# A geometric approach to Poincaré inequality and Minkowski content 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.

The goal of this paper is to continue the study of the relation between the
Poincaré inequality and the lower bounds of Minkowski content of separating
sets, initiated in our previous work [4]. A new shorter proof is provided. An
intermediate tool is the study of the lower bound of another geometric
quantity, called separating ratio. The main novelty is the description of the
relation between the infima of the separating ratio and the Minkowski content
of separating sets. We prove a quantitative comparison between the two infima
in the local quasigeodesic case and equality in the local geodesic one. No
Poincaré assumption is needed to prove it. The main tool employed in the proof
is a new function, called the position function, which allows in a certain
sense to fibrate a set in boundaries of separating sets. We also extend the
proof to measure graphs, where due to the combinatorial nature of the problem,
the approach is more intuitive. In the appendix, we revise some classical
characterizations of the $p$-Poincaré inequality, by proving along the way
equivalence with a notion of $p$-pencil that extends naturally the definition
for $p=1$.

###### Key words and phrases:

Poincaré inequality, doubling measure, separating sets, Minkowski content,
metric measure spaces, pencil of curves

###### 2020 Mathematics Subject Classification:

30L99, 28A75, 49J52

###### Contents

  1. 1 Introduction
    1. 1.1 The position function and the proof of Theorem 1.6
    2. 1.2 Structure of the paper
    3. 1.3 Acknowledgments
  2. 2 Preliminaries
    1. 2.1 Curves in metric spaces
    2. 2.2 Connectivity properties
    3. 2.3 Measure properties and Poincaré inequality
    4. 2.4 Minkowski content
  3. 3 The $p$-set-connectedness and the $p$-set-pencil
  4. 4 The proof of Theorem 1.4
  5. 5 Relation between Minkowski content and separating ratio
    1. 5.1 Path separating sets
    2. 5.2 The position function
    3. 5.3 Equivalence of minima via the position function
  6. 6 The equivalence of the infima on measure graphs
  7. A Revision of conditions equivalent to the Poincaré inequality
    1. A.1 Keith’s module characterization
    2. A.2 Pencil and $A_{p}$-connectedness characterizations

##  1\. Introduction

Doubling metric measure spaces satisfying a $p$-Poincaré inequality for some
$p\geq 1$ are natural objects to study problems in geometric measure theory
and potential theory on metric measure spaces. This was particularly clear
after the theory developed in [6] and [14]. As such, this class has widely
spread and is by now very common in works on analysis in metric spaces.
Although the Poincaré inequality has an analytical formulation, it has been
realised that it is in fact a profoundly geometric condition. Under the
standing doubling assumption, it can be characterized in purely geometric
terms (see [17], [18], [12], [9], [7], [4]). In this paper we continue the
study of the geometric condition formulated by the authors in [4] in terms of
energy associated to separating sets, for the case $p=1$. In particular, among
the several energies studied in [4], we focus on the Minkowski content with
respect to the measure weighted with the Riesz potential. To be precise we
recall the definition of separating sets and of Riesz potential.

######  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
$B_{r}(y)\subseteq\Omega^{c}$. The class of all separating sets from $x$ to
$y$ is denoted by $\SS_{\textup{top}}(x,y)$.

######  Definition 1.2.

Let $({\rm X},{\sf d},\mathfrak{m})$ be a metric measure space. Given
$x,y\in{\rm X}$ and $L\geq 1$, the _$L$ -truncated Riesz potential with poles
at $x,y$_ is

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

where $B_{x,y}^{L}=B_{2L{\sf d}(x,y)}(x)\cup B_{2L{\sf d}(x,y)}(y)$. Its
associated measure is $\mathfrak{m}_{x,y}^{L}=R_{x,y}^{L}\mathfrak{m}$.

Given a couple of points $x,y\in{\rm X}$, we formulate the following
condition.

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

where $(\mathfrak{m}_{x,y}^{L})^{+}(\Omega)$ denotes the Minkowski content of
the set $\Omega$ with respect to the measure $\mathfrak{m}_{x,y}^{L}$, see
Section 2.4 for the details. The notation ${\text{(BMC)}}_{x,y}$ stands for
‘big Minkowski content of separating sets at $x,y$’. We recall that, according
to [4, Theorem 1.2], a doubling metric measure space has a $1$-Poincaré
inequality if and only if ${\text{(BMC)}}_{x,y}$ holds for every $x,y\in{\rm
X}$ with uniform constants. This statement is a quantification of the well-
known heuristic intuition that the validity of the Poincaré inequality is
obstructed by the presence of bottlenecks in the metric space. The Minkowski
content computed via the reference measure weighted with the Riesz potential
is the correct energy to measure the size of boundaries of separating sets.

Typically, all the other geometric characterizations of the Poincaré
inequality in literature are expressed in terms of properties of curves
connecting two points. One of the goals of this paper is to investigate
thoroughly the relation between these classical approaches and the (BMC)x,y
property. With this in mind we consider a definition similar to the one used
in [11] (originated from previous contributions in [2] and [10]) for which we
need an auxiliary quantity. We define the width of a set $A\subseteq{\rm X}$
with respect to the points $x,y\in{\rm X}$ as

$0pt_{x,y}(A):=\inf_{\gamma\in\Gamma_{x,y}}\ell(\gamma\cap A),$

where $\Gamma_{x,y}$ is the set of rectifiable paths connecting $x$ to $y$.
The quantity $0pt_{x,y}(A)$ measures the width of the set $A$ in the following
sense: we consider all the curves (with finite length) connecting $x$ to $y$
and we look at the one whose length inside $A$ is minimal.

######  Definition 1.3.

Let $C>0$ and $L\geq 1$. We say that $({\rm X},{\sf d},\mathfrak{m})$ is
$(C,L)$ $1$-set-connected at $x,y\in{\rm X}$ if

(1) $0pt_{x,y}(A)\leq C\mathfrak{m}_{x,y}^{L}(A)\text{ for all }A\subseteq{\rm
X}\text{ Borel}.$

This definition is inspired by [11], however there are two main differences.
In [11] the definition that the authors use can be rephrased as

(2) $0pt_{x,y}^{\Lambda}(A)\leq C{\sf d}(x,y)\left(M_{L{\sf
d}(x,y)}\chi_{A}(x)+M_{L{\sf d}(x,y)}\chi_{A}(y)\right)\text{ for all
}A\subseteq{\rm X}\text{ Borel},$

where $M_{\rho}(f)(x)$ is the Hardy-Littlewood maximal function of a function
$f\geq 0$ computed at the point $x$ up to scale $\rho>0$ (see [14]) and where
in the definition of $0pt_{x,y}^{\Lambda}(A)$ one only considers
$\Lambda$-quasigeodesics connecting $x$ and $y$, see Section 3 for details.
The first difference is in the class of curves used for defining the width.
Notice that in [11], the finiteness of $\Lambda$ in (2) is crucial to prove
that this condition _for every couple of points_ and with uniform constants
implies a $p$-Poincaré inequality for all $p>1$ (see [11, Theorem 2.19 & Lemma
2.20]). The second difference is in the right hand sides. They reflect the two
equivalent classical pointwise estimates of Heinonen (see [15, Sec. 9])

  * •

$|u(x)-u(y)|\leq C\int{\rm lip}u\,{\mathrm{d}}\mathfrak{m}_{x,y}^{L}$ for all
$u\in{{\rm Lip}}({\rm X})$ and for every $x,y\in{\rm X}$,

  * •

$|u(x)-u(y)|\leq C\,{\sf d}(x,y)\,(M_{L{\sf d}(x,y)}({\rm lip}u)(x)+M_{L{\sf
d}(x,y)}({\rm lip}u)(y))$ for all $u\in{{\rm Lip}}({\rm X})$ and for every
$x,y\in{\rm X}$,

whose validity characterizes the $1$-Poincaré inequality in doubling setting.
We recall that ${\rm lip}\,u(x)$ denotes the local Lipschitz constant of the
function $u$ at $x\in{\rm X}$.
The first result of this paper shows that the pointwise estimates à la
Heinonen for a fixed couple of points $x,y\in{\rm X}$, the (BMC)x,y condition
and Definition 1.3 are all equivalent, if ${\rm X}$ satisfies mild
connectedness assumptions.

######  Theorem 1.4.

Let $({\rm X},{\sf d},\mathfrak{m})$ be a doubling metric measure space which
is path connected and locally $\Lambda$-quasiconvex. Let $x,y\in{\rm X}$. Then
the following conditions are quantitatively equivalent:

  * (i)

there exist $C>0$, $L\geq 1$ such that

(3) $|u(x)-u(y)|\leq C\int{\rm
lip}u\,{\mathrm{d}}\mathfrak{m}_{x,y}^{L}\qquad\text{for all }u\in{{\rm
Lip}}({\rm X});$

  * (ii)

there exist $C>0$, $L\geq 1$ such that

(4) $|u(x)-u(y)|\leq C\int
g\,{\mathrm{d}}\mathfrak{m}_{x,y}^{L}\qquad\text{for all }u\text{ Borel and
all }g\in{\rm UG}(u);$

  * (iii)

the space $({\rm X},{\sf d},\mathfrak{m})$ is $(C,L)$ $1$-set-connected at
$x,y$;

  * (iv)

there exist $C,L$ such that (1) is satisfied by every closed subset
$A\subseteq{\rm X}$;

  * (v)

the space $({\rm X},{\sf d},\mathfrak{m})$ satisfies (BMC)x,y.

Some remarks are in order. First of all, the equivalence between items (i) and
(v) in the statement above is true even dropping the assumption that the
metric space is complete (this will be proved in our forthcoming work [5]).
Secondly, in Theorem 1.4 we just work with _two fixed $x,y$_ at the price of
assuming local connectivity properties of the metric space. By looking at the
previous statement for every couple of points, we achieve the next result,
which is new and improves the mentioned result of [11].

######  Corollary 1.5.

Let $({\rm X},{\sf d},\mathfrak{m})$ be a doubling metric measure space. Then
it satisfies a $1$-Poincaré inequality if and only if there exist $C>0$ and
$L\geq 1$ such that

$0pt_{x,y}(A)\leq C\mathfrak{m}_{x,y}^{L}(A)$

for every $x,y\in{\rm X}$ and every $A\subseteq{\rm X}$ Borel.

Indeed using the same language we introduced above, we can rephrase the
results in [11] for general $p\geq 1$ in terms of our definitions. Their
results read as follows: if ${\rm X}$ satisfies a $p$-Poincaré inequality,
then ${\rm X}$ is $(C,L,\Lambda)$ $p$-set-connected. Conversely, if ${\rm X}$
is $(C,L,\Lambda)$ $p$-set-connected, then it satisfies a $q$-Poincaré
inequality for every $q>p$. We refer to Section 3 for the definition of
$(C,L,\Lambda)$ $p$-set-connectedness and the proof of this statement.
Corollary 1.5 states the full equivalence between the two conditions (even
with infinite $\Lambda$) in case $p=1$.

The proof of Theorem 1.4 is relatively elementary, as it uses classical tools
in analysis on metric spaces and the coarea inequality for the Minkowski
content. We just want to sketch the proof of (iii) implies (v). Given a set
$\Omega$, we apply (iii) to the sets $B_{r}(\Omega)\setminus\Omega$ for $r$
sufficiently small getting

$0pt_{x,y}(B_{r}(\Omega)\setminus\Omega)\leq
C\mathfrak{m}_{x,y}^{L}(B_{r}(\Omega)\setminus\Omega).$

It is straightforward to check that
$0pt_{x,y}(B_{r}(\Omega)\setminus\Omega)\geq r$ if $\Omega\in\SS_{\rm
top}(x,y)$. The conclusion follows by dividing by
$0pt_{x,y}(B_{r}(\Omega)\setminus\Omega)$ and taking the limit as $r\to 0$.

As suggested by the idea above, to compare the $1$-set-connectedness and the
Minkowski content computed with respect to $\mathfrak{m}_{x,y}^{L}$ it is
natural to introduce the following notion, which has the same dimensionality
of the Minkowski content. The separating ratio between $x$ and $y$ of a set
$A\subseteq{\rm X}$ is

(5) ${\rm SR}_{x,y}(A):=\frac{\mathfrak{m}_{x,y}^{L}(A)}{0pt_{x,y}(A)}.$

With this notation the proof of (iv) implies (v) in Theorem 1.4 gives

(6) $\inf_{A\subseteq{\rm X}\text{ closed}}{\rm
SR}_{x,y}(A)\leq\inf_{\Omega\in\SS_{\text{top}}(x,y)}(\mathfrak{m}_{x,y}^{L})^{+}(\Omega)$

and Theorem 1.4 says that a positive lower bound for one of the two infima is
equivalent to a positive lower bound for the other one. However, inspired by
the proof given above, it seems possible to approach directly the equivalence
between (iv) and (v). Indeed we can quantitatively relate, with explicit
constants, the infima of the two functionals. This is the content of the next
result.

