poincaré inequality in geometric terms. Let us notice
that all the previous characterizations are defined in terms of properties of
curves. The novelty of our approach is that our equivalent conditions are not
formulated using curves but using properties related to separating sets. Let
us introduce such a notion.

######  Definition 1.1.

Let $({\rm X},{\sf d},\mathfrak{m})$ be a metric measure space and let
$x,y\in{\rm X}$. A closed set $\Omega$ is a separating set from $x$ to $y$ if
there exists $r>0$ such that $B_{r}(x)\subseteq\textup{int}(\Omega)$ and
$B_{r}(y)\subseteq\Omega^{c}$. We denote by $\SS_{\textup{top}}(x,y)$ the
class of all separating sets from $x$ to $y$.

Here the subscript ‘${\rm top}$’ stands for topological, in contrast with
other notions of separating sets studied in [CC24]. We state the main result
of the paper. It shows that for $p=1$ the Poincaré inequality is equivalent to
have a uniform lower bound on several energies of boundaries of separating
sets. These energies are all defined through the Riesz potential.

######  Theorem 1.2.

Let $({\rm X},{\sf d},\mathfrak{m})$ be a doubling metric measure space. Then
the following conditions are quantitatively equivalent:

  * (PI)

$1$-Poincaré inequality;

  * (BP)

$\exists c>0,L\geq 1$ such that $\int R_{x,y}^{L}\,{\mathrm{d}}{\rm
Per}_{\mathfrak{m}}(\Omega,\cdot)\geq c$ for every $x,y\in{\rm X}$ and
$\Omega\in\SS_{\textup{top}}(x,y)$;

  * (BC)

$\exists c>0,L\geq 1$ such that
$\textup{cap}_{\mathfrak{m}_{x,y}^{L}}(\Omega)\geq c$ for every $x,y\in{\rm
X}$ and $\Omega\in\SS_{\textup{top}}(x,y)$;

  * (BH)

$\exists c>0,L\geq 1$ such that
$\int_{\partial\Omega}R_{x,y}^{L}\,{\mathrm{d}}\mathcal{H}^{{\rm
cod-}{1}}_{\mathfrak{m}}\geq c$ for every $x,y\in{\rm X}$ and
$\Omega\in\SS_{\textup{top}}(x,y)$;

  * (BMC)

$\exists c>0,L\geq 1$ such that $(\mathfrak{m}_{x,y}^{L})^{+,1}(\Omega)\geq c$
for every $x,y\in{\rm X}$ and $\Omega\in\SS_{\textup{top}}(x,y)$;

  * (BAM)

$\exists c>0,L\geq 1$ such that ${\rm
AM}_{1}(\partial\Omega,\mathfrak{m}_{x,y}^{L})\geq c$ for every $x,y\in{\rm
X}$ and $\Omega\in\SS_{\textup{top}}(x,y)$;

  * (BAM⋔)

$\exists c>0,L\geq 1$ such that ${\rm
AM}_{1}^{\pitchfork}(\Omega,\mathfrak{m}_{x,y}^{L})\geq c$ for every
$x,y\in{\rm X}$ and $\Omega\in\SS_{\textup{top}}(x,y)$.

We refer the reader to Section 3 for the definition of all the quantities
involved in the statement. The expert reader can recognize the perimeter, the
variational capacity, the Minkowski content, the codimension-$1$ Hausdorff
measure and the $1$-approximate modulus. The acronyms of each line should be
read accordingly: the letter B stands for Big, therefore for instance (BP)
means ‘Big Perimeter’, while (BH) means ‘Big codimension-$1$ Hausdorff
measure’. Similarly for the other quantities. We refer the reader to Theorem
6.1 where additional equivalent conditions are presented. In particular we
stress the fact that all the conditions above that involve the topological
boundary of a separating set can be checked on its essential boundary.
Moreover we defined other equivalent conditions by defining the perimeter and
the codimension-$1$ Hausdorff measure directly on the metric measure space
$({\rm X},{\sf d},\mathfrak{m}_{x,y}^{L})$.

Let us give an intuition about what these conditions in the main statement
mean. Let us start with (BH) and analyze the special case of $({\rm X},{\sf
d},\mathfrak{m})$ being $s$-Ahlfors regular, with $s\geq 1$. In this case, the
condition (BH) implies that for every $x,y\in{\rm X}$ and every $\Omega\in{\rm
SS}_{\rm top}(x,y)$, $\mathcal{H}^{s-1}(\partial\Omega)>0$. Thus in particular
the Hausdorff dimension of $\partial\Omega$ is at least $s-1$. This says that
the Poincaré inequality implies that the boundary of separating sets are big
in terms of Hausdorff dimension. Actually, the (BH) says more than what just
stated and quantifies how big the $(s-1)$ Hausdorff measure of
$\partial\Omega$ is, taking into account the distance from the poles. This can
be better understood if one thinks about separating sets in the Euclidean case
$\mathbb{R}^{d}$ endowed with Lebesgue measure, which can be regarded as the
simplest example of $1$-PI space. Here, considering two points
$x,y\in\mathbb{R}^{d}$, sufficiently small balls around $x$ are all separating
sets from $x$ to $y$ and, while the $\mathcal{H}^{d-1}$ measure of the spheres
converges to $0$, the weighted Hausdorff measure with the Riesz potential
$R_{x,y}^{L}\,\mathcal{H}^{d-1}$ is uniformly bounded from below.

Let us discuss the condition (BMC) and its relation with the main proof in
[FO19]. In order to prove that $1$-PI spaces supports a $1$-pencil of curves
for each couple of points $x,y\in{\rm X}$, the authors argue as follows: they
discretize the metric measure space by considering a $\delta$-net for $\delta$
small and they build a graph connecting close points in the $\delta$-net. Then
considering all possible cuts in the graph, namely a subset of vertices
containing $x$ and not $y$, they prove that a suitable energy of the cut
weighted with the Riesz potential is uniformly bounded from below. The bound
from below is independent of $x,y,\delta$ and can be regarded as a discretized
version of our characterizations, for instance the closest one being (BMC).
One of the goals of [CC24] is to investigate the relation between boundary of
cuts in the appoximating graphs and boundary of separating sets in the
continuous version.

The conditions (BAM) and (BAM⋔), since they are formulated in terms of modulus
of curves, share some similarities to Keith’s characterization of $p$-PI space
in [Kei03]. First of all we prove along the way, see Remark 6.2, that ${\rm
X}$ is a $1$-PI space if and only if for every $x,y\in{\rm X}$ we have ${\rm
AM}_{1}(\Gamma_{x,y}^{L},\mathfrak{m}_{x,y}^{L})\geq c$ for some $c>0$ and
$L\geq 1$. This is a strengthening of Keith’s characterization for $p=1$. On
the other hand (BAM) and (BAM${}^{\pitchfork})$ involves the class of curves
hitting the boundary of a separating set, which is a much larger family of
curves. However the validity of this condition for every separating set, which
can be equivalently stated as a property of arbitrarily short curves (see
Lemma 3.15), is enough to get a uniform lower bound on ${\rm
AM}_{1}(\Gamma_{x,y}^{L},\mathfrak{m}_{x,y}^{L})$.

There are two natural research directions starting from our result. The first
one is the case $p>1$. We prove in Proposition 6.3 that the natural
$p$-generalization of (BH) is satisfied by a doubling metric measure space
with a $p$-Poincaré inequality. Example 6.4 shows that this version of (BH)
does not imply any Poincaré inequality. It would be interesting to find
additional assumptions on the metric measure space so that the condition
appearing in Proposition 6.3 characterizes $p$-PI spaces. The second direction
is whether it is possible to test each of the properties appearing in Theorem
1.2 on a subclass of more regular separating sets. The regularity is such sets
can be expressed as sets with Lipschitz (or even smooth) boundaries in the
case of Riemannian manifolds or in terms of metric measure regularity
conditions, such as the interior (or exterior) ball condition and similar
properties (see the discussion in [CR24, §4.2]).

###  1.1. Structure of the paper

The paper is structured as follows. In Section 2 we recall definitions and
properties we will need in the presentation. We also study the measure
$\mathfrak{m}_{x,y}^{L}$ and its known relations with the Poincaré inequality.
In Section 3 we introduce the several energies appearing in Theorem 1.2 and we
prove some basic relations between them holding in full generality. In
Sections 4 and 5 we specialize the study of the relations between our
different energies in case of spaces satisfying a Poincaré inequality. These
sections are quite technical, especially Section 5 in which we compare two
possible ways to weight the perimeter measure and the codimension Hausdorff
measure with the Riesz potential. Finally in Section 6 we prove all the
equivalences stated in Theorem 1.2. The core of the arguments are contained in
the proof of the main theorem. In particular the reader can skip the proofs of
Section 5 during a first read.

###  1.2. Acknowledgments

The first author was supported by Academy of Finland, grants no. 321896. He
currently acknowledges the support by the European Union’s Horizon 2020
research and innovation programme (Grant agreement No. 948021). He also thanks
the kind hospitality of the Univerisity of Ottawa and of Augusto Gerolin. We
thank Francesco Nobili for a careful reading of the manuscript.

##  2\. Preliminaries

Let $({\rm X},{\sf d},\mathfrak{m})$ be a metric measure space, i.e. $({\rm
X},{\sf d})$ is a complete and separable metric space and $\mathfrak{m}$ is
nonnegative Borel measure that is finite on bounded sets. Given
$A\subseteq{\rm X}$ and $r\geq 0$ we denote by $B_{r}(A):=\\{x:\,{\sf
d}(x,A)<r\\}$ and by $\overline{B}_{r}(A):=\\{x:\,{\sf d}(x,A)\leq r\\}$, i.e.
respectively the open and closed $r$-neighbourhood of $A$. For instance the
open ball of center $x\in{\rm X}$ and radius $r$ is $B_{r}(x)$. We denote by
$\mathscr{B}({\rm X})$ the class of Borel subsets of ${\rm X}$.
We denote by ${\rm Lip}({\rm X})$ the space of Lipschitz functions on ${\rm
X}$ with values in $\mathbb{R}$. For $u\in{\rm Lip({\rm X})}$ we denote by
${\rm Lip}(u)$ the Lipschitz constant of $u$. The _local Lipschitz constant_
${\rm lip}u\colon{\rm X}\to\mathbb{R}$ of a function $u\colon{\rm
X}\to\mathbb{R}$ as

${\rm lip}u(x):=\varlimsup_{y\to x}\frac{|u(y)-u(x)|}{{\sf
d}(y,x)}=\lim_{\delta\to 0}\sup_{B_{\delta}(x)}\frac{|u(y)-u(x)|}{{\sf
d}(y,x)}$

with the convention that ${\rm lip}u(x)=0$ if $x$ is an isolated point. The
function ${\rm sl}(x,\delta):=\sup_{B_{\delta}(x)}\frac{|u(y)-u(x)|}{{\sf
d}(y,x)}$ is called the slope of $u$ at $x$ at scale $\delta$.

Given a set $A\subseteq{\rm X}$, we define by ${\rm int}(A)$ its topological
interior and ${\rm ext}(A):={\rm int}(A^{c})$. Given a
$\mathfrak{m}$-measurable set $A\subseteq{\rm X}$ we define its measure
theoretic interior $I_{A}$ and its measure theoretic exterior $O_{A}$ as
follows:

$I_{A}:=\left\\{x\in{\rm X}:\lim_{r\to 0}\frac{\mathfrak{m}(B_{r}(x)\cap
A)}{\mathfrak{m}(B_{r}(x))}=1\right\\},\quad O_{A}:=\left\\{x\in{\rm
X}:\lim_{r\to 0}\frac{\mathfrak{m}(B_{r}(x)\setminus
A)}{\mathfrak{m}(B_{r}(x))}=1\right\\}.$

The set ${\rm X}\setminus(I_{A}\cup O_{A})$ is the essential boundary of $A$,
denoted by $\partial^{e}A$. Observe that

$\partial^{e}A=\left\\{x\in{\rm X}:\varlimsup_{r\to
0}\frac{\mathfrak{m}(B_{r}(x)\cap A)}{\mathfrak{m}(B_{r}(x))}>0\text{ and
}\varlimsup_{r\to 0}\frac{\mathfrak{m}(B_{r}(x)\setminus
A)}{\mathfrak{m}(B_{r}(x))}>0\right\\}.$

It is straightforward consequence of the definitions that $I_{A}\supseteq{\rm
int}(A)$ and $O_{A}\supseteq{\rm ext}(A)$, so $\partial^{e}A\subseteq\partial
A$.

###  2.1. Curves in metric spaces

Let $({\rm X},{\sf d})$ be a metric space. The space of continuous curves
$\gamma\colon[0,1]\to{\rm X}$ is denoted by $C([0,1],{\rm X})$. The length of
a curve $\gamma\in C([0,1],{\rm X})$ is

$\ell(\gamma):=\sup\left\\{\sum_{i=0}^{N-1}{\sf
d}(\gamma_{t_{i}},\gamma_{t_{i+1}}),\,\\{t_{i}\\}_{i=1}^{N}\subseteq[0,1],\,N\in\mathbb{N},\,0=t_{0}<t_{1}<\dots<t_{N}=1\right\\}.$

A curve $\gamma\in C([0,1],{\rm X})$ is rectifiable if $\ell(\gamma)<\infty$.
The metric speed of a curve $\gamma$ at a point $t\in[0,1]$ is
$|\dot{\gamma}_{t}|:=\lim_{h\to 0}\frac{{\sf
d}(\gamma_{t+h},\gamma_{t})}{|h|}\in[0,\infty]$ if such a limit exists. A
curve $\gamma$ is absolutely continuous, and we write $\gamma\in{\rm
AC}([0,1],{\rm X})$, if there exists $0\leq h\in L^{1}(0,1)$ such that ${\sf
d}(\gamma_{1},\gamma_{0})\leq\int_{0}^{1}h(t)\,{\mathrm{d}}t$. If
$\gamma\in{\rm AC}([0,1],{\rm X})$, $|\dot{\gamma}_{t}|$ exists and it is
finite for a.e. $t\in[0,1]$ and $(t\mapsto|\dot{\gamma}_{t}|)\in
L^{1}([0,1])$. Moreover, such a function is the minimal $h$ (in the a.e.
sense) that can be chosen in the definition of ${\rm AC}([0,1],{\rm X})$.
The integral of a Borel function $g\colon{\rm X}\to[0,\infty)$ over
$\gamma\in{\rm AC}([0,1],{\rm X})$, is defined as

$\int_{\gamma}g\,{\mathrm{d}}s:=\int_{0}^{1}g(\gamma_{t})\,|\dot{\gamma}_{t}|\,{\mathrm{d}}t.$

Let us fix two points $x,y\in{\rm X}$ and $L\geq 1$. We set

$\Gamma_{x,y}^{L}:=\\{\gamma\in{\rm AC}([0,1],{\rm
X}),\,\gamma_{0}=x,\,\gamma_{1}=y,\,\ell(\gamma)\leq L\,{\sf d}(x,y)\\}.$

An element of $\Gamma_{x,y}^{L}$ is called a $L$-quasigeodesic joining $x$ to
$y$. A curve $\gamma\in C([0,1],{\rm X})$ is a geodesic if $\ell(\gamma)={\sf
d}(x,y)$, i.e. it is a $1$-quasigeodesic joining its endpoints.
The metric space $({\rm X},{\sf d})$ is $L$-quasiconvex if
$\Gamma_{x,y}^{L}\neq\emptyset$ for every $x,y\in{\rm X}$ and it is geodesic
if it is $1$-quasiconvex.
Let $u\colon{\rm X}\to\mathbb{R}$ be a function. A function $g\colon{\rm
X}\to[0,+\infty]$ such that

$|u(\gamma_{1})-u(\gamma_{0})|\leq\int_{\gamma}g\,{\mathrm{d}}s$

for every rectifiable curve $\gamma$ is called an _upper gradient_ of $u$. The
set of upper gradients of $u$ is denoted by ${\rm UG}(u)$.

###  2.2. Modulus and approximate modulus of a family of curves

We recall the notion of $p$-modulus and $p$-approximate modulus of a family of
curves with respect a Borel nonnegative locally finite measure $p$. Let $({\rm
X},{\sf d},\mathfrak{m})$ be a metric measure space, let $p\geq 1$ and let
$\Gamma\subseteq C([0,1]),{\rm X})$. The set of admissible densities for
$\Gamma$ is

${\rm Adm}(\Gamma):=\left\\{\rho\colon{\rm X}\to[0,+\infty]:g\text{ Borel and
}\int_{\gamma}\rho\,{\mathrm{d}}s\geq 1\text{ for all
}\gamma\in\Gamma\right\\}.$

The $p$-modulus of $\Gamma$ with respect to the measure $\mathfrak{m}$ is

${\rm Mod}_{p}(\Gamma,\mathfrak{m}):=\inf_{\rho\in{\rm
Adm}(\Gamma)}\int\rho^{p}\,{\mathrm{d}}\mathfrak{m}.$

In this paper we will deal largely with the approximate modulus. In order to
define it we introduce the set of admissible sequences of densities for
$\Gamma$ as

${\text{Adm-seq}}(\Gamma):=\left\\{\\{\rho_{j}\\}_{j}:\,\rho_{j}\colon{\rm
X}\to[0,+\infty]\text{ Borel and
}\varliminf_{j\to\infty}\int_{\gamma}\rho_{j}\,{\mathrm{d}}s\geq 1\text{ for
all }\gamma\in\Gamma\right\\}.$

The approximate $p$-modulus of $\Gamma$ with respect to the measure
$\mathfrak{m}$ is

${\rm AM}_{p}(\Gamma,\mathfrak{m}):=\inf_{\\{\rho_{j}\\}\in\text{Adm-
seq}(\Gamma)}\varliminf_{j\to+\infty}\int\rho_{j}^{p}\,{\mathrm{d}}\mathfrak{m}$

It is clear from the definition that ${\rm
AM}_{p}(\Gamma,\mathfrak{m})\leq{\rm Mod}_{p}(\Gamma,\mathfrak{m})$ for every
family $\Gamma$. Moreover for $p>1$, ${\rm AM}_{p}(\Gamma,\mathfrak{m})={\rm
Mod}_{p}(\Gamma,\mathfrak{m})$ for every $\Gamma\subseteq C([0,1],{\rm X})$
(cp. for instance [HEMM17, Theorem 1]). The equality is generally false for
$p=1$. Both the modulus and the approximate modulus are outer measures on
$C([0,1],{\rm X})$ (cp. [HEMM17, Theorem 15 and 16]).