######  Theorem 1.6.

Let $({\rm X},{\sf d},\mathfrak{m})$ be a doubling, path-connected, locally
$\Lambda$-quasiconvex metric measure space and let $x,y\in{\rm X}$. Then

(7)
$\Lambda^{-1}\inf_{\Omega\in\SS_{\textup{top}}(x,y)}(\mathfrak{m}_{x,y}^{L})^{+}(\Omega)\leq\inf_{A\subseteq{\rm
X}\,\textup{closed}}{\rm
SR}_{x,y}(A)\leq\inf_{\Omega\in\SS_{\textup{top}}(x,y)}(\mathfrak{m}_{x,y}^{L})^{+}(\Omega).$

In particular, if $({\rm X},{\sf d})$ is path connected and locally geodesic
we have

(8) $\inf_{A\subseteq{\rm X}\,\textup{closed}}{\rm
SR}_{x,y}(A)=\inf_{\Omega\in\SS_{\textup{top}}(x,y)}(\mathfrak{m}_{x,y}^{L})^{+}(\Omega).$

Let us remark that (7) follows by the application of Theorem 1.4 as well, with
a non explicit control of the constant in the first inequality. Moreover, the
second statement in Theorem 1.6 does not follow from Theorem 1.4. Besides the
discussion about the constants, we believe that the importance of the
statement relies on its proof, which is more geometrical rather than
analytical, does not use neither the results nor the methods of Theorem 1.4
and allows a better understanding of the separating ratio and the Minkowski
content. We discuss it in detail in the next subsection.

###  1.1. The position function and the proof of Theorem 1.6

We start the discussion on the proof of Theorem 1.6 with a toy example. We
warn the reader that the sketch of the proof in this example is informal and a
bit imprecise. Let us consider the two dimensional Euclidean space and a
couple of points, say $x$ and $y$. Let us compute the infimum of the
separating ratio between $x$ and $y$. The choice of the dumbbell-shaped domain
$D$ as in Figure 1 as a competitor is not convenient for the minimization of
the separating ratio. Indeed, we can reduce the volume of the set (with
respect to $\mathfrak{m}_{x,y}^{L}$) by keeping unaltered its width. This
procedure leads to the rectangle $R$ in Figure 1. The next step is to chop $R$
in $N$ slimmer rectangles $R_{i}$ of width equal to the $N$-th fraction of the
width of $R$. We then select one of such slimmer rectangles, say $R_{1}$, with
the property that its separating ratio is less than or equal the separating
ratio of $R$. This is a consequence of the following formula:

$\frac{1}{N}\sum_{i=1}^{N}{\rm SR}_{x,y}(R_{i})={\rm SR}_{x,y}(R).$

This formula is trivial in such a case and can be regarded as a discrete
version of coarea formula. By iterating this procedure we find slimmer and
slimmer rectangles whose separating ratio is smaller and smaller such that
they converge to a line $L$. The half-plane with boundary $L$ and containing
$x$ is a topological separating set between $x$ and $y$ and its Minkowski
content can be estimated from above by the separating ratio of the converging
rectangles.

Figure 1. The picture gives an informal explanation of the proof of the
Theorem in the the toy example of the two dimensional Euclidean case for a
specific choice of $x,y$ and $D$ in the definition of separating ratio.

The same ideas can be formalized in the general setting. The main ingredient
is a specific function we defined for such a purpose, called the position
function.
For $A\subseteq{\rm X}$ and $\gamma\in\Gamma_{x,y}$ we define

(9) ${\rm pos}_{\gamma,A}(z):=\inf_{s\in\gamma^{-1}(z)}\ell(\gamma\cap
A,s),\qquad\text{and}\qquad{\rm pos}_{A}(z):=\inf_{\gamma\in\Gamma_{x,y}}{\rm
pos}_{\gamma,A}(z),$

where $\ell(\gamma\cap
A,s):=\int_{0}^{s}\chi_{A}(t)|\dot{\gamma}(t)|\,{\mathrm{d}}t$ is the length
spent by $\gamma$ inside $A$ up to time $s$. See Section 5 for a precise
discussion. The steps sketched in the example above can be formally formulated
as follows.

  * •

Reduction to rectangles. Let $A\subseteq{\rm X}$ be such that
$0pt_{x,y}(A)>0$. Then $A^{\prime}:=A\cap\\{0<{\rm
pos}_{A}(z)<0pt_{x,y}(A)\\}$ satisfies ${\rm SR}_{x,y}(A^{\prime})\leq{\rm
SR}_{x,y}(A).$ A proof of this fact can be obtained with the same methods of
Proposition 5.3.

  * •

Reduction to the Minkowski content. Let $A\subseteq{\rm X}$ be such that
$0pt_{x,y}(A)>0$. Then there exists $t\in(0,0pt_{x,y}(A))$ such that
$\Omega:=\\{{\rm pos}_{A}\leq t\\}$ is a topological separating set from $x$
to $y$ and $\mathfrak{m}^{+}(\Omega)\leq\Lambda\,{\rm SR}_{x,y}(A)$. This is
the way we prove Theorem 1.6.

We remark that $\partial\Omega\subseteq\\{{\rm pos}_{A}=t\\}$ plays the role
of the line $L$ in the toy example. These and other procedures will be
investigated in detail in Section 5.
The key observation of the position function is that it allows to fibrate a
separating set into separating sets by looking at its level sets. Note indeed
that this is the most delicate point: considering again the toy example of the
Euclidean space, it is easy to find an example of a couple of points and
separating set for which for many choice of Lipschitz functions their level
sets are not in general separating sets.

Since we believe that the argument in Section 5 is technical, in order to
isolate the main ideas behind the proof, we propose a similar argument in the
case of measure graphs. This is the content of Section 6. Since the objects
are discrete and the argument is more combinatorial, we explain more in detail
how the continuous argument reduces in this case. To be more precise, we
define the discretized version of separating sets, size, separating ratio and
Minkowski content. Finally, we provide an explicit procedure, that takes in
input a discretized separating set and gives as output another discretized
separating set with no larger separating ratio but with smallest possible
discretized width. For such sets the computation of the discretized separating
ratio coincides with the computation of the discretized Minkowski content.
This provides the exact equality of the minima of the two discretized
functionals, see Theorem 6.5.

An open question is if the equality of infima on measure graphs in Theorem 6.5
could be used to prove equality of the infima in the continuous setting. Every
metric space can be approximated by a suitable locally finite graph by looking
at a $\delta$-net for $\delta\ll 1$ and associating to this set of points a
structure of measure graph (see for instance the construction in [13] and
Remark 6.4). It would be interesting to understand the convergence of the
discrete separating ratio and Minkowski content of the approximations to their
continuous versions, but this is yet to be understood.

###  1.2. 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 recall preliminaries about
curves, doubling measures, Poincaré inequality and Minkowski content. In
Section 3 we introduce the concept of $p$-set-connectedness and $p$-set-pencil
and we prove some relations with the $p$-Poincaré inequality. We also show the
complete equivalence for $p=1$ using Theorem 1.4. In Section 4, we prove
Theorem 1.4 and Corollary 1.5 following the scheme described in the
introduction. In Section 5 we study the relation between the separating ratio
and the Minkowski content of separating sets. The position function is
introduced, its properties are studied and Theorem 1.6 is proved. In Section 6
we introduce the discretized version of separating ratio, Minkowski content
and position function and study the equivalent version of Theorem 1.6 in the
setting of measure graphs. This section is self-contained and contains all the
needed definitions for the discrete case. Finally, Appendix A contains a
revised version of already known characterizations of Poincaré inequality via
a curvewise approach, but formulated for a fixed couple of points. We also
prove that all these conditions can be formulated equivalently either in terms
of quasigeodesics with fixed bounded length or rectifiable curves.

###  1.3. 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). The first
named author wishes to thank David Bate for fruitful discussions. We thank
Tommaso Rossi and Pietro Wald for their comments on a preliminary version of
the manuscript.

##  2\. Preliminaries

In this paper a metric measure space $({\rm X},{\sf d},\mathfrak{m})$ is a
complete and separable metric space $({\rm X},{\sf d})$ equipped with a
nonnegative Borel measure $\mathfrak{m}$ that is finite on bounded sets. Given
$A\subseteq{\rm X}$ and $r\geq 0$ we set $B_{r}(A):=\\{x:\,{\sf d}(x,A)<r\\}$
and $\overline{B}_{r}(A):=\\{x:\,{\sf d}(x,A)\leq r\\}$, they are 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 $\mathbb{R}$. A function $u\colon{\rm X}\to\mathbb{R}$ is
locally Lipschitz if for every $x\in{\rm X}$ there exist $r_{x}>0$ and
$\Lambda_{x}\geq 0$ such that $u\lower 3.0pt\hbox{$|_{B_{r_{x}}(x)}$}$ is
$\Lambda_{x}$-Lipschitz. The set of maps that are locally Lipschitz is denoted
by ${\rm Lip}_{\rm loc}({\rm X})$. Every locally Lipschitz map is Lipschitz on
a compact set.

######  Lemma 2.1 ([3, Theorem 4.2]).

Let $({\rm X},{\sf d})$ be a metric space, $K\subseteq{\rm X}$ be compact and
$u\in{\rm Lip}_{\rm loc}({\rm X})$. Then $u\lower 3.0pt\hbox{$|_{K}$}$ is
Lipschitz.

The _local Lipschitz constant_ ${\rm lip}u\colon{\rm X}\to\mathbb{R}$ of a
function $u\colon{\rm X}\to\mathbb{R}$ is

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

with the convention that ${\rm lip}u(x)=0$ if $x$ is an isolated point.

###  2.1. Curves in metric spaces

Let $C([0,1],{\rm X})$ be the space of continuous curves
$\gamma\colon[0,1]\to{\rm X}$ endowed with the sup distance ${\sf
d}_{\infty}$, i.e. ${\sf
d}_{\infty}(\gamma,\gamma^{\prime}):=\sup_{t\in[0,1]}{\sf
d}(\gamma(t),\gamma^{\prime}(t))$ for $\gamma,\gamma^{\prime}\in C([0,1],{\rm
X})$. Since $({\rm X},{\sf d})$ is complete and separable, so $(C([0,1],{\rm
X}),{\sf d}_{\infty})$ is. The length of a curve is defined as

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

A curve $\gamma\in C([0,1],{\rm X})$ is said to be rectifiable if
$\ell(\gamma)<\infty$. Every rectifiable curve admits a Lipschitz
reparametrization, i.e. there exists an increasing map
$\phi\colon[0,1]\to[0,1]$ with $\phi(0)=0$ and $\phi(1)=1$ such that
$\tilde{\gamma}:=\gamma\circ\phi$ is $\ell(\gamma)$-Lipschitz.
Given $x,y\in{\rm X}$ and $\Lambda\geq 1$ we define the set of
$\Lambda$-quasigeodesics from $x$ to $y$ as

$\Gamma_{x,y}^{\Lambda}:=\\{\gamma\in C([0,1],{\rm
X})\,:\,\gamma(0)=x,\gamma(1)=y,\gamma\text{ is }\,\Lambda{\sf
d}(x,y)\text{-Lipschitz}\\}.$

Observe that, by definition, every $\gamma\in\Gamma_{x,y}^{\Lambda}$ satisfies
$\ell(\gamma)\leq\Lambda{\sf d}(x,y)$. Moreover every curve $\gamma$
satisfying $\ell(\gamma)\leq\Lambda{\sf d}(x,y)$ admits a reparametrization
belonging to $\Gamma_{x,y}^{\Lambda}$. The set $\Gamma_{x,y}^{\Lambda}$ is
compact if ${\rm X}$ is a proper metric space, as follows by Ascoli-Arzelà
Theorem. We also define $\Gamma_{x,y}:=\bigcup_{\Lambda\geq
1}\Gamma_{x,y}^{\Lambda}$. Notice that every rectifiable curve from $x$ to $y$
admits a reparametrization belonging to $\Gamma_{x,y}$.
The concatenation of two curves $\gamma,\eta\in C([0,1],{\rm X})$ with
$\eta(0)=\gamma(1)$ is the curve $\gamma\star\eta\in C([0,1],{\rm X})$ defined
as $(\gamma\star\eta)(t)=\gamma(2t)$ if $0\leq t\leq\frac{1}{2}$ and
$(\gamma\star\eta)(t)=\eta(2t-1)$ if $\frac{1}{2}\leq t\leq 1$. The
concatenation of two Lipschitz curves is still Lipschitz. In particular if
$x\neq y\neq z\neq x$ are points of ${\rm X}$ then
$\gamma\star\eta\in\Gamma_{x,y}$ for every $\gamma\in\Gamma_{x,z}$ and every
$\eta\in\Gamma_{z,y}$. Given a curve $\gamma\in C([0,1],{\rm X})$ we define
its reverse as $-\gamma\in C([0,1],{\rm X})$, where
$(-\gamma)(t)=\gamma(1-t)$. If $\gamma\in C([0,1],{\rm X})$ and $0\leq
s_{1}\leq s_{2}\leq 1$ then the restriction of $\gamma$ to $[s_{1},s_{2}]$ is
the curve $\gamma\lower 3.0pt\hbox{$|_{[s_{1},s_{2}]}$}\in C([0,1],{\rm X})$
defined by $\gamma\lower
3.0pt\hbox{$|_{[s_{1},s_{2}]}$}(t)=\gamma(s_{1}+t(s_{2}-s_{1}))$ if $s_{1}\leq
s_{2}$. If $\gamma$ is Lipschitz then also $\gamma\lower
3.0pt\hbox{$|_{[s_{1},s_{2}]}$}$ is Lipschitz.