###  2.3. The Poincaré inequality and the Riesz potential

A metric measure space $({\rm X},{\sf d},\mathfrak{m})$ is doubling if there
exists $C_{D}>0$ such that for every $x\in{\rm X}$ and $r>0$ we have

$\mathfrak{m}(B_{2r}(x))\leq C_{D}\,\mathfrak{m}(B_{r}(x)).$

A consequence of the definition of the doubling assumption is that $({\rm
X},{\sf d})$ is proper, i.e. closed and bounded sets are compact. A metric
measure space $({\rm X},{\sf d},\mathfrak{m})$ is asymptotically doubling if

$\varlimsup_{r\to
0}\frac{\mathfrak{m}(B_{2r}(x))}{\mathfrak{m}(B_{r}(x))}<\infty$

for $\mathfrak{m}$-a.e.$x\in{\rm X}$. We say it is $C$-asymptotically doubling
if

$\varlimsup_{r\to 0}\frac{\mathfrak{m}(B_{2r}(x))}{\mathfrak{m}(B_{r}(x))}\leq
C$

for $\mathfrak{m}$-a.e.$x\in{\rm X}$.
We say that $({\rm X},{\sf d},\mathfrak{m})$ is $s$-Ahlfors regular with $s>0$
if there exists a constant $C_{\rm AR}>0$ such that for every $x\in{\rm X}$
and every $0<r<{\rm diam}({\rm X})$ we have

(1) $C_{\rm AR}^{-1}\,r^{s}\leq\mathfrak{m}(B_{r}(x))\leq C_{\rm AR}\,r^{s}.$

A metric measure space $({\rm X},{\sf d},\mathfrak{m})$ satisfies a (weak)
$p$-Poincaré inequality if there exists $C_{P}\geq 1$ and $\lambda\geq 1$ such
that

$\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{B_{r}(x)}\left|u-\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{B_{r}(x)}u\,{\mathrm{d}}\mathfrak{m}\right|\,{\mathrm{d}}\mathfrak{m}\leq
C_{P}r\,\left(\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{B_{\lambda r}(x)}({\rm
lip}u)^{p}\,{\mathrm{d}}\mathfrak{m}\right)^{\frac{1}{p}}$

for every $u\in{\rm Lip}({\rm X})$. A metric measure space $({\rm X},{\sf
d},\mathfrak{m})$ is called a $p$-PI space if it is doubling and it satisfies
a $p$-Poincaré inequality. The _structural constants_ of a $p$-PI spaces are
$C_{D},C_{P},\lambda$.

######  Remark 2.1.

If $({\rm X},{\sf d},\mathfrak{m})$ is a $p$-PI space, then, as proved in
[Che99, Theorem 17.1] (see also [HKST15, Theorem 8.3.2]), the metric space
$({\rm X},{\sf d})$ is $L$-quasigeodesic for some $L\geq 1$ depending only on
the structural constants. By considering ${\sf
d}^{\prime}(x,y):=\inf\\{\,\ell(\gamma):\,\gamma_{0}=x,\,\gamma_{1}=y\,\\}$,
it is straightforward to check that

(2) ${\sf d}(x,y)\leq{\sf d}^{\prime}(x,y)\leq L{\sf d}(x,y),$

and that $({\rm X},{\sf d}^{\prime})$ is geodesic. Moreover $({\rm X},{\sf
d}^{\prime},\mathfrak{m})$ is again a $p$-PI space.

The Poincaré inequality on a doubling metric measure space has several
equivalent definitions (cp. [CC24, Theorem 3.4]). In order to state some of
them we recall one of the most important objects of this paper: the Riesz
potential. Let $({\rm X},{\sf d},\mathfrak{m})$ be a metric measure space.
Given $x,y\in{\rm X}$, the _Riesz potential with poles at $x$ and $y$_
$R_{x,y}\colon{\rm X}\to[0,\infty)$ is defined for every $z\in{\rm
X}\setminus\\{x,y\\}$ as

(3) $\displaystyle R_{x,y}(z):$ $\displaystyle=\frac{{\sf
d}(x,z)}{\mathfrak{m}(B_{{\sf d}(x,z)}(x))}+\frac{{\sf
d}(y,z)}{\mathfrak{m}(B_{{\sf d}(y,z)}(y))}=:R_{x}(z)+R_{y}(z).$

Moreover, we define $R_{x,y}(x)=R_{x,y}(y)=0$. For $L\geq 1$ we set
$B_{x,y}^{L}:=B_{2L{\sf d}(x,y)}(x)\cup B_{2L{\sf d}(x,y)}(y)$ and
$\overline{B}_{x,y}^{L}:=\overline{B}_{2L{\sf
d}(x,y)}(x)\cup\overline{B}_{2L{\sf d}(x,y)}(y)$. The $L$-_truncated Riesz
potential with poles at $x,y$_ is

(4) $R_{x,y}^{L}(z):=\chi_{B_{x,y}^{L}}(z)R_{x,y}(z)$

for every $z\in{\rm X}\setminus\\{x,y\\}$. The corresponding Riesz measure is
defined as

(5) $\mathfrak{m}_{x,y}^{L}=R^{L}_{x,y}\,\mathfrak{m}.$

It is a measure on ${\rm X}$ which is supported on $\overline{B}_{x,y}^{L}$.
This measure has been already studied for instance in [Hei01] and [Kei03]. We
can now state two equivalent formulations of the Poincaré inequality on
doubling metric measure spaces.

######  Proposition 2.2 ([Hei01, Theorem 9.5], [CC24, Theorem 3.4]).

Let $({\rm X},{\sf d},\mathfrak{m})$ be a doubling metric measure space. The
following are quantitatively equivalent:

  * ($p$-PI)

$({\rm X},{\sf d},\mathfrak{m})$ satisfies a $p$-Poincaré inequality;

  * ($p$-PtPI)

$\exists C>0$, $L\geq 1$ such that for every $x,y\in{\rm X}$ and every
$u\in{\rm Lip}({\rm X})$ it holds

$|u(x)-u(y)|^{p}\leq C\,{\sf d}(x,y)^{p-1}\,\int_{\rm X}({\rm
lip}u)^{p}\,{\mathrm{d}}\mathfrak{m}_{x,y}^{L}.$

  * ($p$-pencil)

$\exists C>0$, $L\geq 1$ such that for every $x,y\in{\rm X}$ there exists
$\alpha\in\mathcal{P}(\Gamma_{x,y}^{L})$ such that

$\left(\int\int_{\gamma}g\,{\mathrm{d}}s{\mathrm{d}}\alpha\right)^{p}\leq
C\,{\sf d}(x,y)^{p-1}\int_{\rm X}g^{p}\,{\mathrm{d}}\mathfrak{m}_{x,y}^{L}$

for every nonnegative Borel function $g$.

The next result establishes some basic properties of the Riesz measures.

######  Proposition 2.3.

Let $({\rm X},{\sf d},\mathfrak{m})$ be a $C_{D}$-doubling metric measure
space and fix $x,y\in{\rm X}$ and $L\geq 1$. Then

  * (i)

$\mathfrak{m}_{x,y}^{L}({\rm X})\leq 8C_{D}L{\sf d}(x,y)$. In particular
$\mathfrak{m}_{x,y}^{L}$ is a finite Borel measure;

  * (ii)

$\mathfrak{m}_{x,y}^{L}$ is $C_{D}$-asymptotically doubling, in particular
$({\rm X},{\sf d},\mathfrak{m}_{x,y}^{L})$ is a Vitali space.

For the definition of Vitali space we refer to [HKST15, §3.4].

###### Proof.

The first estimate follows by

$\mathfrak{m}_{x,y}^{L}({\rm X})\leq\int_{B_{4L{\sf
d}(x,y)}(x)}R_{x}\,{\mathrm{d}}\mathfrak{m}+\int_{B_{4L{\sf
d}(x,y)}(y)}R_{y}\,{\mathrm{d}}\mathfrak{m}\leq 8C_{D}L{\sf d}(x,y),$

where for the last inequality we rewrited the Riesz potential in dyadic scales
as in [Hei01, p. 72]. Observe that the poles $x,y$ could have positive
$\mathfrak{m}$-measure, so it is important to have defined
$R_{x,y}(x)=R_{x,y}(y)=0$ for this computation.
For the second point we recall that $({\rm X},{\sf d},\mathfrak{m})$ is a
Vitali space since $\mathfrak{m}$ is doubling (cp. [HKST15, Theorem 3.4.3]).
Moreover $R_{x,y}^{L}\in L^{1}(\mathfrak{m})$, as the computation of the first
item shows. Therefore, by [HKST15, p.77], for $\mathfrak{m}$-a.e.$z$ we have
that

$\lim_{r\to 0}\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{B(z,r)}R_{x,y}^{L}\,{\mathrm{d}}\mathfrak{m}=R_{x,y}^{L}(z).$

For these points we have

$\varlimsup_{r\to
0}\frac{\mathfrak{m}_{x,y}^{L}(B_{2r}(z))}{\mathfrak{m}_{x,y}^{L}(B_{r}(z))}\leq
C_{D}\varlimsup_{r\to 0}\frac{\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{B_{2r}(z)}R_{x,y}^{L}\,{\mathrm{d}}\mathfrak{m}}{\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{B_{r}(z)}R_{x,y}^{L}\,{\mathrm{d}}\mathfrak{m}}=C_{D}.$

Since $\mathfrak{m}_{x,y}^{L}$ is absolutely continuous with respect to
$\mathfrak{m}$ it follows that for $\mathfrak{m}_{x,y}^{L}$-a.e. point the
estimate $\varlimsup_{r\to
0}\frac{\mathfrak{m}_{x,y}^{L}(B_{2r}(z))}{\mathfrak{m}_{x,y}^{L}(B_{r}(z))}\leq
C_{D}$ holds, proving that $({\rm X},{\sf d},\mathfrak{m}_{x,y}^{L})$ is
asymptotically doubling. Then it is a Vitali space by [HKST15, Theorem 3.4.3].
∎

The next continuity result for the Riesz potential will be used intensively
along the paper.

######  Lemma 2.4.

If $({\rm X},{\sf d},\mathfrak{m})$ is a geodesic, doubling metric measure
space then the map

$\mathcal{D}(R)\to[0,+\infty],\quad(x,y,z)\mapsto R_{x,y}(z)$

is continuous, where $\mathcal{D}(R):=\\{(x,y,z)\in{\rm X}:z\neq x\neq y\neq
z\\}$.

###### Proof.

Let us take sequences $x_{j}$, $y_{j}$ and $z_{j}$ converging respectively to
$x,y,z$ with the assumption that $z_{j}\neq x_{j}\neq y_{j}\neq z_{j}$ and
$z\neq x\neq y\neq z$. In particular $\min\\{{\sf d}(x,z),{\sf d}(y,z)\\}>0$.
Therefore we can apply [HKST15, Proposition 11.5.3] to get that
$\mathfrak{m}(B_{{\sf d}(x_{j},z_{j})}(x_{j}))$ converges to
$\mathfrak{m}(B_{{\sf d}(x,z)}(x))$, and similarly for $\mathfrak{m}(B_{{\sf
d}(y_{j},z_{j})}(y_{j}))$. This shows that $R_{x_{j},y_{j}}(z_{j})$ converges
to $R_{x,y}(z)$. ∎

##  3\. Several energies to measure boundaries

In this section we introduce several ways to measure the energy of boundaries
of separating sets appearing in Theorem 1.2, where the separating sets have
been defined in Definition 1.1. The first one is the perimeter, which is by
now a classical object of investigation in metric measure geometry (see
[Amb02], [ADM14], [Lah20] for basics and fine properties and for instance
[KLLS19], [CKR23], [BNP23] for related recent problems in analysis on metric
spaces).
Let $({\rm X},{\sf d},\mathfrak{m})$ be a metric measure space. The total
variation of $u\in L^{1}_{\rm loc}(X)$ on an open set $A\subset X$ is

$|Du|(A):=\inf\left\\{\liminf_{i\to\infty}\int_{A}{\rm
lip}u_{i}\,{\mathrm{d}}\mathfrak{m}:\,u_{i}\in{\rm Lip}_{\rm
loc}(A),\,u_{i}\to u\mbox{ in }L^{1}_{\rm loc}(A)\right\\}.$

For an arbitrary set $B\subset X$, we define

$|Du|(B):=\inf\\{|Du|(A):\,B\subset A,\,A\subset X\text{ open}\\}.$

The set function $|Du|\colon\mathscr{B}({\rm X})\to[0,\infty]$ is a Borel
measure on ${\rm X}$ by [Mir03, Theorem 3.4] (see also [ADM14, Lemma 5.2]). We
say that a Borel set $E\subseteq{\rm X}$ has finite perimeter if
$|D\chi_{E}|({\rm X})<\infty$. In this case, we set ${\rm
Per}(E,\cdot):=|D\chi_{E}|$.

It was proved in [ADM14, Theorem 1.1] that

(6) ${\rm
Per}(E,A)=\inf\left\\{\varliminf_{i\to\infty}\int_{A}g_{i}\,d\mathfrak{m}:\,u_{i}\in
L^{1}(A),\,g_{i}\in{\rm UG}(u_{i}),\,u_{i}\to\chi_{E}\mbox{ in
}L^{1}(A)\right\\}$

for all $A\subseteq{\rm X}$ open and bounded. Here the boundedness assumption
is enough for having that $L^{1}_{\rm loc}(A)$-convergence coincides with
$L^{1}(A)$-convergence, the latter being the one considered in the right hand
side term above and in the equivalences in [ADM14]. Whenever we are interested
in stressing the measure on the metric measure space $({\rm X},{\sf
d},\mathfrak{m})$, we will write ${\rm Per}_{\mathfrak{m}}(E,\cdot)$. The
perimeter measure satisfies a coarea inequality.

######  Lemma 3.1.

Let $({\rm X},{\sf d},\mathfrak{m})$ be a metric measure space. If $u\in
L^{1}_{\rm loc}({\rm X})\cap{\rm Lip}({\rm X})$ then

(7) $\int_{-\infty}^{+\infty}\int g\,{\mathrm{d}}{\rm
Per}(\\{u>t\\},\cdot)\,{\mathrm{d}}t\leq\int_{\rm X}g\,{\rm
lip}u\,{\mathrm{d}}\mathfrak{m}$

for every Borel $g\geq 0$.

###### Proof.

The proof follows by the coarea formula for BV functions (see [Mir03]) and
that, since $u\in L^{1}_{{\rm loc}}({\rm X})\cap{\rm Lip}({\rm X})$, ${\rm
lip}u\in{\rm UG}(u)$ and $|Du|\leq{\rm lip}u\,\mathfrak{m}$. ∎

In particular, applying the coarea inequality to the Lipschitz function ${\sf
d}(x,\cdot)$, we get the following well known fact.

######  Lemma 3.2.

Let $({\rm X},{\sf d},\mathfrak{m})$ be a metric measure space and let
$x\in{\rm X}$. Then the ball $B_{r}(x)$ has finite perimeter for almost every
$r>0$.

We end this section with a list of useful well known properties of the
perimeter.

######  Remark 3.3.

Let $({\rm X},{\sf d},\mathfrak{m})$ be a metric measure space. Let
$E,F\subseteq{\rm X}$ be Borel sets. Let $U\subseteq{\rm X}$ be open and let
$B\subseteq{\rm X}$ be Borel. Then:

  * (i)

_Locality_. If $\mathfrak{m}((E\Delta F)\cap U)=0$, then

(8) ${\rm Per}(E,U)={\rm Per}(F,U);$

  * (ii)

_Subadditivity_. It holds ${\rm Per}(E\cup F,B)+{\rm Per}(E\cap F,B)\leq{\rm
Per}(E,B)+{\rm Per}(F,B)$;

  * (iii)

_Complementation_. It holds ${\rm Per}(E,B)={\rm Per}({\rm X}\setminus E,B)$;

  * (iv)

_Support_. It holds ${\rm spt}({\rm Per}(E,\cdot))\subseteq\partial E$.

Let us comment the proof of (iv). The sequence $u_{j}=u$, where $u=1$ on ${\rm
int}(E)$ and $u=0$ on ${\rm ext}(E)$ is admissible for the computation of
${\rm Per}(E,{\rm int}(E)\cup{\rm ext}(E))$, thus giving ${\rm Per}(E,{\rm
int}(E)\cup{\rm ext}(E))=0$.

The next one is the codimension-$p$ Hausdorff measure. For every $p\geq 0$, we
define the gauge function

(9) $h_{p}(B_{r}(x)):=\frac{\mathfrak{m}(B_{r}(x))}{r^{p}}$

for every $x\in{\rm X}$ and $r>0$. We denote by $\mathcal{H}^{{\rm
cod-}{p}}_{\delta}$ the pre-measure with parameter $\delta>0$ defined as

(10) $\mathcal{H}^{{\rm
cod-}{p}}_{\delta}(E):=\inf\left\\{\sum_{j=1}^{\infty}\frac{\mathfrak{m}(B_{r_{j}}(x_{j}))}{r_{j}^{p}}:\,E\subset\bigcup_{j=1}^{\infty}B_{r_{j}}(x_{j}),\,\sup_{j}r_{j}<\delta\right\\}.$

######  Definition 3.4.

The codimension-$p$ Hausdorff measure $\mathcal{H}^{{\rm cod-}{p}}$ is the
Borel regular outer measure defined as

(11) $\mathcal{H}^{{\rm cod-}{p}}(E):=\sup_{\delta>0}\mathcal{H}^{{\rm
cod-}{p}}_{\delta}(E)\,\text{for every }E\subseteq{\rm X}.$

When we want to emphasize the measure with respect to with we are computing
the codimension $p$-Hausdorff measure we write $\mathcal{H}^{{\rm
cod-}{p}}_{\mathfrak{m}}$.

######  Remark 3.5.

The following three properties follow easily from the definition.

  * (i)

If $({\rm X},{\sf d},\mathfrak{m})$ is $s$-Ahlfors regular then we have
$C_{\rm AR}^{-1}\mathcal{H}^{s-p}\leq\mathcal{H}^{{\rm cod-}{p}}\leq C_{\rm
AR}\mathcal{H}^{s-p}$, where $\mathcal{H}^{s-p}$ is the $(s-p)$-dimensional
spherical Hausdorff measure.

  * (ii)

If $({\rm X},{\sf d},\mathfrak{m})$ is a doubling metric measure space then
$\mathcal{H}^{{\rm cod-}{0}}=\mathfrak{m}$.

  * (iii)

The codimension-$p$ Hausdorff measures behave like the classical Hausdorff
measures in the following sense: if $\mathcal{H}^{{\rm cod-}{p}}(E)>0$ then
$\mathcal{H}^{{\rm cod-}{q}}(E)=+\infty$ for all $q>p$, while if
$\mathcal{H}^{{\rm cod-}{p}}(E)<+\infty$ then $\mathcal{H}^{{\rm
cod-}{q}}(E)=0$ for all $q<p$.

We introduce the following proposition, whose proof is a modification of
standard arguments for the Hausdorff measure.

######  Proposition 3.6.

Let $({\rm X},{\sf d},\mathfrak{m})$ be a metric measure space and let $p>0$.
Let $A\subseteq{\rm X}$ be Borel. If $\mathcal{H}^{{\rm
cod-}{p}}_{\mathfrak{m}}(A)<\infty$, then $\mathfrak{m}(A)=0$.

###### Proof.

Assume $\mathfrak{m}(A)>0$. For every covering
$A\subseteq\bigcup_{i=1}^{\infty}B_{r_{i}}(x_{i})$ with $r_{i}<\delta$ we have
$\sum_{i}\frac{\mathfrak{m}(B_{r_{i}}(x_{i}))}{r_{i}^{p}}\geq\frac{1}{\delta^{p}}\,\sum_{i}\mathfrak{m}(B_{r_{i}}(x_{i}))\geq\frac{1}{\delta^{p}}\,\mathfrak{m}(A)$.
This implies that $\mathcal{H}^{{\rm
cod-}{p}}_{\mathfrak{m},\delta}(A)\geq\frac{1}{\delta^{p}}\mathfrak{m}(A)$.
Thus, by considering the limit as $\delta\to 0$, we get $\mathcal{H}^{{\rm
cod-}{p}}_{\mathfrak{m}}(A)=\infty$. ∎

When $p=1$ we have a coarea inequality for the codimension-$1$ Hausdorff
measure. It was studied in [ADMG17, Proposition 5.1], where they state it for
_doubling_ metric measure space. Moreover in [ADMG17] the result is stated for
$g=\chi_{B}$, $B\subseteq{\rm X}$ Borel. It can be extended to the following
version by the classical argument of approximating a nonnegative Borel
function via an increasing sequence of simple functions. We point out that the
result was stated with the addition assumption that
$\mathfrak{m}(\\{f>0\\})<\infty$, but this hypothesis can be removed via a
cut-off argument. Moreover, we also introduce a coarea formula which is suited
for $\mathfrak{m}_{x,y}^{L}$, for which we provide the proof adapting the
arguments in [ADMG17, Proposition 5.1].

######  Proposition 3.7 (Coarea inequality for codimension-$1$ Hausdorff
measure).

Assume that $({\rm X},{\sf d},\mathfrak{m})$ is a doubling metric measure
space. Let $0\leq u\in{\rm Lip}({\rm X})$. Then for every $g\colon{\rm
X}\to[0,+\infty]$ Borel one has

(12) $\int_{0}^{\infty}\int_{\\{u=t\\}}g\,{\mathrm{d}}\mathcal{H}^{{\rm
cod-}{1}}_{\mathfrak{m}}\,{\mathrm{d}}t\leq 4\int_{\rm X}g\,{\rm
lip}u\,{\mathrm{d}}\mathfrak{m}.$

Moreover if $x,y\in{\rm X}$ and $L\geq 1$, we have

(13) $\int_{0}^{\infty}\int_{\\{u=t\\}}g\,{\mathrm{d}}\mathcal{H}^{{\rm
cod-}{1}}_{\mathfrak{m}_{x,y}^{L}}\,{\mathrm{d}}t\leq 4\int_{\rm X}g\,{\rm
lip}u\,{\mathrm{d}}\mathfrak{m}_{x,y}^{L}.$

###### Proof.

Inequality (12) is exactly [ADMG17, Proposition 5.1], where they use the gauge
function $h_{1}/2$ in order to define $\mathcal{H}^{{\rm cod-}{1}}$. That is
why we have an additional $2$ factor. We focus on (13).

Step 1. We claim that, given a Borel set $B\subseteq{\rm X}$,
$\mathfrak{m}_{x,y}^{L}(B)=0$ implies $\mathcal{H}^{{\rm
cod-}{1}}_{\mathfrak{m}_{x,y}^{L}}(B\cap\\{u=t\\})=0$ for a.e. $t$. Observe
that it is sufficient to consider the case $x,y\notin B$. Indeed for a generic
$B$ we set $B^{\prime}=B\setminus\\{x,y\\}$. Then
$\mathfrak{m}_{x,y}^{L}(B)=\mathfrak{m}_{x,y}^{L}(B^{\prime})$ by our
definition of $\mathfrak{m}_{x,y}^{L}$. Moreover $\\{x,y\\}\cap\\{u=t\\}$ is
not empty for at most two values of $t$. So, if $x,y\notin B$ and
$\mathfrak{m}_{x,y}^{L}(B)=0$ then $\mathfrak{m}(B\cap B_{x,y}^{L})=0$, thus
by (12) with $g=\chi_{B}$ we get that $\mathcal{H}^{{\rm
cod-}{1}}_{\mathfrak{m}}(B\cap B_{x,y}^{L}\cap\\{u=t\\})=0$ for a.e. $t$. For
every such $t\neq u(x),u(y)$ we have ${\sf d}(x,\\{u=t\\})>0$ and ${\sf
d}(y,\\{u=t\\})>0$ since $u$ is Lipschitz. Thus, $R_{x,y}^{L}$ is both bounded
from above and below on $\\{u=t\\}\cap B_{x,y}^{L}$, which gives, by the very
definition of codimension Hausdorff measure, $\mathcal{H}^{{\rm
cod-}{1}}_{\mathfrak{m}_{x,y}^{L}}(B\cap\\{u=t\\})=0$ for a.e. $t$.

Step 2. Let $\delta>0$, let $U\subseteq{\rm X}$ be open and bounded,
$S\subseteq U$ be countable dense and consider the fine covering
$\mathcal{F}:=\\{B_{r}(x),r\in\mathbb{Q},\,x\in S,\,r<\min({\sf
d}(x,U^{c}),\delta)\\}$. Since $\mathfrak{m}_{x,y}^{L}$ is a Vitali space
(Proposition 2.3) we find a disjoint subfamily $\\{B_{r_{i}}(x_{i})\\}$ such
that $\mathfrak{m}_{x,y}^{L}(U\setminus\bigcup_{i}B_{r_{i}}(x_{i}))=0$. Call
$\tilde{B}:=\bigcup_{i}B_{r_{i}}(x_{i})$. Then

(14) $\mathcal{H}^{{\rm
cod-}{1}}_{\mathfrak{m}_{x,y}^{L},\delta}(U\cap\\{u=t\\})\leq\mathcal{H}^{{\rm
cod-}{1}}_{\mathfrak{m}_{x,y}^{L},\delta}(\tilde{B}\cap\\{u=t\\})+\mathcal{H}^{{\rm
cod-}{1}}_{\mathfrak{m}_{x,y}^{L},\delta}((U\setminus\tilde{B})\cap\\{u=t\\})\leq\mathcal{H}^{{\rm
cod-}{1}}_{\mathfrak{m}_{x,y}^{L},\delta}(\tilde{B}\cap\\{u=t\\})$

for a.e. $t$, by using Step 1. Thus $\mathcal{H}^{{\rm
cod-}{1}}_{\mathfrak{m}_{x,y}^{L},\delta}(U\cap\\{u=t\\})=\mathcal{H}^{{\rm
cod-}{1}}_{\mathfrak{m}_{x,y}^{L},\delta}(\tilde{B}\cap\\{u=t\\})$. Define
$t_{i}^{-}:=\inf_{B_{r_{i}}(x_{i})}u$ and
$t_{i}^{+}:=\sup_{B_{r_{i}}(x_{i})}u$. If $B_{r_{i}}(x_{i})\cap\\{u=t\\}\neq
0$, then $t\in[t_{i}^{-},t_{i}^{+}]$; thus,

$\mathcal{H}^{{\rm
cod-}{1}}_{\mathfrak{m}_{x,y}^{L},\delta}(\tilde{B}\cap\\{u=t\\})\leq\sum_{i,\,\text{s.t.
}t\in[t_{i}^{-},t_{i}^{+}]}\frac{\mathfrak{m}_{x,y}^{L}(B_{r_{i}}(x_{i}))}{r_{i}}.$

Integrating betweeen $0$ and $\infty$ we obtain

(15) $\int_{0}^{\infty}\mathcal{H}^{{\rm
cod-}{1}}_{\mathfrak{m}_{x,y}^{L},\delta}(\tilde{B}\cap\\{u=t\\})\leq\sum_{i}\frac{\mathfrak{m}_{x,y}^{L}(B_{r_{i}}(x_{i}))}{r_{i}}(t_{i}^{+}-t_{i}^{-})\leq
4\int_{\tilde{B}}{\rm sl}(x,\delta)\,{\mathrm{d}}\mathfrak{m}_{x,y}^{L},$

where in the last inequality we used for $x\in B_{r_{i}}(x_{i})$

$\frac{t_{i}^{+}-t_{i}^{-}}{r_{i}}=\sup_{y,y^{\prime}\in
B_{r_{i}}(x_{i})}\frac{f(y)-f(y^{\prime})}{r_{i}}\leq 4\sup_{y\in
B_{r_{i}}(x_{i})}\frac{|f(y)-f(x)|}{{\sf d}(x,y)}\leq 4{\rm sl}(x,\delta).$

Therefore

$\int_{0}^{\infty}\mathcal{H}^{{\rm
cod-}{1}}_{\mathfrak{m}_{x,y}^{L},\delta}(U\cap\\{u=t\\})\leq 4\int_{U}{\rm
sl}(x,\delta)\,{\mathrm{d}}\mathfrak{m}_{x,y}^{L}$

and taking the limit as $\delta\to 0$, we obtain
$\int_{0}^{\infty}\mathcal{H}^{{\rm
cod-}{1}}_{\mathfrak{m}_{x,y}^{L}}(U\cap\\{u=t\\})\leq\int_{U}{\rm
lip}u\,{\mathrm{d}}\mathfrak{m}$. By monotone class theorem, we extend the
previous inequality to all Borel sets and monotone approximation to all Borel
nonnnegative functions $g$, thus proving (13). ∎

######  Remark 3.8.

It is not clear to us if a coarea-formula holds under the only assumption that
$({\rm X},{\sf d},\mathfrak{m})$ is a Vitali space or even with the stronger
assumption that it is asymptotically doubling. For this reason we adapted the
proof of [ADMG17] for specific case of $\mathfrak{m}_{x,y}^{L}$.

We remark that some version of the coarea inequality for $p>1$ cannot hold
because the $\mathcal{H}^{{\rm cod-}{p}}$ measure of the level sets of a
Lipschitz function is generally $+\infty$.

A third way to measure the energy of a set is the Minkowski content.

######  Definition 3.9.

Let $({\rm X},{\sf d},\mathfrak{m})$ be a metric measure space and let
$A\subseteq{\rm X}$ be Borel. The codimension-$p$ Minkowski content of
$A\subset{\rm X}$ is

$\mathfrak{m}^{+,p}(A):=\varliminf_{r\to
0}\frac{\mathfrak{m}\left(\overline{B}_{r}(A)\setminus A\right)}{r^{p}}.$

The $1$-Minkowski content satisfies a coarea inequality. We write both the
versions of [KL14, Prop. 3.5] and of [ADMG17, Lemma 3.2].

######  Proposition 3.10 (Coarea inequality for Minkowski content).

Let $({\rm X},{\sf d},\mathfrak{m})$ be a metric measure space and suppose
$\mathfrak{m}$ is finite. Then, for every bounded $u\in{\rm Lip}({\rm X})$, we
have

(16) $\int_{-\infty}^{\infty}\mathfrak{m}^{+,1}(\\{u\geq
t\\})\,{\mathrm{d}}t\leq\int_{\rm X}{\rm{\rm lip}}u\,{\mathrm{d}}\mathfrak{m}$

and

(17)
$\int_{-\infty}^{\infty}\mathfrak{m}^{+,1}(\partial\\{u>t\\})\,{\mathrm{d}}t\leq
2\int_{\rm X}{\rm{\rm lip}}u\,{\mathrm{d}}\mathfrak{m}.$

###### Proof.

The first inequality (16) is exactly [ADMG17, Lemma 3.2]. For the second one
we remark that the version of the Minkowski content used in [KL14] is the
following:

$\mathfrak{m}_{\text{KL}}^{+}(A)=\varliminf_{r\to
0}\frac{\mathfrak{m}(B_{r}(A))}{2r}.$

It is clear that $\mathfrak{m}^{+,1}(A)\leq 2\mathfrak{m}_{\text{KL}}^{+}(A)$,
so (17) follows directly from [KL14, Prop. 3.5]. ∎

The next energy we introduce is the variational $p$-capacity.

######  Definition 3.11.

Let $({\rm X},{\sf d},\mathfrak{m})$ be a metric measure space and let
$A\subseteq U$ be two subsets of ${\rm X}$ such that $U$ is open. Let $p\geq
1$. The variational $p$-capacity of the couple $(A,U)$ (see for instance
[BB11, §6.3]) is

(18) $\textup{cap}_{\mathfrak{m}}^{p}(A,U):=\inf\left\\{\left(\int
g^{p}\,{\mathrm{d}}\mathfrak{m}\right)^{\frac{1}{p}},\,g\in\text{UG}(u),\text{
spt}(u)\subset U,u\geq 1\text{ on }A\right\\}.$

The $p$-variational capacity of a subset $A\subseteq{\rm X}$ with respect to
$\mathfrak{m}$ is

$\textup{cap}^{p}_{\mathfrak{m}}(A)=\sup_{A\subset U,U\text{
open}}\textup{cap}^{p}_{\mathfrak{m}}(A,U).$

There are many possible variants of the variational capacity of a set. We
decided for the goals of this manuscript to stick with the one above. By a
classical truncation argument it is enough to take functions $u$ such that
$\chi_{A}\leq u\leq 1$ in the right hand side of (18), see [BB11, §6.3,
p.161].

######  Remark 3.12.

If $A\subseteq{\rm X}$ is compact then every open set $U$ containing $A$
contains also $B_{r}(A)$ for some small enough $r>0$. From the definition of
variational capacity it follows that if $A\subseteq U_{1}\subseteq U_{2}$,
with $U_{1},U_{2}$ open, then
$\textup{cap}^{p}_{\mathfrak{m}}(A,U_{1})\geq\textup{cap}^{p}_{\mathfrak{m}}(A,U_{2})$.
Therefore, if $A\subseteq{\rm X}$ is compact, then

$\textup{cap}^{p}_{\mathfrak{m}}(A)=\sup_{r>0}\textup{cap}^{p}_{\mathfrak{m}}(A,B_{r}(A)).$

The variational capacity of a compact set is always bounded from above by the
lower Minkowski content, while for $p=1$ it is bounded from below by the
perimeter.

######  Lemma 3.13.

Let $({\rm X},{\sf d},\mathfrak{m})$ be a metric measure space and let
$A\subseteq{\rm X}$ be compact. Then

$\textup{cap}^{p}_{\mathfrak{m}}(A)\leq\mathfrak{m}^{+,p}(A)$

for all $p\geq 1$. Moreover

${\rm Per}_{\mathfrak{m}}(A,{\rm X})\leq\textup{cap}^{1}_{\mathfrak{m}}(A).$

###### Proof.

Let us start with the first inequality. By Remark 3.12, it is enough to
estimate from above $\textup{cap}^{p}_{\mathfrak{m}}(A,B_{r}(A))$. We fix
$\tau<1$ and we consider the function $u_{\tau}=\max(1-{\sf d}_{A}(\cdot)/\tau
r,0)$. It is straightforward to check that ${\rm lip}u_{\tau}\leq 1/\tau r$ on
$\overline{B_{r}(A)}\setminus A$ and zero otherwise. Moreover $u_{\tau}$ has
compact support in $B_{r}(A)$. Thus

$\textup{cap}^{p}_{\mathfrak{m}}(A,B_{r}(A))\leq\int({\rm
lip}u_{\tau})^{p}\,{\mathrm{d}}\mathfrak{m}\leq\frac{\mathfrak{m}(\overline{B_{r}(A)}\setminus
A)}{\tau^{p}r^{p}}.$

Taking the infimum over $\tau<1$, we get

$\textup{cap}^{p}_{\mathfrak{m}}(A,B_{r}(A))\leq\frac{\mathfrak{m}(\overline{B_{r}(A)}\setminus
A)}{r^{p}}$

for every $r>0$. Taking the liminf as $r\to 0$, we conclude the proof of the
first part of the statement.
We now assume that $p=1$. We have
$\textup{cap}^{1}_{\mathfrak{m}}(A)=\sup_{r>0}\textup{cap}^{1}_{\mathfrak{m}}(A,B_{r}(A))$.
For every $r>0$ let $g_{r}\in\text{UG}(u_{r})$, with $u_{r}\colon{\rm
X}\to\mathbb{R}$ such that spt$(u_{r})\subset B_{r}(A)$ and $u_{r}=1$ on $A$,
be such that $\int
g_{r}\,{\mathrm{d}}\mathfrak{m}\leq\textup{cap}^{1}_{\mathfrak{m}}(A,B_{r}(A))+r$.
We can choose $u_{r}=1$ on $A$ as we recalled after the definition of the
variational capacity. We observe that $u_{r}$ converges pointwise to
$\chi_{A}$, because $A$ is closed. So $u_{r}\to\chi_{A}$ in $L^{1}({\rm X})$
by dominate convergence. Using Remark 3.3.(iv) we get that ${\rm
Per}_{\mathfrak{m}}(A,{\rm X})={\rm Per}_{\mathfrak{m}}(A,B_{s}(A))$ for every
$s>0$. Thus, since $B_{s}(A)$ is open and bounded for $s>0$, we can use (6)
and have that ${\rm Per}_{\mathfrak{m}}(A,B_{s}(A))\leq\varliminf_{r\to
0}\int_{\rm X}g_{r}\,{\mathrm{d}}\mathfrak{m}$. Thus, we have

${\rm Per}_{\mathfrak{m}}(A,{\rm X})={\rm
Per}_{\mathfrak{m}}(A,B_{s}(A))\leq\varliminf_{r\to 0}\int_{\rm
X}g_{r}\,{\mathrm{d}}\mathfrak{m}\leq\varliminf_{r\to
0}\textup{cap}^{1}_{\mathfrak{m}}(A,B_{r}(A))+r=\textup{cap}^{1}_{\mathfrak{m}}(A),$

concluding the proof. ∎

The last energies we consider are defined through the approximate modulus.

######  Definition 3.14.

Let $({\rm X},{\sf d},\mathfrak{m})$ be a metric measure space and let
$A\subseteq{\rm X}$. We consider the family $\Gamma(A):=\\{\gamma\in
C([0,1],{\rm X}):\,\gamma\cap A\neq\emptyset\\}.$ Let $p\geq 1$. The
approximate $p$-modulus of $A$ is

${\rm AM}_{p}(A,\mathfrak{m}):={\rm AM}_{p}(\Gamma(A),\mathfrak{m}).$

We also introduce the family $\Gamma(A)^{\pitchfork}:=\\{\gamma\in
C([0,1],{\rm X}):\,\gamma_{0}\in I_{A},\gamma_{1}\in O_{A}\\}.$ The
transversal approximate $p$-modulus of $A$ is

${\rm AM}^{\pitchfork}_{p}(A,\mathfrak{m}):={\rm
AM}_{p}(\Gamma(A)^{\pitchfork},\mathfrak{m}).$

The approximate moduli above can be computed with arbitrarily short curves.

######  Lemma 3.15.

Let $({\rm X},{\sf d},\mathfrak{m})$ be a metric measure space, let
$A\subseteq{\rm X}$ and let $\delta>0$. Define the following families of
curves:

$\Gamma(A)_{\delta}:=\\{\gamma\in\Gamma(A):\ell(\gamma)\leq\delta\\}\subseteq\Gamma(A);$
$\Gamma(A)^{\pitchfork}_{\delta}=\\{\gamma\in\Gamma(A)^{\pitchfork}:\ell(\gamma)\leq\delta\\}\subseteq\Gamma(A)^{\pitchfork}.$

Then

${\rm AM}_{p}(A,\mathfrak{m})={\rm
AM}_{p}(\Gamma(A)_{\delta},\mathfrak{m}),\quad{\rm
AM}_{p}^{\pitchfork}(A,\mathfrak{m})={\rm
AM}_{p}(\Gamma(A)^{\pitchfork}_{\delta},\mathfrak{m}).$

###### Proof.

The inclusions $\Gamma(A)_{\delta}\subseteq\Gamma(A)$ and
$\Gamma(A)^{\pitchfork}_{\delta}\subseteq\Gamma(A)^{\pitchfork}$ give ${\rm
AM}_{p}(\Gamma(A)_{\delta},\mathfrak{m})\leq{\rm AM}_{p}(A,\mathfrak{m})$ and
${\rm AM}_{p}(\Gamma(A)^{\pitchfork}_{\delta},\mathfrak{m})\leq{\rm
AM}_{p}^{\pitchfork}(A,\mathfrak{m})$. On the other hand for every
$\gamma\in\Gamma(A)$ we can find a subcurve
$\gamma_{\delta}\in\Gamma(A)_{\delta}$. Therefore an admissible sequence for
$\Gamma(A)_{\delta}$ is admissible also for $\Gamma(A)$. This implies ${\rm
AM}_{p}(A,\mathfrak{m})\leq{\rm AM}_{p}(\Gamma(A)_{\delta},\mathfrak{m})$ and
similarly ${\rm AM}^{\pitchfork}_{p}(A,\mathfrak{m})\leq{\rm
AM}_{p}(\Gamma(A)^{\pitchfork}_{\delta},\mathfrak{m})$. ∎

The approximate modulus is bounded from above by the $p$-Minkowski content
when the set has measure zero.