For every Lipschitz curve $\gamma\colon[0,1]\to{\rm X}$ the function
$|\dot{\gamma}(\cdot)|\colon[0,1]\to[0,\infty)$ defined as
$|\dot{\gamma}(t)|:=\lim_{h\to 0}\frac{{\sf
d}(\gamma(t+h),\gamma(t))}{|h|}\in[0,\infty)$ is almost everywhere well
defined and it is called the metric speed of $\gamma$. In particular this
holds for all curves in $\Gamma_{x,y}$, for every $x,y\in{\rm X}$. Given a
Borel function $g\colon{\rm X}\to[0,\infty)$ and $\gamma\colon[0,1]\to{\rm X}$
Lipschitz we define

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

This quantity does not depend on the possible Lipschitz reparametrizations of
$\gamma$.
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
integral on the right is made along any Lipschitz reparametrization of
$\gamma$. The set of upper gradients of $u$ is denoted by ${\rm UG}(u)$.

###  2.2. Connectivity properties

In the sequel we will use often assume that our metric space $({\rm X},{\sf
d})$ satisfies some connectivity properties that we now define in detail. The
metric space $({\rm X},{\sf d})$ is said to be

  * -

path connected if for every $x,y\in{\rm X}$ there exists a continuous path
$\gamma\colon[0,1]\to{\rm X}$ with $\gamma(0)=x,\gamma(1)=y$;

  * -

rectifiable path connected if $\Gamma_{x,y}\neq\emptyset$ for every
$x,y\in{\rm X}$;

  * -

$\Lambda$-quasigeodesic if $\Gamma_{x,y}^{\Lambda}\neq\emptyset$ for every
$x,y\in{\rm X}$;

  * -

pointwise rectifiable path connected if for every $x\in{\rm X}$ there exists
$r_{x}>0$ such that $\Gamma_{x,y}\neq\emptyset$ for every $y\in B_{r_{x}}(x)$;

  * -

pointwise quasigeodesic at $x\in{\rm X}$ if there exist $r_{x}>0$ and
$\Lambda_{x}\geq 1$ such that $\Gamma_{x,y}^{\Lambda_{x}}\neq\emptyset$ for
every $y\in B_{r_{x}}(x)$;

  * -

pointwise quasigeodesic if it is pointwise quasigeodesic at every $x\in{\rm
X}$;

  * -

locally $\Lambda$-quasigeodesic if for every $x\in{\rm X}$ there exist
$r_{x}>0$ such that $\Gamma_{y,z}^{\Lambda}\neq\emptyset$ for every $y,z\in
B_{r_{x}}(x)$.

There are other possible variants but we will not need them. If $\Lambda=1$ in
the definitions involving quasigeodesics, then we say that the space is
(pointwise or locally or globally) geodesic.
There are trivial relations among the conditions. For instance if $({\rm
X},{\sf d})$ is pointwise quasigeodesic then it is pointwise rectifiable path
connected. Another easy one is the following.

######  Lemma 2.2.

Let $({\rm X},{\sf d})$ be a path connected, pointwise rectifiable path
connected metric space. Then it is rectifiable path connected.

###### Proof.

Let $x,y\in{\rm X}$ and $\gamma\colon[0,1]\to{\rm X}$ be a continuous path
such that $\gamma(0)=x$ and $\gamma(1)=y$. For every $t\in[0,1]$ let
$r_{\gamma(t)}>0$ be the number provided by the definition of pointwise
rectifiable path connectedness. By compactness of $[0,1]$ we can find a finite
set of points $\\{t_{0}=0,t_{1},\ldots,t_{N}=1\\}$ such that ${\sf
d}(\gamma(t_{i}),\gamma(t_{i+1}))\leq r_{\gamma(t_{i})}$. For every
$i=0,\ldots,N-1$ let $\eta_{i}\in\Gamma_{\gamma(t_{i}),\gamma(t_{i+1})}$ and
define $\eta=\eta_{0}\star\eta_{1}\star\cdots\star\eta_{N-1}$. Then
$\eta\in\Gamma_{x,y}$. ∎

Observe that there are locally geodesic spaces that are no connected.

###  2.3. Measure properties and Poincaré inequality

A metric measure space is said to be doubling if there exists $C_{D}>0$ such
that for every $x\in{\rm X}$ and $r>0$ we have

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

We refer to $C_{D}$ as the doubling constant or we say that $({\rm X},{\sf
d},\mathfrak{m})$ is $C_{D}$-doubling. 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.
We recall the definition of the (local) Hardy Littlewood maximal function.
Given $f\in L^{1}_{\rm loc}({\rm X})$ and $R>0$, we define

$M_{R}f(x):=\sup_{0<r<R}\mathchoice{{\vbox{\hbox{$\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)}|f|\,{\mathrm{d}}\mathfrak{m}$

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

(10) $\mathchoice{{\vbox{\hbox{$\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 $u\in{\rm Lip}({\rm X})$. This is equivalent to the following
condition: there exists $C_{P}\geq 1$ and $\lambda\geq 1$ such that

(11) $\mathchoice{{\vbox{\hbox{$\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)}g^{p}\,{\mathrm{d}}\mathfrak{m}\right)^{\frac{1}{p}}$

for every $u\colon{\rm X}\to\mathbb{R}$ Borel and every $g\in{\rm UG}(u)$. The
equivalence between these two different conditions is proved in [17, Theorem
2].

The Poincaré inequality on a doubling metric measure space has several
equivalent definitions that will be studied in big detail in the Appendix.
Most of them are expressed in terms of 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

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

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

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

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

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

It is a measure on ${\rm X}$ which is supported on $\overline{B}_{x,y}^{L}$.
It has been studied for instance in [15], [17] and [4]. We recall one of its
basic properties.

######  Lemma 2.3 ([15, p. 72], [4, Prop. 2.2]).

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

We state the pointwise version of the Poincaré inequality for doubling metric
measure spaces.

######  Proposition 2.4 ([15, Theorem 9.5], [17, Theorem 2]).

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

  * (i)

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

  * (ii)

$\exists C>0$, $L\geq 1$ such that for every $x,y\in{\rm X}$ and every
$u\colon{\rm X}\to\mathbb{R}$ Borel it holds

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

for every $g\in{\rm UG}(u)$.

  * (iii)

$\exists C>0$, $L\geq 1$ such that for every $x,y\in{\rm X}$ and every
$u\colon{\rm X}\to\mathbb{R}$ Lipschitz it holds

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

The pointwise estimate (15) remains quantitatively equivalent to (16) even if
we require it only for a fixed couple of points. But for that we need to
assume some connectivity property of the space, which are ensured for instance
when a global Poincaré inequality holds. The proof will be given in Appendix
A.

######  Lemma 2.5.

Let $({\rm X},{\sf d},\mathfrak{m})$ be a locally $\Lambda$-quasiconvex metric
measure space and let $x,y\in{\rm X}$. Then the following conditions are
quantitatively equivalent:

  * (i)

there exist $C>0,L\geq 1$ such that $|u(x)-u(y)|^{p}\leq C{\sf
d}(x,y)^{p-1}\int_{\rm X}g^{p}(z)\,{\mathrm{d}}\mathfrak{m}_{x,y}^{L}(z)$ for
every $u$ Borel and every $g\in{\rm UG}(u)$;

  * (ii)

there exist $C>0,L\geq 1$ such that $|u(x)-u(y)|^{p}\leq C{\sf
d}(x,y)^{p-1}\int_{\rm X}{\rm
lip}u(z)^{p}\,{\mathrm{d}}\mathfrak{m}_{x,y}^{L}(z)$ for every $u\in{\rm
Lip}({\rm X})$.

###  2.4. Minkowski content

We recall the definition of Minkowski content of a set.

######  Definition 2.6.

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

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

The Minkowski content satisfies a coarea type inequality.

######  Proposition 2.7 ([1, Lemma 3.2]).

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

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

##  3\. The $p$-set-connectedness and the $p$-set-pencil

The goal of this section is to introduce new natural conditions on a metric
measure space and show their relation with the Poincaré inequality. These
conditions are inspired by the one studied in [10] and [11]. Let $({\rm
X},{\sf d},\mathfrak{m})$ be a metric measure space. Given a set
$A\subseteq{\rm X}$ and a rectifiable curve $\gamma\colon[0,1]\to{\rm X}$ we
define the length of $\gamma$ in $A$ as

$\ell(\gamma\cap A):=\int_{\gamma}\chi_{A}\,{\mathrm{d}}s.$

Let $x,y\in{\rm X}$, $x\neq y$ and $A\subseteq{\rm X}$ Borel. We define the
$\Lambda$-width and the width of $A$ as

$0pt^{\Lambda}_{x,y}(A):=\inf_{\gamma\in\Gamma_{x,y}^{\Lambda}}\ell(\gamma\cap
A),\text{ and respectively
}0pt_{x,y}(A):=\inf_{\gamma\in\Gamma_{x,y}}\ell(\gamma\cap A),$

where $\Lambda\geq 1$. We use the usual convention $\inf\emptyset=+\infty$. In
order to simplify the notations we will often consider $0pt_{x,y}$ as the
limit case of $0pt_{x,y}^{\Lambda}$ with $\Lambda=\infty$. The same convention
is used in the following definition.

######  Definition 3.1.

Let $C>0$, $L\geq 1$ and $\Lambda\in[1,+\infty]$. We say that $({\rm X},{\sf
d},\mathfrak{m})$ is $(C,L,\Lambda)$ $p$-set-connected at $x,y\in{\rm X}$ if

(18) $\left(0pt_{x,y}^{\Lambda}(A)\right)^{p}\leq C{\sf
d}(x,y)^{p-1}\mathfrak{m}_{x,y}^{L}(A)\text{ for all }A\subseteq{\rm X}\text{
Borel.}$

If $\Lambda=+\infty$ we will simply say that $({\rm X},{\sf d},\mathfrak{m})$
is $(C,L)$ $p$-set-connected at $x,y\in{\rm X}$.
We say that $({\rm X},{\sf d},\mathfrak{m})$ has a $(C,L,\Lambda)$ $p$-set-
pencil between $x,y\in{\rm X}$ if there exists
$\alpha\in\mathscr{P}(\Gamma_{x,y}^{\Lambda})$, i.e. $\alpha$ is a probability
measure on the space $\Gamma_{x,y}^{\Lambda}$, such that

(19) $\left(\int\ell(\gamma\cap A)\,{\mathrm{d}}\alpha(\gamma)\right)^{p}\leq
C{\sf d}(x,y)^{p-1}\mathfrak{m}_{x,y}^{L}(A)\text{ for all }A\subseteq{\rm
X}\text{ Borel}.$

Also in this case if $\Lambda=+\infty$ we will simply say that $({\rm X},{\sf
d},\mathfrak{m})$ has a $(C,L)$ $p$-set-pencil between $x,y\in{\rm X}$.

These definitions are special cases of properties that are equivalent to the
pointwise Poincaré inequality as studied in Appendix A. For instance the
$(C,L,\Lambda)$ $p$-set-connectedness property correspond to condition (ii) of
Theorem A.3 applied to the functions $\chi_{A}$, when $\Lambda<+\infty$. When
$\Lambda=+\infty$ it is related to condition (iii) of Theorem A.3, which is
formulated in terms of $1$-Lipschitz functions. A similar relation holds
between the $p$-set-pencil conditions and conditions (iv) and (v) of Theorem
A.3.
There are easy relations between the notions introduced above and the validity
of the pointwise estimate (15).

######  Lemma 3.2.

Let $({\rm X},{\sf d},\mathfrak{m})$ be a path connected, pointwise
quasiconvex doubling metric measure space. Let $x,y\in{\rm X}$. Then

  * (i)

if $({\rm X},{\sf d},\mathfrak{m})$ has a $(C,L,\Lambda)$ $p$-set-pencil
between $x,y$ then it has a $(C,L)$ $p$-set-pencil between $x,y$;

  * (ii)

if $({\rm X},{\sf d},\mathfrak{m})$ has a $(C,L,\Lambda)$ $p$-set-pencil
between $x,y$ then it is $(C,L,\Lambda)$ $p$-set-connected at $x,y$;

  * (iii)

if $({\rm X},{\sf d},\mathfrak{m})$ is $(C,L,\Lambda)$ $p$-set-connected at
$x,y$ then it is $(C,L)$ $p$-set-connected at $x,y$;

  * (iv)

if (15) holds at $x,y\in{\rm X}$ then $({\rm X},{\sf d},\mathfrak{m})$ has a
$(C,L,\Lambda)$ $p$-set-pencil between $x,y$, quantitatively.