######  Lemma 3.16.

Let $({\rm X},{\sf d},\mathfrak{m})$ be a metric measure space and let $p\geq
1$. Let $A\subseteq{\rm X}$ be such that $\mathfrak{m}(A)=0$. Then

${\rm AM}_{p}(A,\mathfrak{m})\leq\mathfrak{m}^{+,p}(A).$

###### Proof.

Let $r_{j}>0$ be a generic decreasing sequence converging to $0$. We consider
the functions $\rho_{j}=\frac{1}{r_{j}}\chi_{B_{r_{j}}(A)}$. This sequence is
admissible for $\Gamma(A)$. Indeed, if $\gamma\in\Gamma(A)$ is not constant,
then its image has diameter at least $2r_{j_{0}}$ for some fixed $j_{0}$. For
every $j\geq j_{0}$ we have $\ell(\gamma\cap B_{r_{j}}(A))\geq r_{j}$, since
$\gamma\cap A\neq\emptyset$. Then
$\varliminf_{j\to+\infty}\int_{\gamma}\rho_{j}\geq 1$. Therefore

${\rm
AM}_{p}(A,\mathfrak{m})\leq\varliminf_{j\to+\infty}\int\rho_{j}^{p}\,{\mathrm{d}}\mathfrak{m}=\varliminf_{j\to+\infty}\frac{\mathfrak{m}(B_{r_{j}}(A))}{r_{j}^{p}}.$

By the arbitrariness of the sequence $r_{j}$ and the fact that
$\mathfrak{m}(A)=0$ we get

${\rm AM}_{p}(A,\mathfrak{m})\leq\mathfrak{m}^{+,p}(A).$

∎

We conclude this section with an easy but useful remark.

######  Remark 3.17.

Let ${\sf d},{\sf d}^{\prime}$ be two equivalent metrics on a set ${\rm X}$,
i.e. there exists a constant $C_{0}\geq 1$ such that

(19) $C_{0}^{-1}{\sf d}\leq{\sf d}^{\prime}\leq C_{0}{\sf d},$

and let $\mathfrak{m}$ be a Borel measure on ${\rm X}$. Then

  * (i)

if $({\rm X},{\sf d},\mathfrak{m})$ is $C_{D}$-doubling then $({\rm X},{\sf
d}^{\prime},\mathfrak{m})$ is $C_{D}^{1+2\log_{2}C_{0}}$-doubling.

  * (ii)

for every set $A\subseteq{\rm X}$ and every $p\geq 1$ it holds

$\displaystyle C_{0}^{-1}\cdot{\rm Per}_{\mathfrak{m},{\sf d}}(A)\leq{\rm
Per}_{\mathfrak{m},{\sf d}^{\prime}}(A)\leq C_{0}\cdot{\rm
Per}_{\mathfrak{m},{\sf d}}(A).$ $\displaystyle
C_{0}^{-p}\cdot\mathcal{H}^{{\rm cod-}{p}}_{\mathfrak{m},{\sf
d}}(A)\leq\mathcal{H}^{{\rm cod-}{p}}_{\mathfrak{m},{\sf d}^{\prime}}(A)\leq
C_{0}^{p}\cdot\mathcal{H}^{{\rm cod-}{p}}_{\mathfrak{m},{\sf d}}(A).$
$\displaystyle C_{0}^{-p}\cdot\mathfrak{m}^{+,p}_{\mathfrak{m},{\sf
d}}(A)\leq\mathfrak{m}^{+,p}_{\mathfrak{m},{\sf d}^{\prime}}(A)\leq
C_{0}^{p}\cdot\mathfrak{m}^{+,p}_{\mathfrak{m},{\sf d}}(A).$ $\displaystyle
C_{0}^{-p}\cdot{\rm cap}^{p}_{\mathfrak{m},{\sf d}}(A)\leq{\rm
cap}^{p}_{\mathfrak{m},{\sf d}^{\prime}}(A)\leq C_{0}^{p}\cdot{\rm
cap}^{p}_{\mathfrak{m},{\sf d}}(A).$ $\displaystyle C_{0}^{-p}\cdot{\rm
AM}_{p,{\sf d}}(A,\mathfrak{m})\leq{\rm AM}_{p,{\sf
d}^{\prime}}(A,\mathfrak{m})\leq C_{0}^{p}\cdot{\rm AM}_{p,{\sf
d}}(A,\mathfrak{m}),$

The notations used here are self explaining.

##  4\. Relations between the energies under Poincaré inequality

There are more connections between the several energies we introduced in the
last section when the metric measure space $({\rm X},{\sf d},\mathfrak{m})$ is
doubling and satisfies a $1$-Poincaré inequality. The first one is the
representation formula for the perimeter, as proven in [Amb02, Theorem 4.2]
and [AMJP04, Theorem 4.6]. The proposition, which is stated in a local
version, is already present in the literature, but we are not able to find a
proof. We include it for completeness.

######  Proposition 4.1.

Let $({\rm X},{\sf d},\mathfrak{m})$ be $1$-PI space. Let $U\subseteq{\rm X}$
be open and $A$ be a Borel set such that ${\rm
Per}_{\mathfrak{m}}(A,U)<\infty$. Then

(20) ${\rm Per}_{\mathfrak{m}}(A,\cdot)=\theta\,\mathcal{H}^{{\rm
cod-}{1}}_{\mathfrak{m}}\lower 3.0pt\hbox{$|_{\partial^{e}A\cap U}$},$

as measures on $U$, where $\theta\colon U\to[c_{1},C_{D}]$ and
$c_{1}=c_{1}(C_{D},C_{P},\lambda)>0$.

###### Proof.

Let us consider $f(x):={\sf d}(x,U^{c})$, that is $1$-Lipschitz. Applying the
coarea inequality (7) we have that $\\{f>t\\}$ has finite perimeter for a.e.
$t\in\mathbb{R}$. We take a sequence $t_{n}\to 0$ for which
$U_{n}:=\\{f>t_{n}\\}$ has finite perimeter. By Remark 3.3.(iii) we have

${\rm Per}_{\mathfrak{m}}(U_{n}\cap A,U)\leq{\rm
Per}_{\mathfrak{m}}(U_{n},U)+{\rm Per}_{\mathfrak{m}}(A,U),$

thus ${\rm Per}_{\mathfrak{m}}(U_{n}\cap A,U)<\infty$. Indeed the measure
${\rm Per}_{\mathfrak{m}}(U_{n},\cdot)$ is supported on $\partial U_{n}\subset
U$ by Remark 3.3.(v), so ${\rm Per}_{\mathfrak{m}}(U_{n},U)={\rm
Per}_{\mathfrak{m}}(U_{n},{\rm X})<\infty$. For the same reason we have ${\rm
Per}_{\mathfrak{m}}(U_{n}\cap A,{\rm X})={\rm Per}_{\mathfrak{m}}(U_{n}\cap
A,U)<\infty$. We are in position to apply [Amb02, Theorem 4.2]. So there exist
a constant $c_{1}=c_{1}(C_{D},C_{P},\lambda)$ and Borel functions
$\theta_{n}\colon{\rm X}\to[c_{1},C_{D}]$ such that ${\rm
Per}_{\mathfrak{m}}(A\cap U_{n},\cdot)=\theta_{n}\mathcal{H}^{{\rm
cod-}{1}}_{\mathfrak{m}}\lower 3.0pt\hbox{$|_{\partial^{e}(A\cap U_{n})}$}$.
We denote $\mathcal{H}^{{\rm cod-}{1}}_{\mathfrak{m}}\lower
3.0pt\hbox{$|_{\partial^{e}(A\cap U_{n})}$}$ by $\mu_{n}$.
Step 1: $\mu_{n}(B)=\mu_{m}(B)$ if $m\geq n$ and $B\subseteq U_{n}$.
This follows from the equalities

(21) $\mu_{n}(B)=\mathcal{H}^{{\rm cod-}{1}}_{\mathfrak{m}}(\partial^{e}A\cap
B)=\mu_{m}(B)$

that we now justify. By [BPR20, Proposition 1.16], where the finite perimeter
assumption is not actually used, we get $\partial^{e}(A\cap
U_{n})\subseteq\partial^{e}A\cup\partial^{e}U_{n}$ and similarly for $m$.
Therefore

$\mu_{n}(B)\leq\mathcal{H}^{{\rm cod-}{1}}_{\mathfrak{m}}(\partial^{e}A\cap
B)+\mathcal{H}^{{\rm cod-}{1}}_{\mathfrak{m}}(\partial^{e}U_{n}\cap
B)=\mathcal{H}^{{\rm cod-}{1}}_{\mathfrak{m}}(\partial^{e}A\cap B),$

because $B\cap\partial U_{n}=\emptyset$, and similarly for $m$. On the other
hand we have

$\partial^{e}A=(\partial^{e}A\cap U_{n})\cup(\partial^{e}A\cap
U_{n}^{c})\subseteq\partial^{e}(A\cap U_{n})\cup(\partial^{e}A\cap
U_{n}^{c}),$

where the last containment follows from the fact that if $x\in U_{n}$ then

$\varlimsup_{r\to 0}\frac{\mathfrak{m}(B_{r}(x)\cap
A)}{\mathfrak{m}(B_{r}(x))}=\varlimsup_{r\to
0}\frac{\mathfrak{m}(B_{r}(x)\cap(A\cap
U_{n}))}{\mathfrak{m}(B_{r}(x))},\,\varlimsup_{r\to
0}\frac{\mathfrak{m}(B_{r}(x)\setminus
A)}{\mathfrak{m}(B_{r}(x))}=\varlimsup_{r\to
0}\frac{\mathfrak{m}(B_{r}(x)\setminus(A\cap U_{n}))}{\mathfrak{m}(B_{r}(x))}$

because $U_{n}$ is open, and similarly for $m$. Thus

$\mathcal{H}^{{\rm cod-}{1}}_{\mathfrak{m}}(\partial^{e}A\cap
B)\leq\mathcal{H}^{{\rm cod-}{1}}_{\mathfrak{m}}(\partial^{e}(A\cap U_{n})\cap
B)+\mathcal{H}^{{\rm cod-}{1}}_{\mathfrak{m}}(\partial^{e}A\cap U_{n}^{c}\cap
B)=\mu_{n}(B)$

and similarly for $m$.
Step 2: for every $n$ there exists $V_{n}\subseteq U_{n}$ such that
$\mu_{m}(V_{n})=0$ and every point of $U_{n}\setminus V_{n}$ is a Lebesgue
point of $\theta_{m}$ with respect to the measure $\mu_{m}$, for every $m>n$.
By [Amb02, Corollary 4.5] ${\rm Per}_{\mathfrak{m}}(A\cap U_{m},\cdot)$ is
asymptotically doubling and therefore $\mu_{m}$ is asymptotically doubling as
well. This implies that the metric measure space $({\rm X},{\sf d},\mu_{m})$
is Vitali (cp. [HKST15, Theorem 3.4.3]). By [HKST15, §3.4] the Lebesgue
differentiation theorem holds on $({\rm X},{\sf d},\mu_{m})$. We apply it to
$U_{n}$: we can find a Borel set $N_{m}\subseteq U_{n}$ with
$\mu_{m}(N_{m})=0$ and such that every point of $U_{n}\setminus N_{m}$ is a
Lebesgue point of $\theta_{m}$ with respect to the measure $\mu_{m}$. We
define $V_{n}:=\bigcup_{m>n}N_{m}$ and we compute, for $m>n$,

$\mu_{m}(V_{n})\leq\sum_{\ell>n}\mu_{m}(N_{\ell})=\sum_{\ell>n}\mu_{\ell}(N_{\ell})=0,$

where for the first equality we applied Step 1.
Step 3: $\theta_{m}(x)=\theta_{\ell}(x)$ for every $x\in U_{n}\setminus V_{n}$
and every $m,\ell>n$.
We apply Step 2 to get

$\displaystyle\theta_{m}(x)=\lim_{r\to
0}\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{B_{r}(x)}\theta_{m}\,{\mathrm{d}}\mu_{m}$
$\displaystyle=\lim_{r\to
0}\frac{1}{\mu_{m}(B_{r}(x))}\int_{B_{r}(x)}\theta_{m}\,{\mathrm{d}}\mu_{m}$
$\displaystyle=\lim_{r\to 0}\frac{1}{\mu_{m}(B_{r}(x))}{\rm
Per}_{\mathfrak{m}}(A\cap U_{m},B_{r}(x))$ $\displaystyle=\lim_{r\to
0}\frac{1}{\mu_{m}(B_{r}(x))}{\rm Per}_{\mathfrak{m}}(A,B_{r}(x))$

where we used Remark 3.3.(i) and the fact that $B_{r}(x)\subseteq U_{m}$ if
$r$ is small enough. The same computation holds for $\ell$. Moreover
$B_{r}(x)$ is also included in $U_{n}$ if $r$ is small enough, so
$\mu_{m}(B_{r}(x))=\mu_{\ell}(B_{r}(x))$ if $r$ is small enough and
$m,\ell>n$, by Step 1. This implies that $\theta_{m}(x)=\theta_{\ell}(x)$.

Let us fix some additional notation. We set $\mu:=\mathcal{H}^{{\rm
cod-}{1}}_{\mathfrak{m}}\lower 3.0pt\hbox{$|_{(\partial^{e}A)\cap U}$}$ and
$V:=\bigcup_{n}V_{n}$. For $x\in U\setminus V$ we set
$\theta(x):=\theta_{m}(x)$, where $m>n$ and $x\in U_{n}$.
Step 4: $\mu(V)=0$ and $\theta$ is well defined on $U\setminus V$, defining a
Borel function on $U$ with values on $[c_{1},C_{D}]$.
We have

$\mu(V)=\mathcal{H}^{{\rm
cod-}{1}}_{\mathfrak{m}}\left(\bigcup_{n}\partial^{e}A\cap
V_{n}\right)\leq\sum_{n}\mathcal{H}^{{\rm
cod-}{1}}_{\mathfrak{m}}(\partial^{e}A\cap V_{n})=\sum_{n}\mu_{n+1}(V_{n})=0,$

where in the second to last equality we used (21). We need to show that
$\theta$ is well defined. We fix $x\in U\setminus V$. By definition $x$
belongs to $U_{n}\setminus V_{n}$ for some $n$. Step 3 says that the quantity
$\theta_{m}(x)$ does not depend on the choice of $m>n$. Moreover suppose that
$x$ belongs to $U_{n}\setminus V_{n}$ and $U_{m}\setminus V_{m}$. Choosing
$\ell>m,n$ we get that the assignment $\theta(x)=\theta_{\ell}(x)$ is well
defined and does not depend on $n$ and $m$, so $\theta$ is well defined on
$U\setminus V$. By definition $c_{1}\leq\theta\leq C_{D}$ because each
$\theta_{n}$ has this property. Finally it is not difficult to prove that
$\theta$ is Borel using that each $\theta_{n}$ is Borel.
Step 5: ${\rm Per}_{\mathfrak{m}}(A\cap U,B)=\theta\,\mathcal{H}^{{\rm
cod-}{1}}_{\mathfrak{m}}\lower 3.0pt\hbox{$|_{\partial^{e}A\cap U}$}(B)$ for
every $B\subseteq U$.
First of all we prove the statement for an open set $B$ such that ${\sf
d}(B,U^{c})>0$, so that $B\subseteq U_{n}$ if $n$ is big enough. Using that
$\theta(x)=\theta_{n}(x)$ for $\mu$ and $\mu_{n}$ almost every $x\in B$ and
that $\mu=\mu_{n}$ on $B$ by (21) we get

$(\theta\mu)(B)=\int_{B}\theta\,{\mathrm{d}}\mu=\int_{B}\theta_{n}\,{\mathrm{d}}\mu_{n}={\rm
Per}_{\mathfrak{m}}(A\cap U_{n},B)={\rm Per}_{\mathfrak{m}}(A,B).$

The last equality follows by Remark 3.3.(i). For a generic open $B\subseteq U$
we define $B_{n}=B\cap U_{n}$ and we observe that $B$ is the increasing union
of the sets $B_{n}$. Since both $\mu$ and ${\rm Per}_{\mathfrak{m}}(A,\cdot)$
are Borel measures we get

$(\theta\mu)(B)=\lim_{n\to+\infty}(\theta\mu)(B_{n})=\lim_{n\to+\infty}{\rm
Per}_{\mathfrak{m}}(A,B_{n})={\rm Per}_{\mathfrak{m}}(A,B).$

An application of monotone class theorem gives that $(\theta\mu)(B)={\rm
Per}_{\mathfrak{m}}(A,B)$ for every Borel set $B\subseteq U$. ∎

Given a set, it is possible to check the fact that it has finite perimeter by
checking the finiteness of the codimension-$1$ Hausdorff measure of the
essential boundary: this is the content of [Lah20, Theorem 1.1]. We extend the
characterization by using also the approximate modulus. We specialize also to
the case of separating sets.

######  Proposition 4.2.

Let $({\rm X},{\sf d},\mathfrak{m})$ be a $1$-PI space. Let $U\subseteq{\rm
X}$ be an open set and $A\subseteq{\rm X}$ be a Borel set. Then the following
are equivalent:

  * (i)

${\rm Per}_{\mathfrak{m}}(A,U)<\infty$;

  * (ii)

$\mathcal{H}^{{\rm cod-}{1}}_{\mathfrak{m}}(\partial^{e}A\cap U)<\infty$;

  * (iii)

${\rm AM}_{1}(\partial^{e}A\cap U,\mathfrak{m})<\infty$.

If $x,y\in{\rm X}$, $L\geq 1$, $\min\\{{\sf d}(x,\partial^{e}A),{\sf
d}(y,\partial^{e}A)\\}>0$, $U\subseteq\overline{B}_{x,y}^{L}$ then (i)-(iii)
are equivalent to

  * (iv)

$\mathcal{H}^{{\rm cod-}{1}}_{\mathfrak{m}_{x,y}^{L}}(\partial^{e}A\cap
U)<\infty$;

  * (v)

${\rm AM}_{1}(\partial^{e}A\cap U,\mathfrak{m}_{x,y}^{L})<\infty$.