###### Proof.

The statements (i), (ii) and (iii) are trivial, so we just focus on (iv). We
assume that (15) holds at $x,y$. By item (iv) in Theorem A.3, we can find
$C>0,L\geq 1$ and $\alpha\in\mathscr{P}(\Gamma_{x,y}^{L})$ such that

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

for every $A\subseteq{\rm X}$ Borel. This gives the $(C,L,L)$ $p$-set-pencil
between $x$ and $y$. ∎

As a corollary, we have the following statement.

######  Corollary 3.3.

Let $({\rm X},{\sf d},\mathfrak{m})$ be a doubling metric measure space. If
$({\rm X},{\sf d},\mathfrak{m})$ satisfies a $p$-Poincaré inequality then
there exists $C>0,L,\Lambda\geq 1$ such that $({\rm X},{\sf d},\mathfrak{m})$
has a $(C,L,\Lambda)$ $p$-set-pencil between $x,y$ and it is $(C,L,\Lambda)$
$p$-set-connected at $x,y$, for every $x,y\in{\rm X}$.

###### Proof.

Since $({\rm X},{\sf d},\mathfrak{m})$ satisfies a $p$-Poincaré inequality
then (15) holds for every couple of points $x,y\in{\rm X}$ with constants
depending only on the constants of the Poincaré inequality and the doubling
constant, as recalled in Proposition 2.4. The thesis follows by Lemma 3.2. ∎

There is a partial converse of this corollary, essentially due to [11].

######  Proposition 3.4.

Let $({\rm X},{\sf d},\mathfrak{m})$ be a doubling metric measure space.
Suppose that there exist $C>0,L\geq 1$ such that it is $(C,L,L)$ $p$-set-
connected at $x,y$ for every $x,y\in{\rm X}$. Then it satisfies a $q$-Poincaré
inequality for all $q>p$.

Observe that the assumption $L=\Lambda$ in the statement is not restrictive.

###### Proof.

We claim that $({\rm X},{\sf d},\mathfrak{m})$ is
$(L^{\prime},C^{\prime},p)$-max connected in the sense of [11, Definition
2.12], where $L^{\prime}=2L+1$ and $C^{\prime}=(7CC_{D}L)^{\frac{1}{p}}$.
Let us take $A\subseteq{\rm X}$ with $\max\\{M_{(2L+1){\sf
d}(x,y)}(\chi_{A})(x),M_{(2L+1){\sf d}(x,y)}(\chi_{A})(y)\\}<\tau$,
$\tau\in(0,1]$. For every $\varepsilon>0$ we can find, by (18), a curve
$\gamma_{\varepsilon}\in\Gamma_{x,y}^{L}$ such that
$\ell(\gamma_{\varepsilon}\cap A)\leq C^{\frac{1}{p}}{\sf
d}(x,y)^{\frac{p-1}{p}}\mathfrak{m}_{x,y}^{L}(A)^{\frac{1}{p}}+\varepsilon$.
We recall the classical bound (see [15, p.72]): for every Borel $0\leq g$ and
$\rho>0$, we have
$\int_{B_{\rho}(x)}g\,R_{x}\,{\mathrm{d}}\mathfrak{m}\leq\rho M_{\rho}g(x)$
for every $x\in{\rm X}$. Applying it, we can estimate

(20) $\displaystyle\mathfrak{m}_{x,y}^{L}(A)$
$\displaystyle=\int_{B_{x,y}^{L}}\chi_{A}(z)(R_{x}(z)+R_{y}(z))\,{\mathrm{d}}\mathfrak{m}$
$\displaystyle\leq\int_{B_{(2L+1){\sf
d}(x,y)}(x)}\chi_{A}(z)R_{x}(z)\,{\mathrm{d}}\mathfrak{m}+\int_{B_{(2L+1){\sf
d}(x,y)}(y)}\chi_{A}(z)R_{y}(z)\,{\mathrm{d}}\mathfrak{m}$ $\displaystyle\leq
C_{D}(2L+1){\sf d}(x,y)\left(M_{(2L+1){\sf d}(x,y)}(\chi_{A})(x)+M_{(2L+1){\sf
d}(x,y)}(\chi_{A})(y)\right)$ $\displaystyle\leq 2C_{D}(2L+1){\sf
d}(x,y)\tau.$

So we get $\ell(\gamma_{\varepsilon}\cap
A)\leq(6CC_{D}L)^{\frac{1}{p}}\tau^{\frac{1}{p}}{\sf d}(x,y)+\varepsilon$,
where we used that $L\geq 1$ so $2L+1\leq 3L$. Choosing $\varepsilon$ small
enough we get $\ell(\gamma_{\varepsilon}\cap
A)\leq(7CC_{D}L)^{\frac{1}{p}}\tau^{\frac{1}{p}}{\sf d}(x,y)$. Set
$\gamma:=\gamma_{\varepsilon}$ for this particular choice of $\varepsilon$.
Then we have $\gamma\in\Gamma_{x,y}^{L}\subseteq\Gamma_{x,y}^{L^{\prime}}$ and
$\ell(\gamma\cap A)\leq C^{\prime}\tau^{\frac{1}{p}}{\sf d}(x,y)$. This shows
that $({\rm X},{\sf d},\mathfrak{m})$ is $(L^{\prime},C^{\prime},p)$-max
connected.
The fact that $({\rm X},{\sf d},\mathfrak{m})$ satisfies a $q$-Poincaré
inequality for all $q>p$ follows by [11, Theorem 2.19 & Lemma 2.20]. ∎

######  Remark 3.5.

For the proof of the $q$-Poincaré inequality using the methods of [11] two
things are needed: a bound to the length of the curves involved in the max-
connectedness property of [11], which corresponds to consider the version of
our $p$-set-connectedness with $\Lambda<\infty$, and the validity of (18) _for
all couple_ of points $x,y\in{\rm X}$. Both these properties are in contrast
with the equivalences we show in Appendix A. It is unclear to us if these two
properties are necessary to infer the validity of (15) for $q>p$ at fixed
points $x,y\in{\rm X}$. There are two difficulties in adapting the proof of
Theorem A.3 in this setting. First of all it is not clear how to get a
$p$-set-pencil from a $p$-set-connectedness. Secondly it is not clear if the
existence of a $p$-set-pencil _between $x,y$_ implies (15) for $q>p$ at $x,y$.

In the special case $p=1$ we can answer completely in the affirmative to the
questions raised in the previous remark, and the situation is completely
similar to what we see in Appendix A. In particular in the next theorem, which
is the setwise analogous of Theorem A.3, we see how the equivalences work if
we only know the conditions at two points $x,y$. This is completely new since,
as recalled above, the methods of [11] do not provide a $1$-Poincaré
inequality under a $(L,C,1)$-max connectedness assumption, but only a
$q$-Poincaré inequality for every $q>1$. The other difference, as already
remarked, is that we do not need that the assumption holds for every couple of
points, but just for a fixed one.

######  Theorem 3.6.

Let $({\rm X},{\sf d},\mathfrak{m})$ be a path connected, locally
$\Lambda$-quasiconvex, doubling metric measure space and let $x,y\in{\rm X}$.
The following conditions are quantitatively equivalent:

  * (i)

$({\rm X},{\sf d},\mathfrak{m})$ satisfies (15) at $x,y$ for $p=1$;

  * (ii)

$({\rm X},{\sf d},\mathfrak{m})$ is $(C,L,\Lambda^{\prime})$ $1$-set-connected
at $x,y$;

  * (iii)

$({\rm X},{\sf d},\mathfrak{m})$ is $(C,L)$ $1$-set-connected at $x,y$;

  * (iv)

$({\rm X},{\sf d},\mathfrak{m})$ has a $(C,L,\Lambda^{\prime})$ $1$-set-pencil
between $x,y$;

  * (v)

$({\rm X},{\sf d},\mathfrak{m})$ has a $(C,L)$ $1$-set-pencil between $x,y$.

The proof is a consequence of Theorem 1.4.

###### Proof.

By Lemma 3.2 we know that (i) implies (iv), (iv) implies (v) and (ii), (v)
implies (ii) and (v) implies (iii). So it is enough to show that (iii) implies
(i). For this it is enough to use Theorem 1.4. ∎

##  4\. The proof of Theorem 1.4

In this section we prove Theorem 1.4. The $1$-set connectedness can be
rephrased in the following way. For every subset $A\subseteq{\rm X}$ we define
its $\Lambda$-separating ratio between $x,y$ as

${\rm
SR}_{x,y}^{\Lambda}(A):=\frac{\mathfrak{m}_{x,y}^{L}(A)}{0pt^{\Lambda}_{x,y}(A)}.$

We also use the conventions that ${\rm SR}^{\Lambda}_{x,y}(A)=0$ if
$0pt^{\Lambda}_{x,y}(A)=+\infty$, that happens if and only if
$\Gamma^{\Lambda}_{x,y}=\emptyset$, and that ${\rm
SR}^{\Lambda}_{x,y}(A)=+\infty$ if $0pt_{x,y}^{\Lambda}(A)=0$. As usual when
$\Lambda=\infty$ we simply omit it from the notation.
So $({\rm X},{\sf d},\mathfrak{m})$ is $(C,L,\Lambda)$ $1$-set-connected if
and only if

(21) $\inf_{A\subseteq{\rm X}}{\rm SR}_{x,y}^{\Lambda}(A)\geq\frac{1}{C}.$

On the other hand in [4, Thm. 1.2] the authors showed that the $1$-Poincaré
inequality is equivalent to a lower bound on the Minkowski content of all
separating sets. For the definition of separating sets we refer to Definition
1.1. More precisely it is there proved that a $1$-Poincaré inequality holds if
and only if there exists $c>0$ and $L\geq 1$ such that for every $x,y\in{\rm
X}$ one has

(22)
$\inf_{\Omega\in\SS_{\text{top}}(x,y)}(\mathfrak{m}_{x,y}^{L})^{+}(\Omega)\geq
c.$

The same statement is true if the condition hold only at two fixed points
$x,y$. This is the content of Theorem 1.4 that we reproduce here for the
reader’s convenience.

######  Theorem 1.4.

Let $({\rm X},{\sf d},\mathfrak{m})$ be a path connected, locally
$\Lambda$-quasiconvex, doubling metric measure space and let $x,y\in{\rm X}$.
Then the following conditions are quantitatively equivalent:

  * (i)

there exist $C>0$, $L\geq 1$ such that

(23) $|u(x)-u(y)|\leq C\int{\rm
lip}u\,{\mathrm{d}}\mathfrak{m}_{x,y}^{L}\qquad\text{for all }u\in{{\rm
Lip}}({\rm X});$

  * (ii)

there exist $C>0$, $L\geq 1$ such that

(24) $|u(x)-u(y)|\leq C\int
g\,{\mathrm{d}}\mathfrak{m}_{x,y}^{L}\qquad\text{for all }u\text{ Borel and
all }g\in{\rm UG}(u);$

  * (iii)

the space $({\rm X},{\sf d},\mathfrak{m})$ is $(C,L)$ $1$-set connected at
$x,y$, i.e. $\inf_{A\subseteq{\rm X}}{\rm SR}_{x,y}(A)\geq C^{-1}$;

  * (iv)

there exist $c>0$, $L\geq 1$ such that $\inf_{A\subseteq{\rm X},\,A\textup{
closed}}{\rm SR}_{x,y}(A)\geq C^{-1}$;

  * (v)

the space $({\rm X},{\sf d},\mathfrak{m})$ satisfies (BMC)x,y, i.e.
$\inf_{\Omega\in\SS_{\textup{top}}(x,y)}(\mathfrak{m}_{x,y}^{L})^{+}(\Omega)\geq
c$ for some $L\geq 1$ and $c>0$.