###### Proof.

The implication from (i) to (ii) follows directly from Proposition 4.1 while
(ii) implies (i) is [Lah20, Theorem 1.1]. The equivalence between (ii) and
(iii) is a direct consequence of [HEMM22, Theorem 4.4], where it is shown that
$\mathcal{H}^{{\rm cod-}{1}}_{\mathfrak{m}}(\cdot)$ and ${\rm
AM}_{1}(\cdot,\mathfrak{m})$ are comparable for Borel sets if $({\rm X},{\sf
d},\mathfrak{m})$ is a $1$-PI space.
We now move to the second part. If $A$ is as in the assumptions then there
exists $M>0$ such that $1/M\leq R_{x,y}^{L}\leq M$ on a small neighbourhood of
$\partial^{e}A$. This implies by a direct estimate that $\mathcal{H}^{{\rm
cod-}{1}}_{\mathfrak{m}}(\partial^{e}A\cap U)<\infty$ if and only if
$\mathcal{H}^{{\rm cod-}{1}}_{\mathfrak{m}_{x,y}^{L}}(\partial^{e}A\cap
U)<\infty$. Moreover ${\rm AM}_{1}(\partial^{e}A\cap U,\mathfrak{m})<\infty$
if and only if ${\rm AM}_{1}(\partial^{e}A\cap
U,\mathfrak{m}_{x,y}^{L})<\infty$ because, thanks to Lemma 3.15, both the
approximate moduli can be computed with curves belonging to the above
mentioned tubular neighbourhood of $\partial^{e}A$. ∎

The second relation we explore is between the approximate modulus and the
codimension Hausdorff measure computed with respect to
$\mathfrak{m}_{x,y}^{L}$. First we state it in case of length spaces, later we
will see how to get the statement under the Poincaré inequality assumption.

######  Proposition 4.3.

Let $({\rm X},{\sf d},\mathfrak{m})$ be a doubling metric measure space such
that $({\rm X},{\sf d})$ is geodesic. Then for every $x,y\in{\rm X}$ and every
Borel set $A\subseteq{\rm X}$ with $\min\\{{\sf d}(x,A),{\sf d}(y,A)\\}>0$ it
holds

${\rm AM}_{p}(A,\mathfrak{m}_{x,y}^{L})\leq C_{D}\cdot\mathcal{H}^{{\rm
cod-}{p}}_{\mathfrak{m}_{x,y}^{2L}}(A).$

The same statement is proved in [HEMM22, Theorem 2.1] for the same quantities
_computed with respect to_ $\mathfrak{m}$. We adapt their proof to the measure
$\mathfrak{m}_{x,y}^{L}$ by localizing the argument. In order to get a nice
local estimate we use the assumption that $({\rm X},{\sf d})$ is a length
space to have the continuity of the Riesz potential $R_{x,y}$.

###### Proof.

We fix $A\subseteq{\rm X}$ as in the assumptions. Let us consider $r\ll 1$ so
that $\overline{B}_{r}(A)\subset{\rm X}\setminus\\{x,y\\}$. We show that

(22) ${\rm AM}_{p}(A,\mathfrak{m}_{x,y}^{L})={\rm
AM}_{p}(A\cap\overline{B}_{x,y}^{L},\mathfrak{m}_{x,y}^{L})\leq{\rm
AM}_{p}(A\cap\overline{B}_{x,y}^{L},\mathfrak{m}_{x,y}^{2L})\leq
C_{D}\cdot\mathcal{H}^{{\rm
cod-}{p}}_{\mathfrak{m}_{x,y}^{2L}}(A\cap\overline{B}_{x,y}^{L}),$

which is enough for the thesis. The first equality is true because the
approximate modulus is an outer measure and ${\rm
AM}_{p}(A\cap(\overline{B}_{x,y}^{L})^{c},\mathfrak{m}_{x,y}^{L})=0$. This is
true since $(\overline{B}_{x,y}^{L})^{c}$ is open and we can argue as in Lemma
3.15 to see that ${\rm
AM}_{p}(A\cap(\overline{B}_{x,y}^{L})^{c},\mathfrak{m}_{x,y}^{L})$ can be
computed on a family of curves which is entirely contained in
$(\overline{B}_{x,y}^{L})^{c}$. Therefore every admissible sequence can be
taken to be zero on $\overline{B}_{x,y}^{L}$. The second inequality is
trivial, since $\mathfrak{m}_{x,y}^{L}\leq\mathfrak{m}_{x,y}^{2L}$. We focus
on the third one.
The function $R_{x,y}$ is continuous at every point of ${\rm
X}\setminus\\{x,y\\}$ by Remark 2.4, so it is uniformly continuous on the
compact set $\overline{B}_{r}(A)\cap\overline{B}_{x,y}^{2L}$. For every
$\varepsilon>0$, there exists $\eta>0$ such that if
$z,z^{\prime}\in\overline{B}_{r}(A)\cap\overline{B}_{x,y}^{2L}$ and ${\sf
d}(z,z^{\prime})<\eta$ then $|R_{x,y}(z)-R_{x,y}(z^{\prime})|\leq\varepsilon$.
We can always assume $\eta\leq r$. Moreover $R_{x,y}$ is continuous and
positive on $\overline{B}_{r}(A)\cap\overline{B}_{x,y}^{2L}$, so there exists
a constant $M$ such that

(23) $M^{-1}\leq R_{x,y}\leq M\text{ on
}\overline{B}_{r}(A)\cap\overline{B}_{x,y}^{2L}.$

Assume that $\mathcal{H}^{{\rm
cod-}{p}}_{\mathfrak{m}_{x,y}^{2L}}(A\cap\overline{B}_{x,y}^{L})<\infty$,
otherwise there is nothing to prove. For $j\in\mathbb{N}$ such that
$\frac{1}{j}<\frac{1}{2}\min\\{\eta,L{\sf d}(x,y)\\}$ we choose
$\\{B_{r_{i}^{j}}(z_{i}^{j})\\}_{i}$ such that
$A\cap\overline{B}_{x,y}^{L}\subset\bigcup_{i}B_{r_{i}^{j}}(z_{i}^{j})$,
$r_{i}^{j}<1/j$ and

(24)
$\sum_{i}\frac{\mathfrak{m}_{x,y}^{2L}(B_{r_{i}^{j}}(z_{i}^{j}))}{(r_{i}^{j})^{p}}\leq\mathcal{H}^{{\rm
cod-}{p}}_{\mathfrak{m}_{x,y}^{2L}}(A\cap\overline{B}_{x,y}^{L})+\frac{1}{j}.$

Without loss of generality we can suppose that
$B_{r_{i}^{j}}(z_{i}^{j})\cap(A\cap\overline{B}_{x,y}^{L})\neq\emptyset$, so
$B_{2r_{i}^{j}}(z_{i}^{j})\subseteq\overline{B}_{x,y}^{2L}$. We define

(25)
$\rho_{j}:=\left(\sum_{i}\frac{\chi_{B_{2r_{i}^{j}}(z_{i}^{j})}}{(r_{i}^{j})^{p}}\right)^{\frac{1}{p}}$

and we follow verbatim the proof in [HEMM22, Thm. 2.1] to show that
$\\{\rho_{j}\\}$ is admissible, i.e.
$\varliminf_{j\to\infty}\int_{\gamma}\rho_{j}\,{\mathrm{d}}s\geq 1$ for every
$\gamma\in\Gamma(A\cap\overline{B}_{x,y}^{L})$. Thus, we have

(26) ${\rm
AM}_{p}(\Gamma(A\cap\overline{B}_{x,y}^{L}),\mathfrak{m}_{x,y}^{2L})\leq\varliminf_{j\to\infty}\int\rho_{j}^{p}\,{\mathrm{d}}\mathfrak{m}_{x,y}^{2L}\leq\varliminf_{j\to\infty}\sum_{i}\frac{\mathfrak{m}_{x,y}^{2L}(B_{2r_{i}^{j}}(z_{i}^{j}))}{(r_{i}^{j})^{p}}.$

By uniform continuity and (23) we have

(27)
$\displaystyle\mathfrak{m}_{x,y}^{2L}(B_{2r_{i}^{j}}(z_{i}^{j}))=\int_{B_{2r_{i}^{j}}(z_{i}^{j})}R_{x,y}\,{\mathrm{d}}\mathfrak{m}$
$\displaystyle\leq(R_{x,y}(z_{i}^{j})+\varepsilon)\mathfrak{m}(B_{2r_{i}^{j}}(z_{i}^{j}))$
$\displaystyle\leq R_{x,y}(z_{i}^{j})(1+\varepsilon
M)\mathfrak{m}(B_{2r_{i}^{j}}(z_{i}^{j}))$ $\displaystyle\leq C_{D}\cdot
R_{x,y}(z_{i}^{j})(1+\varepsilon M)\mathfrak{m}(B_{r_{i}^{j}}(z_{i}^{j}))$

and similarly

(28)
$\displaystyle\mathfrak{m}_{x,y}^{2L}(B_{r_{i}^{j}}(z_{i}^{j}))=\int_{B_{r_{i}^{j}}(z_{i}^{j})}R_{x,y}\,{\mathrm{d}}\mathfrak{m}$
$\displaystyle\geq R_{x,y}(z_{i}^{j})(1-\varepsilon
M)\mathfrak{m}(B_{r_{i}^{j}}(z_{i}^{j})).$

Therefore

(29) $\mathfrak{m}_{x,y}^{2L}(B_{2r_{i}^{j}}(z_{i}^{j}))\leq
C_{D}\frac{1+\varepsilon M}{1-\varepsilon
M}\mathfrak{m}_{x,y}^{2L}(B_{r_{i}^{j}}(z_{i}^{j})).$

This, together with (26), gives

(30) $\displaystyle{\rm
AM}_{p}(A\cap\overline{B}_{x,y}^{L},\mathfrak{m}_{x,y}^{2L})\leq
C_{D}\frac{1+\varepsilon M}{1-\varepsilon M}\mathcal{H}^{{\rm
cod-}{p}}_{\mathfrak{m}_{x,y}^{2L}}(A\cap\overline{B}_{x,y}^{L}).$

Taking the limit for $\varepsilon$ going to zero we end the proof of (22). ∎

We now state the inequality between approximate modulus and codimension
Hausdorff measure for PI spaces.

######  Corollary 4.4.

Let $({\rm X},{\sf d},\mathfrak{m})$ be a $p$-PI space. Then for every
$x,y\in{\rm X}$ and every Borel set $A\subseteq{\rm X}$ with $\min\\{{\sf
d}(x,A),{\sf d}(y,A)\\}>0$ it holds

${\rm AM}_{p}(A,\mathfrak{m}_{x,y}^{L})\leq C\cdot\mathcal{H}^{{\rm
cod-}{p}}_{\mathfrak{m}_{x,y}^{2L}}(A),$

where $C=C(C_{D},C_{P},\lambda)$.

###### Proof.

By Remark 2.1 there exists another metric ${\sf d}^{\prime}$ on ${\rm X}$
which is geodesic and $C_{0}$-equivalent to ${\sf d}$, with $C_{0}$ depending
only on the structural constants. Proposition 4.3 implies

${\rm AM}_{p,{\sf d}^{\prime}}(A,\mathfrak{m}_{x,y}^{L})\leq
C^{\prime}\cdot\mathcal{H}^{{\rm cod-}{p}}_{\mathfrak{m}_{x,y}^{2L},{\sf
d}^{\prime}}(A),$

for a constant $C^{\prime}$ depending only on the structural constants, since
$({\rm X},{\sf d}^{\prime},\mathfrak{m})$ is doubling by Remark 3.17. The same
remark gives the thesis. ∎

##  5\. Relations between the different energies weighted with the Riesz
potential

Among the energies we have introduced there are two of them that are measures:
the perimeter and the codimension Hausdorff measure. In Theorem 1.2 we will
consider their weighted versions, where the weight is the Riesz potential
$R_{x,y}^{L}$. There are actually two ways to do so: either to consider the
measures ${\rm Per}_{\mathfrak{m}_{x,y}^{L}}$ and $\mathcal{H}^{{\rm
cod-}{p}}_{\mathfrak{m}_{x,y}^{L}}$ or to take the measures $R_{x,y}^{L}{\rm
Per}_{\mathfrak{m}}$ and $R_{x,y}^{L}\mathcal{H}^{{\rm
cod-}{p}}_{\mathfrak{m}}$. In this section we show that the two approaches are
comparable if the space satisfies a Poincaré inequality. We start by studying
the case of the perimeter. We focus directly to the case of the separating
sets.

######  Lemma 5.1.

Let $({\rm X},{\sf d},\mathfrak{m})$ be a metric measure space. Let
$x,y\in{\rm X}$ and let $\Omega\in\SS_{\textup{top}}(x,y)$. Assume that
$R_{x,y}^{L}$ is continuous on $B_{x,y}^{L}\setminus\\{x,y\\}$. Then

(31) $\frac{1}{2}\,{\rm
Per}_{\mathfrak{m}_{x,y}^{L}}(\Omega,A)\leq\int_{A}R_{x,y}^{L}\,{\mathrm{d}}{\rm
Per}_{\mathfrak{m}}(\Omega,\cdot)\leq 2\,{\rm
Per}_{\mathfrak{m}_{x,y}^{L}}(\Omega,A)$

for every $A\subseteq{\rm X}$ open. Moreover, if ${\rm
Per}_{\mathfrak{m}}(\Omega,B_{x,y}^{L})<\infty$, we have that for every Borel
set $B\subseteq{\rm X}$

(32) $\frac{1}{2}\,{\rm
Per}_{\mathfrak{m}_{x,y}^{L}}(\Omega,B)\leq\int_{B}R_{x,y}^{L}\,{\mathrm{d}}{\rm
Per}_{\mathfrak{m}}(\Omega,\cdot)\leq 2\,{\rm
Per}_{\mathfrak{m}_{x,y}^{L}}(\Omega,B).$

The assumption on the continuity of the Riesz potential is satisfied for
instance if $({\rm X},{\sf d},\mathfrak{m})$ is doubling and $({\rm X},{\sf
d})$ is geodesic, see Lemma 2.4.

###### Proof.

We fix $\varepsilon>0$. We define

$A_{i}^{\varepsilon}:=\left\\{z\in
B_{x,y}^{L}:\,\left(i-1-\frac{1}{4}\right)\varepsilon<R_{x,y}^{L}(z)<\left(i+\frac{1}{4}\right)\varepsilon\right\\}$

for $i\in\mathbb{N}$. By definition we have $A_{i}^{\varepsilon}\subseteq
B_{x,y}^{L}$ and
$\bigcup_{i}A_{i}^{\varepsilon}=B_{x,y}^{L}\setminus\\{x,y\\}$. The sets
$A_{i}^{\varepsilon}$ are open because $R_{x,y}^{L}$ is continuous on
$B_{x,y}^{L}$. Let $A\subseteq{\rm X}$ be open. Using directly the definition
of perimeter for open sets, we get

$\left(i-1-\frac{1}{4}\right)\varepsilon\,{\rm
Per}_{\mathfrak{m}}(E,A_{i}^{\varepsilon}\cap A)\leq{\rm
Per}_{\mathfrak{m}_{x,y}^{L}}(E,A_{i}^{\varepsilon}\cap
A)\leq\left(i+\frac{1}{4}\right)\varepsilon\,{\rm
Per}_{\mathfrak{m}}(E,A_{i}^{\varepsilon}\cap A).$

for every Borel set $E\subseteq{\rm X}$. Let $\Omega\in\SS_{\text{top}}(x,y)$.
Its topological boundary, which contains the support of ${\rm
Per}_{\mathfrak{m}}(\Omega,\cdot)$ by Remark 3.3.(v), has positive distance
from $x$ and $y$. Then we compute

$\displaystyle\int_{A}R_{x,y}^{L}\,{\mathrm{d}}{\rm
Per}_{\mathfrak{m}}(\Omega,\cdot)$
$\displaystyle=\int_{(B_{x,y}^{L}\setminus\\{x,y\\})\cap
A}R_{x,y}^{L}\,{\mathrm{d}}{\rm Per}_{\mathfrak{m}}(\Omega,\cdot)$
$\displaystyle\leq\sum_{i=0}^{\infty}\int_{A_{i}^{\varepsilon}\cap
A}R_{x,y}^{L}\,{\mathrm{d}}{\rm Per}_{\mathfrak{m}}(\Omega,\cdot)$
$\displaystyle\leq\sum_{i=0}^{\infty}\left(i+\frac{1}{4}\right)\varepsilon{\rm
Per}_{\mathfrak{m}}(\Omega,A_{i}^{\varepsilon}\cap A)$
$\displaystyle\leq\sum_{i=0}^{\infty}{\rm
Per}_{\mathfrak{m}_{x,y}^{L}}(\Omega,A^{\varepsilon}_{i}\cap
A)+\sum_{i=0}^{\infty}\frac{3}{2}\varepsilon{\rm
Per}_{\mathfrak{m}}(\Omega,A^{\varepsilon}_{i}\cap A)$ $\displaystyle\leq
2{\rm Per}_{\mathfrak{m}_{x,y}^{L}}(\Omega,A)+3\varepsilon{\rm
Per}_{\mathfrak{m}}(\Omega,B_{x,y}^{L}\cap A).$

In the last inequality, we used that $\sum_{i\text{ even}}{\rm
Per}_{\mathfrak{m}_{x,y}^{L}}(\Omega,A^{\varepsilon}_{i}\cap A)\leq{\rm
Per}_{\mathfrak{m}_{x,y}^{L}}(\Omega,A)$, since $A_{i}^{\varepsilon}\cap
A_{i+2}^{\varepsilon}=\emptyset$ for every $i\in\mathbb{N}$ and similarly for
the set of odd indexes. The same properties have been used for the estimate of
the additional term ${\rm Per}_{\mathfrak{m}}(\Omega,\cdot)$ in the right hand
side. Now we distinguish two cases. If ${\rm Per}_{\mathfrak{m}}(\Omega,A\cap
B_{x,y}^{L})=\infty$, then also ${\rm Per}_{\mathfrak{m}}(\Omega,A)=\infty$
and inequality (31) is trivially satisfied. If ${\rm
Per}_{\mathfrak{m}}(\Omega,A\cap B_{x,y}^{L})<\infty$ we take the limit as
$\varepsilon$ converging $0$ in the previous inequality. In any case

$\int_{A}R_{x,y}^{L}\,{\mathrm{d}}{\rm Per}_{\mathfrak{m}}(\Omega,\cdot)\leq
2{\rm Per}_{\mathfrak{m}_{x,y}^{L}}(\Omega,A).$

In a similar way we get

${\rm Per}_{\mathfrak{m}_{x,y}^{L}}(\Omega,A)\leq
2\int_{A}R_{x,y}^{L}\,{\mathrm{d}}{\rm
Per}_{\mathfrak{m}}(\Omega,\cdot)+3\varepsilon{\rm
Per}_{\mathfrak{m}}(\Omega,A\cap B_{x,y}^{L}).$

If ${\rm Per}_{\mathfrak{m}}(\Omega,A\cap B_{x,y}^{L})=\infty$ then also
$\int_{A}R_{x,y}^{L}\,{\mathrm{d}}{\rm
Per}_{\mathfrak{m}}(\Omega,\cdot)=\infty$ because there exists $m>0$ such that
$R_{x,y}^{L}\geq m$ on $A\cap B_{x,y}^{L}$. Otherwise, if ${\rm
Per}_{\mathfrak{m}}(\Omega,A\cap B_{x,y}^{L})<\infty$, we can let
$\varepsilon$ converge to $0$ and obtain

${\rm Per}_{\mathfrak{m}_{x,y}^{L}}(\Omega,A)\leq
2\int_{A}R_{x,y}^{L}\,{\mathrm{d}}{\rm Per}_{\mathfrak{m}}(\Omega,\cdot),$

thus concluding the proof of (31). The proof of (32) follows by the previous
step and an application of monotone class theorem. ∎

######  Corollary 5.2.

Let $({\rm X},{\sf d},\mathfrak{m})$ be a $p$-PI space. Let $x,y\in{\rm X}$
and let $\Omega\in\SS_{\textup{top}}(x,y)$. Then

$\frac{1}{C}\,{\rm Per}_{\mathfrak{m}_{x,y}^{L}}(\Omega,{\rm X})\leq\int_{\rm
X}R_{x,y}^{L}\,{\mathrm{d}}{\rm Per}_{\mathfrak{m}}(\Omega,\cdot)\leq C\,{\rm
Per}_{\mathfrak{m}_{x,y}^{L}}(\Omega,{\rm X}),$

where $C=C(C_{D},C_{P},\lambda)$.

###### Proof.

Using Remark 2.1 we find a new metric ${\sf d}^{\prime}$ on ${\rm X}$ which is
geodesic and $C_{0}$-equivalent to ${\sf d}$, with $C_{0}$ depending only on
the structural constants. Lemma 5.1 can be applied to the metric measure space
$({\rm X},{\sf d}^{\prime},\mathfrak{m})$, because of Lemma 2.4. We conclude
by applying Remark 3.17. ∎

We now move to a similar comparison between the two possible versions of the
codimension Hausdorff measures. Let us fix some notation. Let $f\colon{\rm
X}\times[0,+\infty)\to[0,+\infty)$ be a gauge function. For $A\subseteq{\rm
X}$ and $\delta>0$ we set

$\mathcal{H}_{\delta}^{f}(A):=\inf\left\\{\sum_{i=1}^{\infty}f(z_{i},r_{i})\text{
s.t. }A\subseteq\bigcup_{i}B_{r_{i}}(z_{i}),\,r_{i}\leq\delta\right\\}$

and

$\mathcal{H}^{f}(A):=\lim_{\delta\to 0}\mathcal{H}_{\delta}^{f}(A).$

######  Lemma 5.3.

$\mathcal{H}^{f}$ is a measure on the Borel sets.

###### Proof.

From the definition it is an outer measure on ${\rm X}$. Moreover if
$A,B\subseteq{\rm X}$ are separated, i.e. ${\sf d}(A,B)>0$, then
$\mathcal{H}^{f}(A\cup B)=\mathcal{H}^{f}(A)+\mathcal{H}^{f}(B)$. By [EG15,
§1.1, Theorem 5] every Borel set is measurable. ∎

Let $p\geq 1$, $x,y\in{\rm X}$ and $L\geq 1$. We consider two gauge functions:

$f(z,r):=\frac{\mathfrak{m}_{x,y}^{L}(B_{r}(z))}{r^{p}},\qquad\qquad
g(z,r):=R_{x,y}^{L}(z)\cdot\frac{\mathfrak{m}(B_{r}(z))}{r^{p}}.$

######  Lemma 5.4.

Let $({\rm X},{\sf d},\mathfrak{m})$ be a proper metric measure space. Let
$x,y\in{\rm X}$, $L,p\geq 1$. Let $A\subseteq{\rm X}$ such that $\min\\{{\sf
d}(x,A),{\sf d}(y,A)\\}>0$. Assume $R_{x,y}$ is continuous on
$\overline{B}_{x,y}^{L}\setminus\\{x,y\\}$. Then

(33) $\mathcal{H}^{{\rm
cod-}{p}}_{\mathfrak{m}_{x,y}^{L}}(A)=\mathcal{H}^{f}(A)=\mathcal{H}^{g}(A).$

###### Proof.

The first equality is the definition of $\mathcal{H}^{{\rm
cod-}{p}}_{\mathfrak{m}_{x,y}^{L}}(A)$. We focus on the other equality. We fix
$A\subseteq{\rm X}$ as in the assumptions. Let us consider $r\ll 1$ so that
$\overline{B}_{r}(A)\subset{\rm X}\setminus\\{x,y\\}$. By assumption the
function $R_{x,y}$ is continuous at every point of
$\overline{B}_{x,y}^{L}\setminus\\{x,y\\}$, so it is uniformly continuous on
the compact set $\overline{B}_{r}(A)\cap\overline{B}_{x,y}^{L}$. For every
$\varepsilon>0$, there exists $\eta>0$ such that if
$z,z^{\prime}\in\overline{B}_{r}(A)\cap\overline{B}_{x,y}^{L}$ and ${\sf
d}(z,z^{\prime})<\eta$ then $|R_{x,y}(z)-R_{x,y}(z^{\prime})|\leq\varepsilon$.
We can always assume $\eta\leq r$. Moreover $R_{x,y}$ is continuous and
positive on $\overline{B}_{r}(A)\cap\overline{B}_{x,y}^{L}$, so there exists a
constant $M$ such that

(34) $M^{-1}\leq R_{x,y}\leq M\text{ on
}\overline{B}_{r}(A)\cap\overline{B}_{x,y}^{L}.$

We prove first that $\mathcal{H}^{f}(A)\leq\mathcal{H}^{g}(A)$. Fix
$\varepsilon>0$ and find the corresponding $\eta>0$ as above. Fix also an
arbitrary $\tau>0$ and consider the set
$A_{\tau}:=A\cap\overline{B}_{x,y}^{L-\tau}$. We fix
$\delta\leq\min\\{\eta,\frac{\tau{\sf d}(x,y)}{2}\\}$ and we take a collection
$\\{B_{r_{i}}(z_{i})\\}_{i=1}^{\infty}$ such that
$A_{\tau}\subset\cup_{i=1}^{\infty}B_{r_{i}}(z_{i})$ and $r_{i}\leq\delta$. We
can suppose that ${\sf d}(z_{i},A_{\tau})\leq\frac{\tau{\sf d}(x,y)}{2}$,
otherwise $B_{r_{i}}(z_{i})\cap A_{\tau}=\emptyset$ and we can remove
$B_{r_{i}}(z_{i})$ from the covering. Therefore $B_{r_{i}}(z_{i})\subseteq
B_{x,y}^{L}$ and $R_{x,y}^{L}=R_{x,y}$ on $B_{x,y}^{L}$. By uniform continuity
we have

(35)
$\left|\int_{B_{r_{i}}(z_{i})}R_{x,y}^{L}\,{\mathrm{d}}\mathfrak{m}-R_{x,y}^{L}(z_{i})\mathfrak{m}(B_{r_{i}}(z_{i}))\right|\leq\varepsilon\mathfrak{m}(B_{r_{i}}(z_{i}))$

Thus, summing up, we get

(36)
$\left|\sum_{i=1}^{\infty}\frac{\int_{B_{r_{i}}(z_{i})}R_{x,y}^{L}\,{\mathrm{d}}\mathfrak{m}}{r_{i}^{p}}-\sum_{i=1}^{\infty}\frac{R_{x,y}^{L}(z_{i})\mathfrak{m}(B_{r_{i}}(z_{i}))}{r_{i}^{p}}\right|\leq\varepsilon\sum_{i=1}^{\infty}\frac{\mathfrak{m}(B_{r_{i}}(z_{i}))}{r_{i}^{p}}.$

From (36) and the fact that $1\leq MR_{x,y}^{L}(z_{i})$ for every $i$, we get

(37)
$\sum_{i=1}^{\infty}\frac{\int_{B_{r_{i}}(z_{i})}R_{x,y}^{L}\,{\mathrm{d}}\mathfrak{m}}{r_{i}^{p}}\leq\left(1+M\varepsilon\right)\,\sum_{i=1}^{\infty}\frac{R_{x,y}^{L}(z_{i})\mathfrak{m}(B_{r_{i}}(z_{i}))}{r_{i}^{p}}.$

Taking the infimum over all possible coverings with balls of radius smaller
than $\delta$, this gives
$\mathcal{H}_{\delta}^{f}(A_{\tau})\leq(1+M\varepsilon)\,\mathcal{H}_{\delta}^{g}(A_{\tau})$.
Taking the limit as $\delta\to 0$, we get that
$\mathcal{H}^{f}(A_{\tau})\leq(1+M\varepsilon)\,\mathcal{H}^{g}(A_{\tau})$ and
then, as $\varepsilon$ goes to zero, we have
$\mathcal{H}^{f}(A_{\tau})\leq\mathcal{H}^{g}(A_{\tau})$. Now we can pass to
the limit for $\tau$ going to zero to conclude that
$\mathcal{H}^{f}(A)\leq\mathcal{H}^{g}(A)$, because both $\mathcal{H}^{f}$ and
$\mathcal{H}^{g}$ are measures. The converse inequality
$\mathcal{H}^{g}(A)\leq\mathcal{H}^{f}(A)$ follows by considering the other
inequality given by (36) and using a similar argument.
∎

The next step is to link $\mathcal{H}^{g}$ to $R_{x,y}^{L}\mathcal{H}^{{\rm
cod-}{p}}_{\mathfrak{m}}$.

######  Lemma 5.5.

Let $({\rm X},{\sf d},\mathfrak{m})$ be a proper metric measure space. Let
$x,y\in{\rm X}$, $L,p\geq 1$. Let $A\subseteq{\rm X}$ such that $\min\\{{\sf
d}(x,A),{\sf d}(y,A)\\}>0$. Assume $R_{x,y}$ is continuous on
$\overline{B}_{x,y}^{L}\setminus\\{x,y\\}$. Then

(38) $\frac{1}{2}\int_{A}R_{x,y}^{L}\,{\mathrm{d}}\mathcal{H}^{{\rm
cod-}{p}}_{\mathfrak{m}}\leq\mathcal{H}^{g}(A)\leq
2\int_{A}R_{x,y}^{L}\,{\mathrm{d}}\mathcal{H}^{{\rm cod-}{p}}_{\mathfrak{m}}.$

###### Proof.

Let us start with the second inequality. We fix $A\subseteq{\rm X}$ as in the
assumptions. Let us consider $r\ll 1$ so that $\overline{B}_{r}(A)\subset{\rm
X}\setminus\\{x,y\\}$. As recalled above the function $R_{x,y}$ is continuous
at every point of $\overline{B}_{x,y}^{L}\setminus\\{x,y\\}$, so it is
uniformly continuous on the compact set
$\overline{B}_{r}(A)\cap\overline{B}_{x,y}^{L}$. We fix
$\varepsilon,\varepsilon^{\prime}>0$. Then there exists $\eta>0$ such that if
${\sf d}(z,z^{\prime})\leq\eta$ with
$z,z^{\prime}\in\overline{B}_{r}(A)\cap\overline{B}_{x,y}^{L}$ then
$|R_{x,y}(z)-R_{x,y}(z^{\prime})|\leq\varepsilon^{\prime}$. We define
$A_{\tau}:=A\cap\overline{B}_{x,y}^{L-\tau}$. Let us define
$E_{j}^{\varepsilon}:=\\{z\in
B_{x,y}^{L}:(j-1-\frac{1}{4})\varepsilon<R_{x,y}^{L}(z)<(j+\frac{1}{4})\varepsilon\\}$.
Notice that $\bigcup_{j}E_{j}^{\varepsilon}=B_{x,y}^{L}\setminus\\{x,y\\}$.
Let us fix $0<\delta\leq\min\\{r,\eta,\frac{\tau{\sf d}(x,y)}{2}\\}$. We
define
$\varepsilon_{j}:=(\left(j+\frac{1}{4}\right)\varepsilon+\varepsilon^{\prime})^{-1}\,2^{-j}\varepsilon$.
Let us consider a collection $\\{B_{r_{i}^{j}}(z_{i}^{j})\\}_{i}$ with
$r^{i}_{j}\leq\delta$ for every $i$ and such that

(39) $E_{j}^{\varepsilon}\cap
A_{\tau}\subseteq\bigcup_{i=1}^{\infty}B_{r_{i}^{j}}(z_{i}^{j}),\qquad\sum_{j}\frac{\mathfrak{m}(B_{r_{i}^{j}}(z_{i}^{j}))}{(r_{i}^{j})^{p}}\leq\mathcal{H}^{{\rm
cod-}{p}}_{\mathfrak{m},\delta}(E_{j}^{\varepsilon}\cap
A_{\tau})+\varepsilon_{j}.$

Notice that $A_{\tau}\subseteq\bigcup_{i,j}B_{r_{i}^{j}}(z_{i}^{j})$, thus it
is a competitor for the estimate of $\mathcal{H}^{g}_{\delta}(A_{\tau})$. We
compute

(40) $\displaystyle\mathcal{H}^{g}_{\delta}(A_{\tau})$
$\displaystyle\leq\sum_{i,j}^{\infty}R_{x,y}^{L}(z_{i}^{j})\frac{\mathfrak{m}(B_{r_{i}^{j}}(z_{i}^{j}))}{(r_{i}^{j})^{p}}\leq\sum_{j}\left(\left(j+\frac{1}{4}\right)\varepsilon+\varepsilon^{\prime}\right)\sum_{i=1}^{\infty}\frac{\mathfrak{m}(B_{r_{i}^{j}}(z_{i}^{j}))}{(r_{i}^{j})^{p}}$
(41)
$\displaystyle\leq\sum_{j}\left(\left(j+\frac{1}{4}\right)\varepsilon+\varepsilon^{\prime}\right)\left(\mathcal{H}^{{\rm
cod-}{p}}_{\mathfrak{m},\delta}(E_{j}^{\varepsilon}\cap
A_{\tau})+\varepsilon_{j}\right)$ (42)
$\displaystyle\leq\sum_{j}\left(\left(j+\frac{1}{4}\right)\varepsilon+\varepsilon^{\prime}\right)\mathcal{H}^{{\rm
cod-}{p}}_{\mathfrak{m}}(E_{j}^{\varepsilon}\cap
A_{\tau})+\sum_{j}2^{-j}\varepsilon$ (43)
$\displaystyle=\sum_{j}\int_{E_{j}^{\varepsilon}\cap
A_{\tau}}\left(\left(j+\frac{1}{4}\right)\varepsilon+\varepsilon^{\prime}\right)\,{\mathrm{d}}\mathcal{H}^{{\rm
cod-}{p}}_{\mathfrak{m}}+\varepsilon$ (44)
$\displaystyle\leq\sum_{j}\int_{E_{j}^{\varepsilon}\cap
A_{\tau}}\left(R_{x,y}^{L}+\varepsilon^{\prime}+\frac{3}{2}\varepsilon\right)\,{\mathrm{d}}\mathcal{H}^{{\rm
cod-}{p}}_{\mathfrak{m}}+\varepsilon$ (45) $\displaystyle\leq\sum_{j\text{
even}}\int_{E_{j}^{\varepsilon}\cap
A_{\tau}}\left(R_{x,y}^{L}+\varepsilon^{\prime}+\frac{3}{2}\varepsilon\right)\,{\mathrm{d}}\mathcal{H}^{{\rm
cod-}{p}}_{\mathfrak{m}}$ (46) $\displaystyle+\sum_{j\text{
odd}}\int_{E_{j}^{\varepsilon}\cap
A_{\tau}}\left(R_{x,y}^{L}+\varepsilon^{\prime}+\frac{3}{2}\varepsilon\right)\,{\mathrm{d}}\mathcal{H}^{{\rm
cod-}{p}}_{\mathfrak{m}}+\varepsilon$ (47) $\displaystyle\leq
2\int_{A_{\tau}}R_{x,y}^{L}\,{\mathrm{d}}\mathcal{H}^{{\rm
cod-}{p}}_{\mathfrak{m}}+(2\varepsilon^{\prime}+3\varepsilon)\mathcal{H}^{{\rm
cod-}{p}}_{\mathfrak{m}}(A_{\tau})+\varepsilon.$

By taking the limit as $\delta\to 0$ and as $\tau\to 0$, we get

(48) $\mathcal{H}^{g}(A)\leq
2\int_{A}R_{x,y}^{L}\,{\mathrm{d}}\mathcal{H}^{{\rm
cod-}{p}}_{\mathfrak{m}}+(2\varepsilon^{\prime}+3\varepsilon)\mathcal{H}^{{\rm
cod-}{p}}_{\mathfrak{m}}(A)+\varepsilon.$

Now we distinguish two cases. If $\mathcal{H}^{{\rm
cod-}{p}}_{\mathfrak{m}}(A)=\infty$, then also
$\int_{A}R_{x,y}^{L}\,{\mathrm{d}}\mathcal{H}^{{\rm
cod-}{p}}_{\mathfrak{m}}=\infty$ and there is nothing to prove. If
$\mathcal{H}^{{\rm cod-}{p}}_{\mathfrak{m}}(A)<\infty$, we can take the limit
as $\varepsilon$, $\varepsilon^{\prime}$ converging to $0$, concluding the
proof of the right inequality in (38).
Let us prove the first inequality. We fix
$\varepsilon,\varepsilon^{\prime}>0$, $\eta>0$ and $A_{\tau}$ as above. We
notice that $A_{\tau}$ intersects a finite number of sets
$E_{j}^{\varepsilon}$ because $R_{x,y}^{L}$ is bounded on $A_{\tau}$.
Therefore we can find $\delta^{\prime}>0$ such that $\mathcal{H}^{{\rm
cod-}{p}}_{\mathfrak{m}}(A_{\tau}\cap
E_{j}^{\varepsilon})\leq\mathcal{H}^{{\rm
cod-}{p}}_{\mathfrak{m},\delta^{\prime}}(A_{\tau}\cap
E_{j}^{\varepsilon})+\varepsilon_{j}$ for every $j$, where
$\varepsilon_{j}:=(\left(j+\frac{1}{4}\right)\varepsilon)^{-1}\,2^{-j}\varepsilon$.
Given $\delta<\min\\{r,\eta,\delta^{\prime},\frac{\tau{\sf d}(x,y)}{2}\\}$ we
take set of balls $B_{r_{i}^{j}}(z_{i}^{j})$ such that $A_{\tau}\cap
E_{j}^{\varepsilon}\subseteq\bigcup_{i}B_{r_{i}^{j}}(z_{i}^{j})$,
$r_{i}^{j}\leq\delta$ and
$\sum_{i}R_{x,y}^{L}(z_{i}^{j})\frac{\mathfrak{m}(B_{r_{i}^{j}}(z_{i}^{j}))}{(r_{i}^{j})^{p}}\leq\mathcal{H}^{g}(A_{\tau}\cap
E_{j}^{\varepsilon})+\varepsilon$. Computing as before we get

$\displaystyle\int_{A_{\tau}}R_{x,y}^{L}\,{\mathrm{d}}\mathcal{H}^{{\rm
cod-}{p}}_{\mathfrak{m}}\leq\sum_{j}\int_{A_{\tau}\cap
E_{j}^{\varepsilon}}R_{x,y}^{L}\,{\mathrm{d}}\mathcal{H}^{{\rm
cod-}{p}}_{\mathfrak{m}}\leq\sum_{j}\left(j+\frac{1}{4}\right)\varepsilon\,(\mathcal{H}^{{\rm
cod-}{p}}_{\mathfrak{m},\delta^{\prime}}(A_{\tau}\cap
E_{j}^{\varepsilon})+\varepsilon_{j}).$

The balls $B_{r_{i}^{j}}(z_{i}^{j})$ are admissible for the computation of
$\mathcal{H}^{{\rm cod-}{p}}_{\mathfrak{m},\delta^{\prime}}(A_{\tau}\cap
E_{j}^{\varepsilon})$, so we get

$\displaystyle\int_{A_{\tau}}R_{x,y}^{L}\,{\mathrm{d}}\mathcal{H}^{{\rm
cod-}{p}}_{\mathfrak{m}}$
$\displaystyle\leq\sum_{j}\sum_{i}\left(R_{x,y}^{L}(z_{i}^{j})+\frac{3}{2}\varepsilon+\varepsilon^{\prime}\right)\,\frac{\mathfrak{m}(B_{r_{i}^{j}}(z_{i}^{j}))}{(r_{i}^{j})^{p}}+\varepsilon$
$\displaystyle\leq\sum_{j}\left(1+\left(\frac{3}{2}\varepsilon+\varepsilon^{\prime}\right)M\right)\mathcal{H}^{g}(A_{\tau}\cap
E_{j}^{\varepsilon})+\varepsilon,$

where $M$ is such that $M^{-1}\leq R_{x,y}^{L}\leq M$ on
$\overline{B}_{r}(A_{\tau})\cap\overline{B}_{x,y}^{L}$. By treating separately
odd and even indexes, arguing as in the first part of the proof, we get

$\displaystyle\int_{A_{\tau}}R_{x,y}^{L}\,{\mathrm{d}}\mathcal{H}^{{\rm
cod-}{p}}_{\mathfrak{m}}\leq
2\left(1+\left(\frac{3}{2}\varepsilon+\varepsilon^{\prime}\right)M\right)\mathcal{H}^{g}(A_{\tau})+\varepsilon.$

Now we can take $\varepsilon,\varepsilon^{\prime},\tau\to 0$ to get the
thesis. ∎

Putting everything together we finally get the equivalent of Corollary 5.2 for
the codimension Hausdorff measures.