###### Proof.

The equivalence between (i) and (ii) is proved in Theorem A.3, see also Lemma
2.5. By Lemma 3.2 we know that (ii) implies (iii). It is also clear that (iii)
implies (iv). We assume (iv) and we take a separating set
$\Omega\in\SS_{\textup{top}}(x,y)$. If
$(\mathfrak{m}_{x,y}^{L})^{+}(\Omega)=+\infty$ there is nothing to prove, so
we suppose $(\mathfrak{m}_{x,y}^{L})^{+}(\Omega)<+\infty$, which implies
$\mathfrak{m}_{x,y}^{L}(\partial\Omega)=0$. Consider the set
$A_{r}:=\overline{B}_{r}(\Omega)\setminus\textup{int}(\Omega)$, which is a
closed subset of ${\rm X}$. Observe that $0pt_{x,y}(A_{r})\geq r$ if
$r<\min\\{{\sf d}(\partial\Omega,x),{\sf d}(\partial\Omega,y)\\}$. Indeed
every curve $\gamma\in\Gamma_{x,y}$ has to go from the inside of $\Omega$ to
the outside of $\overline{B}_{r}(\Omega)$, because of our choice of $r$.
Therefore its length inside $A_{r}$ is at least $r$. We can now compute

$(\mathfrak{m}_{x,y}^{L})^{+}(\Omega)=\varliminf_{r\to
0}\frac{\mathfrak{m}_{x,y}^{L}(\overline{B}_{r}(\Omega)\setminus\Omega)}{r}=\varliminf_{r\to
0}\frac{\mathfrak{m}_{x,y}^{L}(A_{r})}{r}\geq\varliminf_{r\to
0}\frac{\mathfrak{m}_{x,y}^{L}(A_{r})}{0pt_{x,y}(A_{r})}\geq\inf_{\begin{subarray}{c}A\subseteq{\rm
X}\\\ A\text{ closed}\end{subarray}}{\rm SR}_{x,y}(A)\geq c.$

The second equality follows by the fact that
$\mathfrak{m}_{x,y}^{L}(\partial\Omega)=0$, while the last inequality follows
by our assumption (iv). Since this is true for every
$\Omega\in\SS_{\textup{top}}(x,y)$ we get (v).
The proof of (v) implies (i) is exactly the proof of the last implication in
[4, Theorem 6.1]. For reader’s convenience we report it here. Let $u\in{\rm
Lip}({\rm X})$ and let $x,y\in{\rm X}$. We can assume that $u(x)<u(y)$
otherwise there is nothing to prove. The sets $\Omega_{t}:=\\{u\geq t\\}$
belong to $\SS_{\text{top}}(x,y)$ for all $t\in(u(x),u(y))$. We can apply the
coarea inequality (17) with respect to the measure $\mathfrak{m}_{x,y}^{L}$ to
get

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

Therefore (i) follows with $C=1/c$. ∎

######  Remark 4.1.

We stress that the assumption that the metric space $({\rm X},{\sf d})$ is
path connected and locally $\Lambda$-quasiconvex only enters in the proof of
(i) implies (ii) and (ii) implies (iii).

###### Proof of Corollary 1.5.

If $({\rm X},{\sf d},\mathfrak{m})$ satisfies a $1$-Poincaré inequality then
$({\rm X},{\sf d})$ is $\Lambda$-quasiconvex ([16, Proposition 8.3.2]) and
(15) holds. So $({\rm X},{\sf d},\mathfrak{m})$ is $(C,L)$ $1$-set-connected
at every couple of points by Theorem 1.4. Viceversa by the proof of Theorem
1.4 and Remark 4.1 we conclude that if $({\rm X},{\sf d},\mathfrak{m})$ is
$(C,L)$ $1$-set-connected at every couple of points then it satisfies (16) at
every couple of points. By Proposition 2.4 this implies that $({\rm X},{\sf
d},\mathfrak{m})$ satisfies a $1$-Poincaré inequality. ∎

##  5\. Relation between Minkowski content and separating ratio

In the previous section, in particular in Theorem 1.4, we proved the
equivalence between the pointwise estimate (15) at two fixed points $x,y$ and
a positive lower bound on either the separating ratio as expressed in (21) or
the Minkowski content of topological separating sets as expressed in (22). In
particular a key step in the proof of Theorem 1.4 was to show that the
following inequality is always true:

(25)
$\inf_{\Omega\in\SS_{\textup{top}}(x,y)}(\mathfrak{m}_{x,y}^{L})^{+}(\Omega)\geq\inf_{A\subseteq{\rm
X}\,\text{closed}}{\rm SR}_{x,y}(A).$

The goal of this section is to study in detail these two infima. In this
section we do not need assumptions on the measure $\mathfrak{m}$, except that
it has compact support. So we will deal with the following general situation:
$({\rm X},{\sf d},\mathfrak{m})$ is a metric measure space and, for given
$x,y\in{\rm X}$ and $A\subseteq{\rm X}$, we set

${\rm SR}_{x,y}(A):=\frac{\mathfrak{m}(A)}{0pt_{x,y}(A)}$

with the same conventions as above, i.e. ${\rm SR}_{x,y}(A)=0$ if
$0pt_{x,y}(A)=+\infty$ and ${\rm SR}_{x,y}(A)=+\infty$ if $0pt_{x,y}(A)=0$.
The relevant application is when the measure is $\mathfrak{m}_{x,y}^{L}$ for
some doubling measure $\mathfrak{m}$ and some $L\geq 1$.

###  5.1. Path separating sets

In order to proceed in this direction it is useful to define another class of
separating sets that we call path-separating sets. It is the class

$\SS_{\textup{path}}(x,y):=\\{A\subseteq{\rm
X}\,:\,A\cap\gamma\neq\emptyset\text{ for all }\gamma\in\Gamma_{x,y}\\}.$

For completeness we discuss the relation between topological separating sets
and closed path-separating sets. This is not strictly necessary for the proof
of the equivalence between the minima of the two functionals in Section 5.3.

######  Lemma 5.1.

Let $({\rm X},{\sf d})$ be a metric space and let $x,y\in{\rm X}$. If
$\Omega\in\SS_{\textup{top}}(x,y)$ then
$\partial\Omega\in\SS_{\textup{path}}(x,y)$.

###### Proof.

Let $\Omega\in\SS_{\text{top}}(x,y)$ and let $\gamma\in\Gamma_{x,y}$. The sets
$\textup{int}(\Omega)$ and $\Omega^{c}$ are open, so are
$[0,1]\cap\gamma^{-1}(\textup{int}(\Omega))$ and
$[0,1]\cap\gamma^{-1}(\Omega^{c})$. They cannot cover the whole $[0,1]$ by
connectedness. So there exists $t\in[0,1]$ such that
$\gamma(t)\in\partial\Omega$. This shows that
$\partial\Omega\cap\gamma\neq\emptyset$ for all $\gamma\in\Gamma_{x,y}$, i.e.
$\partial\Omega\in\SS_{\text{path}}(x,y)$. ∎

The next is a partial converse of this statement.

######  Proposition 5.2.

Let $({\rm X},{\sf d})$ be a pointwise quasiconvex metric space and let
$x,y\in{\rm X}$. If $A\in\SS_{\textup{path}}(x,y)$ is closed and satisfies
$\min\\{{\sf d}(x,A),{\sf d}(y,A)\\}>0$ then there exists
$\Omega\in\SS_{\textup{top}}(x,y)$ such that $\partial\Omega\subseteq A$.

###### Proof.

We claim that the set $\Omega$ defined as the closure of

$\tilde{\Omega}:=\\{z\in{\rm X}\,:\,\exists\gamma\in\Gamma_{x,z}\text{ such
that }\gamma\cap A=\emptyset\\}$

satisfies the thesis. First of all $\tilde{\Omega}$ is open. Indeed let
$z\in\tilde{\Omega}$, let $\gamma\in\Gamma_{x,z}$ such that $\gamma\cap
A=\emptyset$ and let $r={\sf d}(z,A)>0$ since $A$ is closed. Let $\Lambda_{z}$
and $r_{z}$ be the constants of the pointwise quasiconvexity at $z$. We claim
that $B_{r^{\prime}}(z)\subseteq\tilde{\Omega}$ if
$r^{\prime}<\min\\{r_{z},\frac{r}{\Lambda_{z}}\\}$. For this take $w\in
B_{r^{\prime}}(z)$ and a curve $\gamma_{w}\in\Gamma_{z,w}^{\Lambda_{z}}$.
Observe that $\gamma_{w}\cap A=\emptyset$ since ${\sf
d}(\gamma_{w}(t),z)\leq\ell(\gamma_{w})\leq\Lambda_{z}r^{\prime}<r$, so
$\gamma_{w}(t)\notin A$ for every $t$. The concatenation
$\gamma\star\gamma_{w}$ belongs to $\Gamma_{x,w}$ and does not intersect $A$,
i.e. $w\in\tilde{\Omega}$.
Secondly $x\in\tilde{\Omega}$ because ${\sf d}(x,A)>0$, so $\tilde{\Omega}$
contains a small ball around $x$ since it is open.
Hence the claim follows if we prove that
$\partial\Omega=\partial\tilde{\Omega}\subseteq A$. Indeed if this is the case
then $y\in\tilde{\Omega}^{c}$ and ${\sf d}(y,\partial\Omega)\geq{\sf
d}(y,A)>0$, showing that there is a small ball around $y$ contained in
$\Omega^{c}$.
It remains to prove that $\partial\tilde{\Omega}\subseteq A$. Let $z$ be a
point of $\partial\tilde{\Omega}$. By definition we can find points
$z_{n}\in\tilde{\Omega}$ and $w_{n}\in\tilde{\Omega}^{c}$ converging to $z$.
Let $r_{z},\Lambda_{z}$ be the constants of the pointwise quasiconvexity at
$z$. We can suppose that $n$ is big enough to have $z_{n},w_{n}\in
B_{r_{z}}(z)$. Hence for every $n$ we fix a curve
$\gamma_{n}\in\Gamma_{x,z_{n}}$ such that $\gamma_{n}\cap A=\emptyset$, a
curve $\eta_{n}\in\Gamma_{z_{n},z}^{\Lambda_{z}}$ and a curve
$\xi_{n}\in\Gamma_{z,w_{n}}^{\Lambda_{z}}$. Finally we consider the
concatenation $\beta_{n}=\gamma_{n}\star\eta_{n}\star\xi_{n}$. By definition
$\beta_{n}\in\Gamma_{x,w_{n}}$, so $\beta_{n}\cap A\neq\emptyset$. Since
$\gamma_{n}\cap A=\emptyset$ we can find a point $v_{n}$ of $A$ on the curve
$\eta_{n}\star\xi_{n}$. By definition ${\sf
d}(z,v_{n})\leq\Lambda_{z}\max\\{{\sf d}(z,z_{n}),{\sf d}(z,w_{n})\\}\to 0$ as
$n$ goes to infinity. This implies that $z\in A$ since $A$ is closed. ∎

###  5.2. The position function

For us, the main object associated to a set is its position function. For a
given set $A\subseteq{\rm X}$, a Lipschitz curve $\gamma\in C([0,1],{\rm X})$
and two parameters $s_{1}\leq s_{2}\in[0,1]$, we define

(26) $\ell(\gamma\cap A,s_{1},s_{2}):=\ell(\gamma\lower
3.0pt\hbox{$|_{[s_{1},s_{2}]}$}\cap
A)=\int_{s_{1}}^{s_{2}}\chi_{A}(\gamma(t))|\dot{\gamma}(t)|\,{\mathrm{d}}t.$

The second equality follows from the definition of $\gamma\lower
3.0pt\hbox{$|_{[s_{1},s_{2}]}$}$. Also the next useful property follows
directly from the definitions:

(27) $\ell((\gamma\star\eta)\cap A,s_{1},s_{2})=\ell(\gamma\cap
A,\min\\{1,2s_{1}\\},\min\\{1,2s_{2}\\})+\ell(\eta\cap
A,\max\\{0,2s_{1}-1\\},\max\\{0,2s_{2}-1\\}).$

For simplicity, we also define $\ell(\gamma\cap A,s):=\ell(\gamma\cap A,0,s)$.
In this case the formula above reduces to

(28) $\ell((\gamma\star\eta)\cap A,s)=\ell(\gamma\cap
A,0,\min\\{1,2s\\})+\ell(\eta\cap A,0,\max\\{0,2s-1\\}).$

For $A\subseteq{\rm X}$ and $\gamma\in\Gamma_{x,y}$ we define the _position
function along the path $\gamma$ of the set $A$_ as

(29) ${\rm pos}_{\gamma,A}\colon{\rm X}\to[0,\infty],\qquad{\rm
pos}_{\gamma,A}(z):=\inf_{s\in\gamma^{-1}(z)}\ell(\gamma\cap A,s).$

with the usual convention that ${\rm pos}_{\gamma,A}(z)=\infty$ if
$z\notin{\rm Im}(\gamma)$. In other words ${\rm pos}_{\gamma,A}(z)$ is the
length spent by $\gamma$ inside $A$ before reaching for the first time the
point $z$. We define the _position function with respect to the set $A$_ as

(30) ${\rm pos}_{A}\colon{\rm X}\to[0,\infty],\qquad{\rm
pos}_{A}(z):=\inf_{\gamma\in\Gamma_{x,y}}{\rm pos}_{\gamma,A}(z).$

Therefore the quantity ${\rm pos}_{A}(z)$ denotes the minimal length that a
curve between $x$ and $y$ has to spend inside $A$ before passing through $z$.
The position function satisfies an interesting property: it allows to
naturally fibrate a set into path-separating subsets by looking at its level
sets. In order to prove it we introduce the following useful notation. Let $A$
and $\gamma\in\Gamma_{x,y}$, in particular $\gamma$ is Lipschitz. For
$\tau\in[0,\ell(\gamma\cap A)]$, we define $(\gamma\cap A)_{\tau}\in{\rm X}$
as

(31) $(\gamma\cap A)_{\tau}=\gamma(\bar{s}),\qquad\text{where
}\bar{s}:=\min\\{s:\,\ell(\gamma\cap A,s)\geq\tau\\}.$

We remark that, since $\gamma$ is Lipschitz then the function
$s\mapsto\ell(\gamma\cap A,s)$ is continuous and the minimum above is well
defined. In the next proposition we will use the properties of the function
$[0,\ell(\gamma\cap A)]\ni\tau\mapsto{\rm pos}_{A}((\gamma\cap
A)_{\tau})\in[0,\infty]$. Figure 2 illustrates the behaviour of this function
in a specific example.

Figure 2. Let us consider $\gamma$ and $A\subseteq\mathbb{R}^{2}$ as in the
picture. In such a case the width of $A$ with respect to $x,y$ is equal to
$\delta_{1}+\delta_{2}$ and in particular is realized by the straight curve
connecting $x$ to $y$. The function ${\rm pos}_{A}((\gamma\cap A)_{\tau})$
computes the position of the point reached when the curve travels a piece of
length $\tau$ inside $A$. Its graph is reported in the picture. In this case
the level set of the position function ${\rm pos}_{A}$ fibrates the set $A$ by
straight vertical lines.

######  Proposition 5.3.

Let $({\rm X},{\sf d})$ be a metric space and let $x,y\in{\rm X}$. Let
$A\in\SS_{\textup{path}}(x,y)$ be closed. Then the sets $\\{{\rm
pos}_{A}=t\\}\cap A$ belong to $\SS_{\textup{path}}(x,y)$ for all
$t\in[0,0pt_{x,y}(A)]$.

We will see in the proof that the assumption $A\in\SS_{\textup{path}}(x,y)$ is
needed in order to guarantee that $\\{{\rm pos}_{A}=t\\}\cap A$ is not empty
for all $t\in[0,0pt_{x,y}(A)]$. The assumption that $A$ is closed is made in
order to simplify the proof. A similar statement holds for every separating
set.

###### Proof.

If $\Gamma_{x,y}=\emptyset$ then every subset of ${\rm X}$ is a path
separating set, thus the statement is trivial. So we can assume that
$\Gamma_{x,y}\neq\emptyset$. Let $A\in\SS_{\text{path}}(x,y)$ be closed and
${\rm pos}_{A}\colon{\rm X}\to[0,+\infty]$ be its position function.

Claim 1: For every $\gamma\in\Gamma_{x,y}$ there exists $z\in\gamma\cap A$
such that ${\rm pos}_{A}(z)\geq 0pt_{x,y}(A)$.
We define $s_{\max}:=\max\\{s:\,s\in\gamma^{-1}(A)\\}$, which is well defined
since $\gamma\cap A\neq\emptyset$ and $A$ is closed. We prove that the point
$z:=\gamma(s_{\max})$ verifies the claim. By contradiction we suppose that
${\rm pos}_{A}(z)<0pt_{x,y}(A)$. For every $\varepsilon>0$ we consider a path
$\gamma_{\varepsilon}\in\Gamma_{x,y}$ such that ${\rm
pos}_{\gamma_{\varepsilon},A}(z)\leq{\rm pos}_{A}(z)+\varepsilon$. By
definition $z=\gamma_{\varepsilon}(s_{\varepsilon})$ for some
$s_{\varepsilon}$ such that

(32) $\ell(\gamma_{\varepsilon}\cap A,s_{\varepsilon})<{\rm
pos}_{\gamma_{\varepsilon},A}(z)+\varepsilon.$

We define $\eta_{\varepsilon}=\gamma_{\varepsilon}\lower
3.0pt\hbox{$|_{[0,s_{\varepsilon}]}$}\star\gamma\lower
3.0pt\hbox{$|_{[s_{\max},1]}$}\in\Gamma_{x,y}$. We compute, using (26) and
(28),

(33) $\ell(\eta_{\varepsilon}\cap A)=\ell(\eta_{\varepsilon}\cap
A,0,1)=\ell(\gamma_{\varepsilon}\lower
3.0pt\hbox{$|_{[0,s_{\varepsilon}]}$}\cap A)+\ell(\gamma\lower
3.0pt\hbox{$|_{[s_{\max},1]}$}\cap A)\leq{\rm pos}_{A}(z)+2\varepsilon,$

because $\ell(\gamma\lower 3.0pt\hbox{$|_{[s_{\max},1]}$}\cap A)=0$ by
definition of $s_{\max}$. Therefore

(34) $0pt_{x,y}(A)\leq\ell(\eta_{\varepsilon}\cap A)\leq{\rm
pos}_{A}(z)+2\varepsilon.$

If $\varepsilon$ is chosen to be sufficiently small, we have that ${\rm
pos}_{A}(z)+2\varepsilon<0pt_{x,y}(A)$, thus giving a contradiction.

Claim 2: for every $\gamma\in\Gamma_{x,y}$ we have

(35) $|{\rm pos}_{A}((\gamma\cap A)_{\tau_{1}})-{\rm pos}_{A}((\gamma\cap
A)_{\tau_{2}})|\leq|\tau_{1}-\tau_{2}|,\,\text{for
}0\leq\tau_{1},\tau_{2}\leq\ell(\gamma\cap A).$

We fix $\tau_{1},\tau_{2}\in[0,\ell(\gamma\cap A)]$ and we assume that
$\tau_{1}\geq\tau_{2}$. By definition $(\gamma\cap
A)_{\tau_{1}}=\gamma({\bar{s}_{1}})$ and $(\gamma\cap
A)_{\tau_{2}}=\gamma({\bar{s}_{2}})$ for some
${\bar{s}_{1}}\geq{\bar{s}_{2}}\in[0,1]$. We split the claim in two parts. We
first prove that ${\rm pos}_{A}((\gamma\cap A)_{\tau_{1}})-{\rm
pos}_{A}((\gamma\cap A)_{\tau_{2}})\leq\tau_{1}-\tau_{2}$. For every
$\varepsilon$, we consider $\eta_{\varepsilon}\in\Gamma_{x,y}$ and
$s_{\varepsilon}\in[0,1]$ such that $(\gamma\cap
A)_{\tau_{2}}=\eta_{\varepsilon}(s_{\varepsilon})$ and

$\ell(\eta_{\varepsilon}\cap A,s_{\varepsilon})\leq{\rm pos}_{A}((\gamma\cap
A)_{\tau_{2}})+\varepsilon.$

We define $\gamma_{\varepsilon}=\eta_{\varepsilon}\lower
3.0pt\hbox{$|_{[0,s_{\varepsilon}]}$}\star\gamma\lower
3.0pt\hbox{$|_{[\bar{s}_{2},1]}$}\in\Gamma_{x,y}$. We now define
$\bar{t}_{1}:=\frac{\bar{s}_{1}-\bar{s}_{2}}{1-\bar{s}_{2}}$. It satisfies
$\gamma\lower
3.0pt\hbox{$|_{[\bar{s}_{2},1]}$}(\bar{t}_{1})=\gamma(\bar{s}_{1})=(\gamma\cap
A)_{\tau_{1}}$. Finally if we set $\tilde{t}_{1}=\frac{1+\bar{t}_{1}}{2}$ we
get $\gamma_{\varepsilon}(\tilde{t}_{1})=(\gamma\cap A)_{\tau_{1}}$. Now we
compute, using again (28) and the definition of the points $(\gamma\cap
A)_{\tau_{1}},(\gamma\cap A)_{\tau_{2}}$,

(36) $\displaystyle{\rm pos}_{A}((\gamma\cap
A)_{\tau_{1}})\leq\ell(\gamma_{\varepsilon}\cap A,0,\tilde{t}_{1})$
$\displaystyle=\ell(\eta_{\varepsilon}\cap
A,0,s_{\varepsilon})+\ell(\gamma\cap A,\bar{s}_{2},\bar{s}_{1})$
$\displaystyle\leq{\rm pos}_{A}((\gamma\cap
A)_{\tau_{2}})+\varepsilon+\tau_{1}-\tau_{2}.$

By taking the limit as $\varepsilon\to 0$, we conclude that ${\rm
pos}_{A}((\gamma\cap A)_{\tau_{1}})-{\rm pos}_{A}((\gamma\cap
A)_{\tau_{2}})\leq\tau_{1}-\tau_{2}$.
For the other inequality the procedure is similar and we just sketch it. For
every $\varepsilon>0$ we consider $\eta_{\varepsilon}\in\Gamma_{x,y}$ and
$s_{\varepsilon}\in[0,1]$ such that $(\gamma\cap
A)_{\tau_{1}}=\eta_{\varepsilon}(s_{\varepsilon})$ and

$\ell(\eta_{\varepsilon}\cap A,s_{\varepsilon})\leq{\rm pos}_{A}((\gamma\cap
A)_{\tau_{1}})+\varepsilon.$

This time we define the curve
$\gamma_{\varepsilon}=\eta_{\varepsilon}\star(-\gamma\lower
3.0pt\hbox{$|_{[\bar{s}_{2},\bar{s}_{1}]}$})\star\gamma\lower
3.0pt\hbox{$|_{[\bar{s}_{2},1]}$}\in\Gamma_{x,y}$ and arguing as above we
arrive to

(37) $\displaystyle{\rm pos}_{A}((\gamma\cap A)_{\tau_{2}})\leq{\rm
pos}_{A}((\gamma\cap A)_{\tau_{1}})+\varepsilon+\tau_{1}-\tau_{2}.$

By taking $\varepsilon\to 0$ we conclude the proof of the claim.

Claim 3: $\\{{\rm pos}_{A}=t\\}\in\SS_{\text{path}}(x,y)$ for every
$t\in[0,0pt_{x,y}(A)]$.
Let us fix $t_{0}\in[0,0pt_{x,y}(A)]$. We have to prove that
$\gamma\cap\\{{\rm pos}_{A}=t_{0}\\}\neq\emptyset$ for every
$\gamma\in\Gamma_{x,y}$. If $t_{0}=0$ the claim is true because every
$\gamma\in\Gamma_{x,y}$ intersects $A$, since $A\in\SS_{\text{path}}(x,y)$ and
$(\gamma\cap A)_{0}$ satisfies ${\rm pos}_{A}((\gamma\cap A)_{0})=0$. The
function

$[0,\ell(\gamma\cap A)]\ni\tau\mapsto{\rm pos}_{A}((\gamma\cap
A)_{\tau})\in[0,\infty)$

is $1$-Lipschitz by Claim 2. Moreover, by Claim 1, there exists a point
$z\in\gamma\cap A$ such that ${\rm pos}_{A}(z)\geq 0pt_{x,y}(A)$, in
particular there exists $\tau_{\max}\in[0,\ell(\gamma\cap A)]$ such that ${\rm
pos}_{A}((\gamma\cap A)_{\tau_{\max}})\geq 0pt_{x,y}(A)$. Thus, by mean value
theorem, there exists a $\tau\in[0,\tau_{\max}]$ such that ${\rm
pos}_{A}((\gamma\cap A)_{\tau})=t_{0}$. ∎

In general we cannot use this decomposition unless we know more properties of
the position function of separating sets. The next proposition shows that it
is more regular under some connectedness properties of the space.

######  Proposition 5.4.

Let $({\rm X},{\sf d})$ be a rectifiable path connected metric space and let
$x,y\in{\rm X}$. Let $A\subseteq{\rm X}$. Then

  * (i)

${\rm pos}_{A}(z)<+\infty$ for every $z\in{\rm X}$;

  * (ii)

if ${\rm X}$ is pointwise $\Lambda_{z}$-quasiconvex at $z\in{\rm X}$ then
${\rm lip}({\rm pos}_{A})(z)\leq\Lambda_{z}$;

  * (iii)

if ${\rm X}$ is pointwise quasiconvex then ${\rm pos}_{A}$ is continuous and
${\rm lip}({\rm pos}_{A})=0$ on $\overline{A}^{c}$;

  * (iv)

if ${\rm X}$ is (locally) $\Lambda$-quasiconvex then ${\rm pos}_{A}$ is
(locally) $\Lambda$-Lipschitz.