######  Corollary 5.6.

Let $({\rm X},{\sf d},\mathfrak{m})$ be a $q$-PI space for some $q\geq 1$. Let
$x,y\in{\rm X}$, $L,p\geq 1$ and $\Omega\in\SS_{\textup{top}}(x,y)$. Then

$\frac{1}{C}\,\mathcal{H}^{{\rm
cod-}{p}}_{\mathfrak{m}_{x,y}^{L}}(\partial\Omega)\leq\int_{\partial\Omega}R_{x,y}^{L}\,{\mathrm{d}}\mathcal{H}^{{\rm
cod-}{p}}_{\mathfrak{m}}\leq C\,\mathcal{H}^{{\rm
cod-}{p}}_{\mathfrak{m}_{x,y}^{L}}(\partial\Omega),$

where $C=C(C_{D},C_{P},\lambda)$.

###### Proof.

It follows by the usual combination of Remarks 2.1 and 3.17 together with the
lemmas 5.4 and 5.5. ∎

##  6\. Proof of the main theorem

We now provide the proof of our main result, that we write here in full
generality.

######  Theorem 6.1.

Let $({\rm X},{\sf d},\mathfrak{m})$ be a doubling metric measure space. Then
the following conditions are quantitatively equivalent:

  * (PI)

$1$-Poincaré inequality;

  * (BP)

$\exists c>0,L\geq 1$ such that $\int R_{x,y}^{L}\,{\mathrm{d}}{\rm
Per}_{\mathfrak{m}}(\Omega,\cdot)\geq c$ for every $x,y\in{\rm X}$ and
$\Omega\in\SS_{\textup{top}}(x,y)$;

  * (BP${}_{\textup{R}}$)

$\exists c>0,L\geq 1$ such that ${\rm
Per}_{\mathfrak{m}_{x,y}^{L}}(\Omega)\geq c$ for every $x,y\in{\rm X}$ and
$\Omega\in\SS_{\textup{top}}(x,y)$;

  * (BC)

$\exists c>0,L\geq 1$ such that
$\textup{cap}_{\mathfrak{m}_{x,y}^{L}}(\Omega)\geq c$ for every $x,y\in{\rm
X}$ and $\Omega\in\SS_{\textup{top}}(x,y)$;

  * (BMC)

$\exists c>0,L\geq 1$ such that $(\mathfrak{m}_{x,y}^{L})^{+,1}(\Omega)\geq c$
for every $x,y\in{\rm X}$ and $\Omega\in\SS_{\textup{top}}(x,y)$;

  * (BH)

$\exists c>0,L\geq 1$ such that
$\int_{\partial\Omega}R_{x,y}^{L}\,{\mathrm{d}}\mathcal{H}^{{\rm
cod-}{1}}_{\mathfrak{m}}\geq c$ for every $x,y\in{\rm X}$ and
$\Omega\in\SS_{\textup{top}}(x,y)$;

  * (BHe)

$\exists c>0,L\geq 1$ such that
$\int_{\partial^{e}\Omega}R_{x,y}^{L}\,{\mathrm{d}}\mathcal{H}^{{\rm
cod-}{1}}_{\mathfrak{m}}\geq c$ for every $x,y\in{\rm X}$ and
$\Omega\in\SS_{\textup{top}}(x,y)$;

  * (BH${}_{\textup{R}}$)

$\exists c>0,L\geq 1$ such that ${\mathcal{H}^{{\rm
cod-}{1}}_{\mathfrak{m}_{x,y}^{L}}}(\partial\Omega)\geq c$ for every
$x,y\in{\rm X}$ and $\Omega\in\SS_{\textup{top}}(x,y)$;

  * (BH${}^{e}_{\textup{R}}$)

$\exists c>0,L\geq 1$ such that ${\mathcal{H}^{{\rm
cod-}{1}}_{\mathfrak{m}_{x,y}^{L}}}(\partial^{e}\Omega)\geq c$ for every
$x,y\in{\rm X}$ and $\Omega\in\SS_{\textup{top}}(x,y)$;

  * (BAM)

$\exists c>0,L\geq 1$ such that ${\rm
AM}_{1}(\partial\Omega,\mathfrak{m}_{x,y}^{L})\geq c$ for every $x,y\in{\rm
X}$ and $\Omega\in\SS_{\textup{top}}(x,y)$;

  * (BAMe)

$\exists c>0,L\geq 1$ such that ${\rm
AM}_{1}(\partial^{e}\Omega,\mathfrak{m}_{x,y}^{L})\geq c$ for every
$x,y\in{\rm X}$ and $\Omega\in\SS_{\textup{top}}(x,y)$;

  * (BAM⋔)

$\exists c>0,L\geq 1$ such that ${\rm
AM}_{1}^{\pitchfork}(\Omega,\mathfrak{m}_{x,y}^{L})\geq c$ for every
$x,y\in{\rm X}$ and $\Omega\in\SS_{\textup{top}}(x,y)$.

With respect to the acronyms in the introduction, we added the superscript $e$
when we deal with the essential boundary and the subscript R when we consider
the metric measure space $({\rm X},{\sf d},\mathfrak{m}_{x,y}^{L})$, namely
with the weighted measure with Riesz potential $\mathfrak{m}_{x,y}^{L}$. The
scheme in Figure 1 outlines the strategy of the proof.

PIBP$\text{BP}_{\text{R}}$BMCBC$\text{BH}^{e}$BH$\text{BH}^{e}_{\text{R}}$$\text{BH}_{\text{R}}$$\text{BAM}^{\pitchfork}$$\text{BAM}^{e}$$\text{BMC}_{0}$BAM
Figure 1. Logical scheme of the proof of Theorem 6.1.

The node BMC0 appearing at the bottom right is the following auxiliary
property:

  * (BMC0)

$\exists c>0,L\geq 1$ such that
$(\mathfrak{m}_{x,y}^{L})^{+,1}(\partial\Omega)\geq c$ for every $x,y\in{\rm
X}$ and $\Omega\in\SS_{\textup{top}}(x,y)$ such that
$\mathfrak{m}_{x,y}^{L}(\partial\Omega)=0$.

###### Proof.

The first cycle of implications we prove is the bottom one.

(PI) $\Rightarrow$ (BAM⋔) and (BAMe). Let $x,y\in{\rm X}$ and let
$\alpha\in\mathcal{P}(\Gamma_{x,y}^{L})$ be a $1$-pencil as in Proposition 2.2
with constants $C>0,L\geq 1$ depending quantitatively on the structural
constants. Let us estimate the approximate $1$-modulus of the family
$\Gamma_{x,y}^{L}$. For every sequence of functions $\rho_{j}$ such that
$\varliminf_{j\to+\infty}\int_{\gamma}\rho_{j}\geq 1$ for every
$\gamma\in\Gamma_{x,y}^{L}$ we have

$\displaystyle
1\leq\int\varliminf_{j\to+\infty}\int_{\gamma}\rho_{j}\,{\mathrm{d}}\alpha\leq\varliminf_{j\to+\infty}\int\int_{\gamma}\rho_{j}\,{\mathrm{d}}\alpha\leq
C\varliminf_{j\to+\infty}\int_{\rm
X}\rho_{j}\,{\mathrm{d}}\mathfrak{m}_{x,y}^{L}$

by Fatou’s lemma. By the arbitrariness of the sequence $\rho_{j}$ we have
${\rm AM}_{1}(\Gamma_{x,y}^{L},\mathfrak{m}_{x,y}^{L})\geq c$, where $c=1/C$.
Let us fix now $\Omega\in\SS_{\text{top}}(x,y)$. By definition
$\Gamma_{x,y}^{L}\subseteq\Gamma(\Omega)^{\pitchfork}$ because
$x=\gamma_{0}\in I_{\Omega}$ and $y=\gamma_{1}\in O_{\Omega}$ for every
$\gamma\in\Gamma_{x,y}^{L}$. Therefore ${\rm
AM}_{1}^{\pitchfork}(\Omega,\mathfrak{m}_{x,y}^{L})\geq c$. This shows (BAM⋔).
In order to prove (BAMe) we fix $x,y\in{\rm X}$ and we consider again a
$1$-pencil $\alpha\in\mathcal{P}(\Gamma_{x,y}^{L})$ with constants $C$ and
$L$. Then ${\rm AM}_{1}(\Gamma_{x,y}^{L},\mathfrak{m}_{x,y}^{L})\geq c$ as
shown before, where $c=1/C$. We choose $L^{\prime}\geq L$ such that both
$B_{2L^{\prime}{\sf d}(x,y)}(x)$ and $B_{2L^{\prime}{\sf d}(x,y)}(y)$ have
finite $\mathfrak{m}$-perimeter and so also ${\rm
Per}_{\mathfrak{m}}(B_{x,y}^{L^{\prime}},{\rm X})<\infty$ by Remark 3.3.(iii):
this is possible by Lemma 3.2. We can choose $L^{\prime}$ arbitrarily close to
$L$ and to fix the constants we suppose $L^{\prime}<L+1$. A straightforward
computation shows that ${\rm
AM}_{1}(\Gamma_{x,y}^{L^{\prime}},\mathfrak{m}_{x,y}^{L^{\prime}})\geq c$ as
well. We fix a separating set $\Omega\in\SS_{\text{top}}(x,y)$ and we consider
two cases: either ${\rm Per}_{\mathfrak{m}}(\Omega,B_{x,y}^{L+1})<\infty$ or
not. If ${\rm Per}_{\mathfrak{m}}(\Omega,B_{x,y}^{L+1})=\infty$ then
Proposition 4.2 implies that ${\rm
AM}_{1}(\partial^{e}\Omega,\mathfrak{m}_{x,y}^{L+1})=\infty$, so there is
nothing else to prove. Suppose now ${\rm
Per}_{\mathfrak{m}}(\Omega,B_{x,y}^{L+1})<\infty$. We claim that ${\rm
Per}_{\mathfrak{m}}(\Omega\cap B_{x,y}^{L^{\prime}},{\rm X})<\infty$. For this
we argue as follows. By [BPR20, Proposition 1.16] we have
$\partial^{e}(\Omega\cap
B_{x,y}^{L^{\prime}})\subseteq\partial^{e}\Omega\cup\partial^{e}B_{x,y}^{L^{\prime}}$.
Together with the fact that $\partial^{e}(\Omega\cap
B_{x,y}^{L^{\prime}})\subseteq\overline{B}_{x,y}^{L^{\prime}}\subseteq
B_{x,y}^{L+1}$ we have

(49) $\partial^{e}(\Omega\cap
B_{x,y}^{L^{\prime}})\subseteq((\partial^{e}\Omega)\cap
B_{x,y}^{L+1})\cup\partial^{e}B_{x,y}^{L^{\prime}}.$

We remark that in [BPR20] the sets are assumed to have finite perimeter but
this assumption is not used in the proof. Therefore

$\mathcal{H}^{{\rm cod-}{1}}_{\mathfrak{m}}(\partial^{e}(\Omega\cap
B_{x,y}^{L^{\prime}}))\leq\mathcal{H}^{{\rm
cod-}{1}}_{\mathfrak{m}}((\partial^{e}\Omega)\cap
B_{x,y}^{L+1})+\mathcal{H}^{{\rm
cod-}{1}}_{\mathfrak{m}}(\partial^{e}B_{x,y}^{L^{\prime}}).$

Both terms on the right are finite because of Proposition 4.2 and our
assumptions. Hence also the left hand side term is finite, implying that ${\rm
Per}_{\mathfrak{m}}(\Omega\cap B_{x,y}^{L^{\prime}},{\rm X})<\infty$, again by
Proposition 4.2. So $\Omega\cap B_{x,y}^{L^{\prime}}$ is a bounded set of
finite perimeter in ${\rm X}$. By [LS17, Corollary 6.4] we have ${\rm
Mod}_{1}(\Gamma(\Omega\cap
B_{x,y}^{L^{\prime}})^{\pitchfork}\setminus\Gamma(\partial^{e}(\Omega\cap
B_{x,y}^{L^{\prime}})),\mathfrak{m})=0$. Hence ${\rm AM}_{1}(\Gamma(\Omega\cap
B_{x,y}^{L^{\prime}})^{\pitchfork}\setminus\Gamma(\partial^{e}(\Omega\cap
B_{x,y}^{L^{\prime}})),\mathfrak{m})=0$. Now we observe that
$\Gamma_{x,y}^{L^{\prime}}\subseteq\Gamma(\Omega\cap
B_{x,y}^{L^{\prime}})^{\pitchfork}$, because $\gamma_{0}=x\in I_{\Omega\cap
B_{x,y}^{L^{\prime}}}$ and $\gamma_{1}=y\in O_{\Omega\cap
B_{x,y}^{L^{\prime}}}$ for every $\gamma\in\Gamma_{x,y}^{L^{\prime}}$. Thus

(50) ${\rm
AM}_{1}(\Gamma_{x,y}^{L^{\prime}}\setminus\Gamma(\partial^{e}(\Omega\cap
B_{x,y}^{L^{\prime}})),\mathfrak{m})=0$

as well. Notice also that every curve $\gamma\in\Gamma_{x,y}^{L^{\prime}}$
cannot intersect $\partial^{e}B_{x,y}^{L^{\prime}}\subseteq\partial
B_{x,y}^{L^{\prime}}$ because ${\sf d}(x,\gamma_{t})\leq\ell(\gamma)\leq
L^{\prime}{\sf d}(x,y)$ for every $t\in[0,1]$, while every point of $\partial
B_{x,y}^{L^{\prime}}$ is at distance at least $2L^{\prime}{\sf d}(x,y)$ from
$x$. However, by (50) and (49), ${\rm AM}_{1}$-a.e. curve of
$\Gamma_{x,y}^{L^{\prime}}$, computed with respect to $\mathfrak{m}$, has to
intersect
$((\partial^{e}\Omega)\cap\overline{B}_{x,y}^{L^{\prime}})\cup\partial^{e}B_{x,y}^{L^{\prime}}$.
This means that ${\rm AM}_{1}$-a.e. curve of $\Gamma_{x,y}^{L^{\prime}}$ has
to intersect $(\partial^{e}\Omega)\cap\overline{B}_{x,y}^{L^{\prime}}$, i.e.

${\rm
AM}_{1}(\Gamma_{x,y}^{L^{\prime}}\setminus\Gamma((\partial^{e}\Omega)\cap\overline{B}_{x,y}^{L^{\prime}}),\mathfrak{m})=0.$

Since $\min\\{{\sf d}(x,\partial\Omega),{\sf d}(y,\partial\Omega)\\}>0$ then
there exists $M\geq 1$ such that $R_{x,y}^{L^{\prime}}\leq M$ on a
neighbourhood of $(\partial\Omega)\cap\overline{B}_{x,y}^{L^{\prime}}$.
Therefore, by a direct computation and an application of Lemma 3.15, we also
have

${\rm
AM}_{1}(\Gamma_{x,y}^{L^{\prime}}\setminus\Gamma((\partial^{e}\Omega)\cap\overline{B}_{x,y}^{L^{\prime}}),\mathfrak{m}_{x,y}^{L^{\prime}})=0.$

Using that ${\rm AM}_{1}(\cdot,\mathfrak{m}_{x,y}^{L^{\prime}})$ is an outer
measure and is monotone with respect to inclusion of families of paths we get

${\rm AM}_{1}(\Gamma(\partial^{e}\Omega),\mathfrak{m}_{x,y}^{L+1})\geq{\rm
AM}_{1}(\Gamma(\partial^{e}\Omega),\mathfrak{m}_{x,y}^{L^{\prime}})\geq{\rm
AM}_{1}(\Gamma_{x,y}^{L^{\prime}},\mathfrak{m}_{x,y}^{L^{\prime}})\geq c.$

This shows (BAMe) with constants $L+1$ and $c$.

Each of (BAM⋔) and (BAMe) $\Rightarrow$ (BAM). This is immediate from the fact
that both $\Gamma(\partial^{e}\Omega)$ and $\Gamma(\Omega)^{\pitchfork}$ are
included in $\Gamma(\partial\Omega)$.

(BAM) $\Rightarrow$ (BMC0). This is a direct consequence of Lemma 3.16.

(BMC0) $\Rightarrow$ (PI). Let $u\colon{\rm X}\to\mathbb{R}$ be a bounded
Lipschitz function such that $u\geq 0$ and let $x,y\in{\rm X}$. We want to
prove the pointwise estimate (($p$-PtPI)). If $u(x)=u(y)$ there is nothing to
prove. Otherwise we can suppose, without loss of generality, that $u(x)<u(y)$.
We first apply the coarea inequality for the codimension $1$-Hausdorff measure
relative to the measure $\mathfrak{m}_{x,y}^{L}$ (Proposition 3.7) to get

$\int_{0}^{+\infty}\mathcal{H}^{{\rm
cod-}{1}}_{\mathfrak{m}_{x,y}^{L}}(\\{u=t\\})\,{\mathrm{d}}t\leq 2\int_{\rm
X}{\rm lip}u\,{\mathrm{d}}\mathfrak{m}_{x,y}^{L}.$

Since $\int_{\rm X}{\rm lip}u\,{\mathrm{d}}\mathfrak{m}^{L}_{x,y}\leq{\rm
Lip}(u)\mathfrak{m}^{L}_{x,y}({\rm X})<\infty$, the right hand side is finite,
then $\mathcal{H}^{{\rm cod-}{1}}_{\mathfrak{m}_{x,y}^{L}}(\\{u=t\\})$ is
finite as well for almost every $t$. Then
$\mathfrak{m}_{x,y}^{L}(\\{u=t\\})=0$ for almost every $t$, by Proposition
3.6. We focus on the sets $\Omega_{t}:=\\{u\geq t\\}$ for $u(x)<t<u(y)$.
Clearly $\Omega_{t}\in\SS_{\text{top}}(x,y)$ for every such $t$. Moreover
$\partial\Omega_{t}\subseteq\\{u=t\\}$, so it has
$\mathfrak{m}_{x,y}^{L}$-measure zero for almost every $t$. In particular for
almost every $t\in(u(x),u(y))$ we can apply the assumption  (BMC0) to get
$(\mathfrak{m}_{x,y}^{L})^{+,1}(\partial\Omega_{t})\geq c$. We are now in
position to apply the coarea inequality for the Minkowski content as stated in
(17) to get

$c\,|u(x)-u(y)|\leq\int_{u(x)}^{u(y)}(\mathfrak{m}_{x,y}^{L})^{+,1}(\partial\Omega_{t})\,{\mathrm{d}}t\leq
2\int_{\rm X}{\rm lip}u\,{\mathrm{d}}\mathfrak{m}_{x,y}^{L}.$

The pointwise estimate follows with $C=2/c$, for Lipschitz, bounded,
nonnegative functions. By translation invariance we immediately have the same
inequality for all bounded, Lipschitz functions. Let ${u\in\rm Lip}({\rm X})$.
Let $u_{n}=\max\\{\min\\{u,n\\},-n\\}$. Notice that $u_{n}\in{\rm Lip}({\rm
X})$ and it is bounded. We have that $|u_{n}(x)-u_{n}(y)|$ converges to
$|u(x)-u(y)|$ and ${\rm lip}u_{n}\leq{\rm lip}u$. Thus, applying the estimate
to $u_{n}$ and taking the limit we conclude.