###### Proof.

The first assertion is trivial, so we move to the second one. Let us fix
$z\in{\rm X}$ and $r_{z}>0$ such that
$\Gamma_{z,w}^{\Lambda_{z}}\neq\emptyset$ for every point $w\in B_{r_{z}}(z)$.
Let us fix $w\in B_{r_{z}}(z)$ and $\varepsilon>0$. Let
$\eta_{\varepsilon}\in\Gamma_{x,y}$ and $s_{\varepsilon}\in[0,1]$ be such that
$z=\eta_{\varepsilon}(s_{\varepsilon})$ and ${\rm
pos}_{A}(z)\leq\ell(\eta_{\varepsilon}\cap A,s_{\varepsilon})+\varepsilon$. We
also take a curve $\gamma_{w}\in\Gamma_{z,w}^{\Lambda_{z}}$. We define the
curve $\gamma=\eta_{\varepsilon}\lower
3.0pt\hbox{$|_{[0,s_{\varepsilon}]}$}\star\gamma_{w}\star(-\gamma_{w})\star\eta_{\varepsilon}\lower
3.0pt\hbox{$|_{[s_{\varepsilon},1]}$}\in\Gamma_{x,y}$. By the usual
application of (28) we get

${\rm pos}_{A}(w)\leq\ell(\eta_{\varepsilon}\cap
A,s_{\varepsilon})+\Lambda_{z}{\sf d}(z,w)\leq{\rm pos}_{A}(z)+\Lambda_{z}{\sf
d}(z,w)+\varepsilon,$

and, by the arbitrariness of $\varepsilon$,

${\rm pos}_{A}(w)\leq{\rm pos}_{A}(z)+\Lambda_{z}{\sf d}(z,w).$

Similarly we also obtain

${\rm pos}_{A}(z)\leq{\rm pos}_{A}(w)+\Lambda_{z}{\sf d}(z,w).$

Therefore we can prove (ii) since

${\rm lip}({\rm pos}_{A})(z)=\lim_{r\to 0}\sup_{w\in B_{r}(z)}\frac{|{\rm
pos}_{A}(w)-{\rm pos}_{A}(z)|}{{\sf d}(z,w)}\leq\Lambda_{z}.$

The same technique shows (iv). The continuity statement in (iii) can be proved
exactly as in [8, Lemma 2.2]. It remains to show that ${\rm lip}({\rm
pos}_{A})(z)=0$ if $z\in\overline{A}^{c}$, in case ${\rm X}$ is pointwise
quasiconvex. Let $r_{z},\Lambda_{z}$ be as in the definition of pointwise
quasiconvexity at $z$. For every $\varepsilon>0$ let
$\gamma_{\varepsilon}\in\Gamma_{x,y}$ and $s_{\varepsilon}$ such that
$z=\gamma_{\varepsilon}(s_{\varepsilon})$ and $\ell(\gamma\cap
A,s_{\varepsilon})\leq{\rm pos}_{A}(z)+\varepsilon$. Take
$r<\min\\{r_{z},\frac{{\sf d}(z,\overline{A})}{\Lambda_{z}}\\}$. For every
$w\in B_{r}(z)$ take a curve $\gamma_{w}\in\Gamma_{z,w}^{\Lambda_{z}}$. By
construction $\gamma_{w}$ avoids $A$ and so, arguing as in the first part of
the proof, we deduce that ${\rm pos}_{A}(z)={\rm pos}_{A}(w)$. Since this is
true for every $w\in B_{r}(z)$ we conclude that ${\rm lip}({\rm
pos}_{A})(z)=0$. ∎

###  5.3. Equivalence of minima via the position function

In this section we use the properties of the position function of a path
separating set to show the equivalence of the minima of the separating ratio
and the Minkowski content. Observe that this result, especially in the locally
geodesic setting, sharpens Theorem 1.4. If we apply it to the measure
$\mathfrak{m}_{x,y}^{L}$, where $\mathfrak{m}$ is doubling, we get Theorem
1.6.

######  Theorem 5.5.

Let $({\rm X},{\sf d},\mathfrak{m})$ be a path connected, locally
$\Lambda$-quasiconvex metric measure space and suppose $\mathop{\rm
supp}\nolimits(\mathfrak{m})$ is compact. Let $x,y\in\mathop{\rm
supp}\nolimits(\mathfrak{m})$. Then

(38)
$\Lambda^{-1}\inf_{\Omega\in\SS_{\textup{top}}(x,y)}\mathfrak{m}^{+}(\Omega)\leq\inf_{A\subseteq{\rm
X}\,\textup{closed}}\frac{\mathfrak{m}(A)}{0pt_{x,y}(A)}\leq\inf_{\Omega\in\SS_{\textup{top}}(x,y)}\mathfrak{m}^{+}(\Omega).$

In particular, if $({\rm X},{\sf d})$ is path connected and locally geodesic
we have

(39)
$\inf_{\Omega\in\SS_{\textup{top}}(x,y)}\mathfrak{m}^{+}(\Omega)=\inf_{A\subseteq{\rm
X}\,\textup{closed}}\frac{\mathfrak{m}(A)}{0pt_{x,y}(A)}.$

###### Proof.

We reproduce here the proof of the right inequality. Let
$\Omega\in\SS_{\textup{top}}(x,y)$. We can suppose
$\mathfrak{m}^{+}(\Omega)<+\infty$ otherwise there is nothing to prove. In
particular we have $\mathfrak{m}^{+}(\Omega)<+\infty$, which implies
$\mathfrak{m}(\partial\Omega)=0$. We consider the sets
$A_{r}:=\overline{B}_{r}(\Omega)\setminus\textup{int}(\Omega)$, which is a
closed subset of ${\rm X}$. As in the proof of Theorem 1.4 we get
$0pt_{x,y}(A_{r})\geq r$ if $r<\min\\{{\sf d}(\partial\Omega,x),{\sf
d}(\partial\Omega,y)\\}$. Therefore

$\mathfrak{m}^{+}(\Omega)=\varliminf_{r\to
0}\frac{\mathfrak{m}(\overline{B}_{r}(\Omega)\setminus\Omega)}{r}=\varliminf_{r\to
0}\frac{\mathfrak{m}(A_{r})}{r}\geq\varliminf_{r\to
0}\frac{\mathfrak{m}(A_{r})}{0pt_{x,y}(A_{r})}\geq\inf_{\begin{subarray}{c}A\subseteq{\rm
X}\\\ A\text{ closed}\end{subarray}}{\rm SR}_{x,y}(A).$

In order to show the first inequality we proceed as follows. First of all
notice that $\Gamma_{x,y}\neq\emptyset$ because $({\rm X},{\sf d})$ is
rectifiable path connected, so $0pt_{x,y}(A)<+\infty$ for every
$A\subseteq{\rm X}$. Now we fix an arbitrary closed subset $A\subseteq{\rm
X}$. We can suppose $0pt_{x,y}(A)>0$, otherwise ${\rm SR}_{x,y}(A)=+\infty$
and there is nothing to prove. In particular $A\in\SS_{\text{path}}(x,y)$. Let
${\rm pos}_{A}$ be the position function associated to $A$. It is locally
$\Lambda$-Lipschitz by Proposition 5.4. Therefore it is Lipschitz when
restricted to the compact set $\mathop{\rm supp}\nolimits(\mathfrak{m})$
because of Lemma 2.1. By the coarea inequality of Proposition 2.7 we get

$\int_{0}^{0pt_{x,y}(A)}\mathfrak{m}^{+}(\\{{\rm pos}_{A}\leq
t\\})\,{\mathrm{d}}t\leq\int_{\rm X}{\rm lip}({\rm
pos}_{A})\,{\mathrm{d}}\mathfrak{m}.$

We notice that, for $t\in(0,0pt_{x,y}(A))$, the sets $\\{{\rm pos}_{A}\leq
t\\}$ belong to $\SS_{\text{top}}(x,y)$ because ${\rm pos}_{A}$ is Lipschitz
on $\mathop{\rm supp}\nolimits(\mathfrak{m})$, $x,y\in\mathop{\rm
supp}\nolimits(\mathfrak{m})$ and ${\rm pos}_{A}(x)=0$, ${\rm
pos}_{A}(y)=0pt_{x,y}(A)$. Moreover ${\rm lip}({\rm pos}_{A})=0$ on $A^{c}$
since $A$ is closed, by Proposition 5.4. Hence, again by Proposition 5.4, we
have

$\int_{0}^{0pt_{x,y}(A)}\mathfrak{m}^{+}(\\{{\rm pos}_{A}\leq
t\\})\,{\mathrm{d}}t\leq\int_{A}{\rm lip}({\rm
pos}_{A})\,{\mathrm{d}}\mathfrak{m}\leq\Lambda\mathfrak{m}(A).$

Dividing by $0pt_{x,y}(A)$ we conclude that we can find some
$t\in(0,0pt_{x,y}(A))$ such that

$\mathfrak{m}^{+}(\\{{\rm pos}_{A}\leq
t\\}\leq\Lambda\frac{\mathfrak{m}(A)}{0pt_{x,y}(A)}=\Lambda\cdot{\rm
SR}_{x,y}(A).$

By the arbitrariness of $A$ we finally have

$\frac{1}{\Lambda}\inf_{\Omega\in\SS_{\textup{top}}(x,y)}\mathfrak{m}^{+}(\Omega)\leq\inf_{A\subseteq{\rm
X}\,\textup{closed}}\frac{\mathfrak{m}(A)}{0pt_{x,y}(A)}.$

∎

The proof of Theorem 5.5 relies on the coarea inequality for the Minkowski
content and a special role is played by the position function. In the next
section we will show an analogous statement in case of discrete metric measure
spaces, where one can use directly the properties of the discrete analogous of
the position function. We believe that the proof in the discrete setting
highlights the main geometric intuitions behind the equivalence of the minima
of the two functionals.

##  6\. The equivalence of the infima on measure graphs

Let us consider a graph, i.e. a couple $({\rm V},E)$, where ${\rm V}$ is a
set, called the _set of vertices_ and $E\subseteq({\rm V}\times{\rm
V})/\mathfrak{S}_{2}$ is called the _set of edges_. $\mathfrak{S}_{2}$ is the
permutation group of order 2 and its non-trivial element acts on ${\rm
V}\times{\rm V}$ as $(v,w)\mapsto(w,v)$. We say that two points $v,w$ are
adjacent, and we write $v\sim w$, if $(v,w)\in E$. We say that $({\rm V},E)$
is a countable graph if ${\rm V}$ is countable. A path in ${\rm V}$ is an
element of the set

$\mathcal{P}:=\left\\{\\{q_{i}\\}_{i=0}^{N}:N\in\mathbb{N},\,q_{i}\in{\rm
V},\,(q_{i},q_{i+1})\in E\right\\}.$

The set of paths connecting two points $v,w\in{\rm V}$ is

$\mathcal{P}_{v,w}:=\left\\{\\{q_{i}\\}_{i=0}^{N}\in\mathcal{P}:q_{0}=v,q_{N}=w\right\\}.$

The concatenation of two paths ${\sf
c}=\\{q_{i}\\}_{i=0}^{N}\in\mathcal{P}_{v,w}$ and ${\sf
c}^{\prime}=\\{q_{i}^{\prime}\\}_{i=0}^{N^{\prime}}\in\mathcal{P}_{w,z}$ is
the path

${\sf c}\star{\sf
c}^{\prime}=(q_{0},\ldots,q_{N}=q_{0}^{\prime},q_{1}^{\prime},\ldots,q_{N^{\prime}})\in\mathcal{P}_{v,z}.$

Given ${\sf c}\in\mathcal{P}$ and a subset $A\subseteq{\rm X}$, the set $({\sf
c}\cap A)$ is naturally ordered by the order of elements of ${\sf c}$. Given
$j\in\mathbb{N}$, we denote by $({\sf c}\cap A)_{j}$ the $j$-th element of
$({\sf c}\cap A)$ with respect to this order.

We define the analogous version in the discrete setting of the width.

######  Definition 6.1 (Discrete width and discrete separating sets).

Let $({\rm V},E)$ be a graph and let $v,w\in{\rm V}$. The width with respect
to $v,w$ of a subset $A\subseteq{\rm V}$ is

${\rm disc\text{-}width}_{v,w}(A):=\inf\\{\\#({\sf c}\cap A):{\sf
c}\in\mathcal{P}_{v,w}\\}$

with the convention that ${\rm disc\text{-}width}_{v,w}(A)=0$ if
$\mathcal{P}_{v,w}=\emptyset$. A set $A\subseteq V$ is a separating set
between $v$ and $w$ if ${\rm disc\text{-}width}_{v,w}(A)\geq 1$, i.e. if every
path between $v$ and $w$ intersects $A$. The class of separating sets between
$v$ and $w$ is denoted by ${\rm disc\text{-}SS}(v,w)$. For instance if
$\mathcal{P}_{v,w}=\emptyset$ then every subset of ${\rm V}$ is separating, in
particular $\emptyset\in\mathcal{P}_{v,w}$.

Given a countable graph $({\rm V},E)$, we consider a set function
$\mu\colon{\rm V}\to(0,+\infty)$. We extend $\mu$ to a measure on $2^{\rm V}$
in the natural way by summing up over the singletons. A measure graph is a
triple $({\rm V},E,\mu)$ as above. The analogous of the separating ratio is
defined as follows.