We now move to the right cycle of implications.

(PI) $\Rightarrow$ (BH${}^{e}_{\text{R}}$) $\Rightarrow$ (BH${}_{\text{R}}$)
$\Rightarrow$ (PI). We have already seen that (PI) implies  (BAMe). Therefore
Corollary 4.4 gives directly  (BH${}^{e}_{\text{R}}$). It is also obvious that
(BH${}^{e}_{\text{R}}$) implies  (BH${}_{\text{R}}$). We now suppose
(BH${}_{\text{R}}$) holds, we take a Lipschitz function $u\colon{\rm
X}\to\mathbb{R}$, $u\geq 0$ and two points $x,y\in{\rm X}$. We want to prove
the pointwise estimate (($p$-PtPI)). If $u(x)=u(y)$ there is nothing to prove,
otherwise once again we can suppose without loss of generality that
$u(x)<u(y)$. The sets $\Omega_{t}:=\\{u\geq t\\}$ belong to
$\SS_{\text{top}}(x,y)$ for all $t\in(u(x),u(y))$. Moreover
$\partial\Omega_{t}\subseteq\\{u=t\\}$. So we can apply the coarea inequality
(12) of Proposition 3.7 with respect to the measure $\mathfrak{m}_{x,y}^{L}$
to get

$c\,|u(x)-u(y)|\leq\int_{u(x)}^{u(y)}\mathcal{H}^{{\rm
cod-}{1}}_{\mathfrak{m}_{x,y}^{L}}(\\{u=t\\})\,{\mathrm{d}}t\leq 2\int_{\rm
X}{\rm lip}u\,{\mathrm{d}}\mathfrak{m}_{x,y}^{L}.$

Once again we can use this coarea inequality because of Proposition 2.3.
Therefore the pointwise estimate follows with $C=2/c$ for Lipschitz,
nonnegative functions. Arguing as before we get the same estimate for all
Lipschitz functions.

The next step is the left cycle of implications.

(PI) $\Rightarrow$ (BHe) $\Rightarrow$ (BH) $\Rightarrow$ (PI). We already
know that (PI) implies  (BH${}^{e}_{\text{R}}$). We immediately deduce  (BHe)
by Corollary 5.6. This gives  (BH). It remains to prove (BH) $\Rightarrow$
(PI). We take a Lipschitz function $u\colon{\rm X}\to\mathbb{R}$, $u\geq 0$
and two points $x,y\in{\rm X}$. We want to prove the pointwise estimate
(($p$-PtPI)). We apply the coarea inequality (12) of Proposition 3.7 with
respect to the measure $\mathfrak{m}$ and Borel function $R_{x,y}^{L}$ to get

$c\,|u(x)-u(y)|\leq\int_{u(x)}^{u(y)}\int_{\\{u=t\\}}R_{x,y}^{L}\,{\mathrm{d}}\mathcal{H}^{{\rm
cod-}{1}}_{\mathfrak{m}}\,{\mathrm{d}}t\leq 2\int_{\rm X}R_{x,y}^{L}\,{\rm
lip}u\,{\mathrm{d}}\mathfrak{m}.$

We now go to (PI) $\Leftrightarrow$ (BP). Suppose  (PI) holds. We already know
that  (BHe) holds. We fix $x,y\in{\rm X}$ and
$\Omega\in\SS_{\text{top}}(x,y)$. If ${\rm
Per}_{\mathfrak{m}}(\Omega,B_{x,y}^{L})=\infty$ there is nothing to prove. So
we can restrict to the case ${\rm Per}(\Omega,B_{x,y}^{L})<\infty$. By
Proposition 4.1 we know that ${\rm
Per}_{\mathfrak{m}}(\Omega,\cdot)=\theta\mathcal{H}^{{\rm
cod-}{1}}_{\mathfrak{m}}\lower 3.0pt\hbox{$|_{\partial^{e}\Omega}$}(\cdot)$ as
measures on $B_{x,y}^{L}$, with $c_{1}\leq\theta\leq C_{D}$ and $c_{1}$
depends only on the structural constants. Therefore

$\displaystyle\int_{\rm X}R_{x,y}^{L}\,{\mathrm{d}}{\rm
Per}_{\mathfrak{m}}(\Omega,\cdot)=\int_{B_{x,y}^{L}}R_{x,y}^{L}\,{\mathrm{d}}{\rm
Per}_{\mathfrak{m}}(\Omega,\cdot)$ $\displaystyle\geq
c_{1}\int_{\partial^{e}\Omega\cap
B_{x,y}^{L}}R_{x,y}^{L}\,{\mathrm{d}}\mathcal{H}^{{\rm
cod-}{1}}_{\mathfrak{m}}$
$\displaystyle=c_{1}\int_{\partial^{e}\Omega}R_{x,y}^{L}\,{\mathrm{d}}\mathcal{H}^{{\rm
cod-}{1}}_{\mathfrak{m}}\geq c_{1}c.$

This shows  (BP) with constants $c_{1}c$ and $L$.
Suppose now (BP) holds. We want to prove the pointwise estimate (($p$-PtPI)).
Let $x,y\in{\rm X}$ and $u\in{\rm Lip}({\rm X})$. Assume without loss of
generality that $u(x)<u(y)$. Consider $u(x)<t<u(y)$ and $\Omega_{t}:=\\{u\leq
t\\}\in{\rm SS}_{{\rm top}}(x,y)$. Then

$c\leq\int R_{x,y}^{L}\,{\mathrm{d}}{\rm Per}_{\mathfrak{m}}(\\{u\leq
t\\},\cdot)=\int R_{x,y}^{L}\,{\mathrm{d}}{\rm
Per}_{\mathfrak{m}}(\\{u>t\\},\cdot).$

where the last equality follows by Remark 3.3.(iii). Hence, using (7),

$c\,|u(x)-u(y)|\leq\int_{u(x)}^{u(y)}\int R_{x,y}^{L}\,{\mathrm{d}}{\rm
Per}_{\mathfrak{m}}(\\{u>t\\},\cdot)\,{\mathrm{d}}t\leq 2\int
R_{x,y}^{L}\,{\rm lip}u\,{\mathrm{d}}\mathfrak{m}.$

The remaining cycle of implications is (PI) $\Rightarrow$ (BP${}_{\text{R}}$)
$\Rightarrow$ (BC) $\Rightarrow$ (BMC) $\Rightarrow$ (PI).
Suppose (PI) holds. We already know that (BP) holds with constants $c>0$,
$L\geq 1$. Let $x,y\in{\rm X}$ and $\Omega\in{\rm SS}_{\rm top}(x,y)$. We
apply Corollary 5.2 to get

$c\leq\int R_{x,y}^{L}{\mathrm{d}}{\rm Per}_{\mathfrak{m}}(\Omega,\cdot)\leq
C^{\prime}\,{\rm Per}_{\mathfrak{m}_{x,y}^{L}}(\Omega,B_{x,y}^{L})\leq
C^{\prime}\,{\rm Per}_{\mathfrak{m}_{x,y}^{L}}(\Omega,{\rm X}).$

Therefore  (BP${}_{\text{R}}$) holds with constants $c/C^{\prime}$ and $L$.
The implications  (BP${}_{\text{R}}$) $\Rightarrow$ (BC) $\Rightarrow$ (BMC)
follow directly from Lemma 3.13, because the separating sets are closed and
their intersection with $\overline{B}_{x,y}^{L}$ is compact. To be precise, in
the application of Lemma 3.13 we use the fact that the perimeter, the capacity
and the Minkowski content of a set $A$ computed in the metric measure space
$({\rm X},{\sf d},\mathfrak{m}_{x,y}^{L})$ are equal to the perimeter, the
capacity and the Minkowski content of the set $A\cap{\rm
supp}\,\mathfrak{m}_{x,y}^{L}=A\cap\overline{B}_{x,y}^{L}$ computed in the
same metric measure space.
Now assume  (BMC) holds and we prove  (PI). Let $u\colon{\rm X}\to\mathbb{R}$,
$u\geq 0$ be a bounded Lipschitz function and let $x,y\in{\rm X}$. We want to
prove the pointwise estimate (($p$-PtPI)). We can assume as usual that
$u(x)<u(y)$. The sets $\Omega_{t}:=\\{u\geq t\\}$ belong to
$\SS_{\text{top}}(x,y)$ for all $t\in(u(x),u(y))$. So we can apply the coarea
inequality (16) of Proposition 3.10 with respect to the measure
$\mathfrak{m}_{x,y}^{L}$ to get

$c\,|u(x)-u(y)|\leq\int_{u(x)}^{u(y)}(\mathfrak{m}_{x,y}^{L})^{+,1}(\\{u\geq
t\\})\,{\mathrm{d}}t\leq\int_{\rm X}{\rm
lip}u\,{\mathrm{d}}\mathfrak{m}_{x,y}^{L}.$

Therefore the pointwise estimate follows with $C=1/c$ for Lipschitz,
nonnegative, bounded functions. As before we get the same estimate for all
Lipschitz functions. ∎

######  Remark 6.2 (Comparison with Keith’s characterizations).

In the proof of the implication (PI) implies (BAM⋔), we prove that (PI)
implies the following condition: there exists $c>0$ and $L\geq 1$ such that
for every $x,y\in{\rm X}$ we have

${\rm AM}_{1}(\Gamma_{x,y}^{L},\mathfrak{m}_{x,y}^{L})\geq c.$

This condition in the proof implies (BAM⋔) which implies (PI). Thus the above
condition is equivalent to (PI) and share a close similarity with Keith’s
characterizations in [Kei03]. Indeed, Keith proved that, given $p\geq 1$, a
doubling metric measure space satisfies a $p$-Poincaré inequality if and only
if

${\rm Mod}_{p}(\Gamma_{x,y}^{L},\mathfrak{m}_{x,y}^{L})\geq c.$

Looking at this condition for $p=1$, we have that our condition is apriori
weaker than Keith’s one, but it turns out to be equivalent.

We remark that the same scheme of proof shows the following property, holding
for every space satisfying a $p$-Poincaré inequality.

######  Proposition 6.3.

Let $({\rm X},{\sf d},\mathfrak{m})$ be a $p$-PI space, $p\geq 1$. Then there
exist $c>0$, $L\geq 1$, depending only on $(C_{D},C_{P},\lambda)$, such that
for every $x,y\in{\rm X}$ and every $\Omega\in\SS_{\textup{top}}(x,y)$ it
holds

$\int_{\partial\Omega}R_{x,y}^{L}\,{\mathrm{d}}\mathcal{H}^{{\rm
cod-}{p}}_{\mathfrak{m}}\geq c.$

In particular $\mathcal{H}^{{\rm cod-}{p}}_{\mathfrak{m}}(\partial\Omega)>0$.

###### Proof.

The same proof of  (PI) implies  (BAM⋔), where we use a $p$-pencil instead of
a $1$-pencil, shows that we can find $c>0$, $L\geq 1$ depending only on the
data such that

${\rm AM}_{p}^{\pitchfork}(\Omega,\mathfrak{m}_{x,y}^{L})\geq c$

for every $\Omega\in\SS_{\text{top}}(x,y)$. Therefore we also have

${\rm AM}_{p}(\partial\Omega,\mathfrak{m}_{x,y}^{L})\geq c.$

Corollary 4.4, together with Corollary 5.6 gives the thesis. The last part of
the statement is obvious. ∎

The converse is false in full generality for $p>1$ as the next example shows.

######  Example 6.4.

Let $({\rm X},{\sf d},\mathfrak{m})$ be the standard Sierpinski carpet. It is
an $s$-Ahlfors regular metric measure space for $s=\frac{\log 8}{\log 3}$. It
does not satisfy a $p$-Poincaré inequality for every $p\geq 1$. Since the
space is connected then every separating set has no empty boundary. Therefore
for every $s^{\prime}>s$ we have $\mathcal{H}^{{\rm
cod-}{s^{\prime}}}(\partial\Omega)\geq
C_{\text{AR}}^{-1}\,\mathcal{H}^{s-s^{\prime}}(\partial\Omega)=+\infty$, where
we used Remark 3.5 and the fact that $\partial\Omega$ is not empty. In
particular $\int_{\partial\Omega}R_{x,y}^{L}\,{\mathrm{d}}\mathcal{H}^{{\rm
cod-}{s^{\prime}}}_{\mathfrak{m}}=+\infty$ for every
$\Omega\in\SS_{\textup{top}}(x,y)$ and for every $x,y\in{\rm X}$.

Let us comment with some examples for which one of the equivalent properties
that are considered in Theorem 1.2 can be used to give necessary and sometimes
sufficient conditions for the validity of a $1$-Poincaré inequality.

######  Example 6.5 ($1$-Ahlfors regular spaces).

Let $({\rm X},{\sf d},\mathfrak{m})$ be a $1$-Ahlfors regular space. By Remark
3.5 there exists a constant $C$ depending only on the Ahlfors regularity
constant $C_{\textup{AR}}$ such that $C^{-1}\mathcal{H}^{{\rm
cod-}{1}}\leq\mathcal{H}^{0}\leq C\mathcal{H}^{{\rm cod-}{1}}$. The following
properties are quantitatively equivalent:

  * (i)

$\exists L\geq 1$ such that ${\rm X}$ is $L$-quasiconvex;

  * (ii)

$\exists L\geq 1$ such that $({\rm X},{\sf d})$ is $L$-biLipschitz equivalent
to a geodesic space;

  * (iii)

$({\rm X},{\sf d},\mathfrak{m})$ satisfies a $1$-Poincaré inequality.

We only need to verify that (i) implies (iii). To do so we verify the (BH)
property. Let us consider $x,y\in{\rm X}$ and $\Omega\in{\rm SS}_{\rm
top}(x,y)$. Let $\gamma$ be a $L$-quasigeodesic connecting $x$ to $y$, so that
$\gamma\subseteq B_{x,y}^{L}$. Moreover
$\gamma\cap\partial\Omega\neq\emptyset$, thus $\gamma\cap(\partial\Omega\cap
B_{x,y}^{L})\neq\emptyset$. By Ahlfors-regularity we have
$C_{\textup{AR}}^{-1}\chi_{B_{x,y}^{L}}\leq R^{L}_{x,y}\chi_{B_{x,y}^{L}}\leq
C_{\textup{AR}}\chi_{B_{x,y}^{L}}$. Thus,

$\int_{\partial\Omega}R^{L}_{x,y}{\mathrm{d}}\mathcal{H}^{{\rm cod-}{1}}\geq
C_{\textup{AR}}^{-1}\,\mathcal{H}^{0}(B_{x,y}^{L}\cap\partial\Omega)\geq
C_{\textup{AR}}^{-1}.$

Examples of $1$-Ahlfors regular spaces include graphs with an upper bound on
the valency of the vertices and a lower bound on the length of the edges.
Similar kind of simplicial complexes have been studied for instance in [CS22].
A general simplicial complex, even with bounded geometry, does not satisfy in
general a $1$-Poincaré inequality. The explicit computation of the best
possible Poincaré inequality for simplicial complexes of bounded geometry can
be done. One of the motivations for our study was the computation of the best
possible Poincaré inequality for geodesically complete metric spaces with
curvature bounded above. At the moment this problem is still unsolved.

######  Example 6.6 (Necessary condition in the Ahlfors regular case).

Let $({\rm X},{\sf d},\mathfrak{m})$ be a $1$-PI space, thus (BH) holds. Then
let $x\in{\rm X}$ and $\Omega=\overline{B}(x,r)$. For every $y$ such that
${\sf d}(x,y)>r$ we have that $\Omega\in{\rm SS}_{\rm top}(x,y)$. For such a
given $y$, $\partial\Omega\subseteq B_{x,y}^{L}$ and (BH) implies
$\int_{\partial\Omega}R_{x,y}\,{\mathrm{d}}\mathcal{H}^{{\rm cod-}{1}}\geq C$.
Thus

$\displaystyle C$ $\displaystyle\leq\int_{\partial
B_{r}(x)}R_{x,y}\,{\mathrm{d}}\mathcal{H}^{{\rm cod-}{1}}=\int_{\partial
B_{r}(x)}\frac{{\sf d}(x,z)}{\mathfrak{m}(B_{{\sf
d}(x,z)}(x))}\,{\mathrm{d}}\mathcal{H}^{{\rm cod-}{1}}+\int_{\partial
B_{r}(x)}\frac{{\sf d}(y,z)}{\mathfrak{m}(B_{{\sf
d}(y,z)}(y))}\,{\mathrm{d}}\mathcal{H}^{{\rm cod-}{1}}$
$\displaystyle\leq\frac{r}{\mathfrak{m}(B_{r}(x))}\,\mathcal{H}^{{\rm
cod-}{1}}(\partial B_{r}(x))+\int_{\partial B_{r}(x)}\frac{{\sf
d}(x,y)+r}{\mathfrak{m}(B_{{\sf d}(x,y)-r}(y))}\,{\mathrm{d}}\mathcal{H}^{{\rm
cod-}{1}}$

We further assume now that ${\rm X}$ is $s$-Alhfors regular for $s\geq 1$ with
constant $C_{\text{AR}}>0$. Then there exists $C_{1}=C_{1}(C,C_{\text{AR}})>0$
such that

(51) $C_{1}\leq r^{1-s}\,\mathcal{H}^{s-1}(\partial B_{r}(x))+\frac{{\sf
d}(x,y)+r}{({\sf d}(x,y)-r)^{s}}\,\mathcal{H}^{s-1}(\partial B_{r}(x)).$

Suppose moreover that $({\rm X},{\sf d})$ is not compact and choose $y_{n}$
such that ${\sf d}(x,y_{n})\to\infty$. Then we get

$\mathcal{H}^{s-1}(\partial B_{r}(x))\geq C_{1}\,r^{s-1}\qquad\text{for every
}r>0\text{ and every }x\in{\rm X}.$

This is a necessary condition for the validity of the $1$-Poincaré inequality
of a Ahlfors regular, non compact metric measure space.
More generally, if $({\rm X},{\sf d},\mathfrak{m})$ is Ahlfors regular and
$1$-PI, but not necessarely non-compact, we get an infinitesimal condition.
Indeed, by taking the limit as $r\to 0$ in (51), we have that for every
$x\in{\rm X}$

$\varliminf_{r\to 0}\frac{\mathcal{H}^{s-1}(\partial B_{r}(x))}{r^{s-1}}\geq
C_{2},$

with $C_{2}$ being equal to $C_{1}$ if $s>1$ and $C_{2}=C_{1}/2$ if $s=1$. The
case $s<1$ is excluded because ${\rm X}$ has to be quasigeodesic.

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