######  Definition 6.2 (Discrete separating ratio).

Let $({\rm V},E,\mu)$ be a measure graph and let $v,w\in{\rm V}$. The discrete
separating ratio of a subset $A\subseteq V$ with respect to $v,w$ is

${\rm disc\text{-}SR}_{v,w}(A):=\frac{\mu(A)}{{\rm
disc\text{-}width}_{v,w}(A)}$

with the convention that ${\rm disc\text{-}SR}_{v,w}(A)=0$ if ${\rm
disc\text{-}width}_{v,w}(A)=+\infty$, i.e. if $\mathcal{P}_{v,w}=\emptyset$,
and ${\rm disc\text{-}SR}_{v,w}(A)=+\infty$ if ${\rm
disc\text{-}width}_{v,w}(A)=0$, i.e. if $A\notin{\rm disc\text{-}SS}(v,w)$.

######  Remark 6.3.

We first remark that our construction and our result do not require a metric
on a graph; in this way, we highlight the combinatorial aspect of the theory
in this discrete setting.

######  Remark 6.4.

A classical example of a graph is given by an approximation of a metric space
in terms of nets. Let $({\rm X},{\sf d})$ be a metric space. For every $r>0$,
let ${\rm X}_{r}$ be a maximal $r$-separated subset of ${\rm X}$: this means
that the distance of every two points of ${\rm X}_{r}$ is at least $r$ and
that ${\rm X}_{r}$ is maximal with this property. It is standard that such a
set is $2r$-dense, i.e. every point of ${\rm X}$ is at distance at most $2r$
from a point of ${\rm X}_{r}$. Let us construct the graph $({\rm
V}_{r},E_{r})$ in the following way: ${\rm V}_{r}={\rm X}_{r}$ as a set and
$v\sim w$ if and only if ${\sf d}(v,w)\leq 3r$. This discretization procedure
of a metric space is standard.
Assume now that $({\rm X},{\sf d},\mathfrak{m})$ is a doubling metric measure
space and let $({\rm V}_{r},E_{r})$ be as above. Now consider
$\mu_{r}\colon{\rm V}_{r}\to(0,+\infty)$ defined by
$\mu_{r}(v):=\mathfrak{m}(B_{3r}(v))$. This defines a measure graph $({\rm
V}_{r},E_{r},\mu_{r})$. For every $A\subseteq{\rm X}$ one can define the set
$A_{r}:=\\{v\in{\rm V}_{r}\,:\,{\sf d}(v,A)<2r\\}$. Observe that, as subsets
of ${\rm X}$, we have $A_{r}\subseteq B_{2r}(A)$ and $A\subseteq
B_{2r}(A_{r})$. In particular

$\mathfrak{m}(B_{r}(A))\leq\mathfrak{m}(B_{3r}(A_{r}))\leq\sum_{v\in
A_{r}}\mathfrak{m}(B_{3r}(v))=\mu_{r}(A_{r})\leq C_{D}^{2}\sum_{v\in
A_{r}}\mathfrak{m}(B_{r}(v))\leq C_{D}^{2}\mathfrak{m}(B_{3r}(A)).$

Therefore, if for simplicity we restrict to the sets with $\mathfrak{m}(A)=0$,
the last computation implies that

$\frac{\mu_{r}(A_{r})}{r}\approx\mathfrak{m}^{+}(A)$

as $r$ goes to $0$. Similarly, at least if $({\rm X},{\sf d})$ is
$\Lambda$-quasigeodesic, we have

$r\cdot{\rm disc\text{-}width}_{x,y}(A_{r})\approx 0pt_{x,y}(A).$

Those are qualitative relations between the quantities defined in the discrete
and the continuous settings. We will discuss the construction also later in
Remark 6.10.

In view of the observations above, it is clear that the next result is the
discrete analogous of Theorem 5.5.

######  Theorem 6.5.

Let $({\rm V},E,\mu)$ be a measure graph and let $v,w\in{\rm V}$. Then:

$\inf_{A\subseteq{\rm V}}{\rm disc\text{-}SR}_{v,w}(A)=\inf_{A\in{\rm
disc\text{-}SS}(v,w)}\mu(A).$

The proof uses the analogous of the position function in the discrete setting.
Let $A\subseteq{\rm X}$ and ${\sf
c}=\\{q_{i}\\}_{i=0}^{N}\in\mathcal{P}_{v,w}$ be a path. The position function
with respect to the set $A$ and the path ${\sf c}$ is the function ${\rm
pos}_{{\sf c},A}\colon{\rm V}\to\mathbb{N}\cup\\{\infty\\}$ defined by

${\rm pos}_{{\sf
c},A}(z):=\min_{i_{0}\in\mathbb{N}\,:\,z=q_{i_{0}}}\\#(\\{q_{i}\\}_{i=0}^{i_{0}}\cap
A).$

If $z\notin{\sf c}$ then ${\rm pos}_{{\sf c},A}(z)=+\infty$. The position
function with respect of the set $A$ is

${\rm pos}_{A}\colon{\rm V}\to\mathbb{N}\cup\\{\infty\\},\quad
z\mapsto\min_{{\sf c}\in\mathcal{P}_{v,w}}{\rm pos}_{{\sf c},A}(z).$

For instance ${\rm pos}_{A}(z)=1$ if there exists ${\sf
c}\in\mathcal{P}_{v,w}$ such that $z$ is the first point of intersection of
${\sf c}$ with $A$. Figure 3 gives an intuition about this definition.

Figure 3. For the given set $A$ made of red points, the point $z_{1}$ has
position $1$, i.e. ${{\rm pos}_{A}}(z_{1})=1$, while ${\rm pos}_{A}(z_{2})=2$.
Indeed $z_{1}$ has position $2$ with respect to the blue path, but it has
position $1$ with respect to the green one. Instead there are no paths for
which $z_{2}$ is the first intersection point with $A$.

The analogous of Proposition 5.3 allows to fibrate a set $A$ by means of level
sets of the position function associated to $A$. Figure 4 shows such a
procedure for the graph and the set $A$ in Figure 3. This is is the key step
in the proof of Theorem 6.5. Notice that, unlike the continuous case, we do
not need additional properties of the position function related to the
connectivity properties of the metric space.

Figure 4. The construction of the sets $A_{i}$’s as in the proof.

###### Proof of Theorem 6.5.

If $\mathcal{P}_{v,w}=\emptyset$ then the thesis is true. Indeed for every
subset of $A$ we have ${\rm disc\text{-}SR}_{v,w}(A)=0$ by definition. On the
other hand $\emptyset\in{\rm disc\text{-}SS}(v,w)$, so also the right hand
side is zero. We can now suppose $\mathcal{P}_{v,w}\neq\emptyset$. The central
step is to show that

(40) $\inf_{A\subseteq{\rm V}}{\rm
disc\text{-}SR}_{v,w}(A)=\inf_{\begin{subarray}{c}A\subseteq{\rm V}\\\ {\rm
disc\text{-}width}_{v,w}(A)=1\end{subarray}}{\rm disc\text{-}SR}_{v,w}(A).$

Indeed, if (40) holds, then the thesis follows by the definition of ${\rm
disc\text{-}SR}_{v,w}(A)$, since

${\rm disc\text{-}SR}_{v,w}(A)=\mu(A)\qquad\text{ if }{\rm
disc\text{-}width}_{v,w}(A)=1.$

It is clear that the right hand side of (40) is always bigger than or equal to
the left hand side, so it is enough to show the other inequality.
Let us take $A\subseteq{\rm V}$. If $A\notin{\rm disc\text{-}SS}(v,w)$ then
${\rm disc\text{-}SR}_{v,w}(A)=+\infty$ and there is nothing to prove. Hence
we can suppose that $A\in{\rm disc\text{-}SS}(v,w)$. It is enough to prove
that there exists $A^{\prime}\in{\rm disc\text{-}SS}(v,w)$ with ${\rm
disc\text{-}width}_{v,w}(A^{\prime})=1$ such that ${\rm
disc\text{-}SR}(A)\geq{\rm disc\text{-}SR}(A^{\prime})$. We consider the sets

$A_{i}:=\\{z\in A\text{ s.t. }{\rm pos}_{A}(z)=i\\}.$

Similarly to what we did in the proof of Proposition 5.3 we claim that
$A_{i}\in{\rm disc\text{-}SS}(v,w)$ and that ${\rm
disc\text{-}width}_{v,w}(A_{i})=1$ for $i=1,\ldots,{\rm
disc\text{-}width}_{v,w}(A)$.

Claim 1. For every ${\sf c}\in\mathcal{P}_{v,w}$ there exists $z\in{\sf c}\cap
A$ such that ${\rm pos}_{A}(z)\geq{\rm disc\text{-}width}_{v,w}(A)$.
Let $z$ be the last point of the set ${\sf c}\cap A$. Suppose ${\rm
pos}(z)<{\rm disc\text{-}width}_{v,w}(A)$ and take a path ${\sf
c}_{z}\in\mathcal{P}_{v,w}$ such that ${\rm pos}_{{\sf c}_{z},A}(z)={\rm
pos}_{A}(z)$. Consider the concatenation of ${\sf c}_{z}$ up to $z$ and ${\sf
c}$ after $z$. This is a path belonging to $\mathcal{P}_{v,w}$ that intersects
$A$ in at most ${\rm pos}_{A}(z)<{\rm disc\text{-}width}_{v,w}(A)$ points, a
contradiction.

Claim 2. For every ${\sf c}\in\mathcal{P}_{v,w}$ we have ${\rm pos}(({\sf
c}\cap A)_{j+1})\leq{\rm pos}(({\sf c}\cap A)_{j})+1$ for every $j$.
Call $z:=({\sf c}\cap A)_{j}$ and $p:={\rm pos}(({\sf c}\cap A)_{j})$. Let
${\sf c}_{z}\in\mathcal{P}_{v,w}$ be such that $z=({\sf c}_{z}\cap A)_{p}$.
Consider the concatenation of ${\sf c}_{z}$ up to $z$ and ${\sf c}$ after $z$.
This new curve belongs to $\mathcal{P}_{v,w}$ and ${\rm pos}(({\sf c}\cap
A)_{j+1})\leq p+1$ by construction.

Claim 3. $A_{i}\in{\rm disc\text{-}SS}(v,w)$ for all $i=1,\ldots,{\rm
disc\text{-}width}_{v,w}(A)$.
Let ${\sf c}\in\mathcal{P}_{v,w}$. Since ${\rm disc\text{-}width}_{v,w}(A)\geq
1$ then the set ${\sf c}\cap A$ is not empty. The first point of intersection
$({\sf c}\cap A)_{1}$ clearly satisfies ${\rm pos}_{A}(({\sf c}\cap
A)_{1})=1$. By Claim 1 we can find a point $z\in{\sf c}$ with ${\rm
pos}_{A}(z)\geq{\rm disc\text{-}width}_{v,w}(A)$. The function ${\rm pos}_{A}$
can increase along consecutive points of ${\sf c}\cap A$ by at most $1$ by
Claim 2. So for every $i=1,\ldots,{\rm disc\text{-}width}_{v,w}(A)$ there must
be some point of ${\sf c}\cap A$ with position $i$, i.e. ${\sf c}\cap
A_{i}\neq\emptyset.$

Claim 4. ${\rm disc\text{-}width}_{v,w}(A_{i})=1$ for all $i=1,\ldots,{\rm
disc\text{-}width}_{v,w}(A)$.
Let ${\sf c}\in\mathcal{P}_{v,w}$ be arbitrary and $i\in\\{1,\ldots,{\rm
disc\text{-}width}_{v,w}(A)\\}$. Call $z$ the last point of intersection of
${\sf c}\cap A_{i}$: it exists because of Claim 3. Let ${\sf
c}_{z}\in\mathcal{P}_{v,w}$ be such that $z=({\sf c}_{z}\cap A)_{i}$. Consider
the concatenation of ${\sf c}_{z}$ up to $z$ and ${\sf c}$ after $z$. This is
a path belonging to $\mathcal{P}_{v,w}$ that intersects $A_{i}$ only in $z$.
Indeed in the part of ${\sf c}$ after $z$ there are no more intersections with
$A_{i}$ by hypothesis. On the other hand it is clear that ${\rm pos}(({\sf
c}_{z}\cap A)_{j})\leq j<i$ for all $j<i$. This shows that ${\rm
disc\text{-}width}_{v,w}(A_{i})=1$.

The sets $A_{i}$, $i=1,\ldots,{\rm disc\text{-}width}_{v,w}(A)$, are disjoint
and contained in $A$, so there exists one of them with
$\mu(A_{i})\leq\frac{\mu(A)}{{\rm disc\text{-}width}_{v,w}(A)}$. Setting
$A^{\prime}=A_{i}$ for such value of $i$ we get

${\rm disc\text{-}SR}_{v,w}(A^{\prime})=\mu(A_{i})\leq\frac{\mu(A)}{{\rm
disc\text{-}width}_{v,w}(A)}={\rm disc\text{-}SR}_{v,w}(A).$

As claimed at the beginning this is enough to conclude. ∎

######  Remark 6.6.

In the discrete argument the coarea inequality has been replaced by the easier
mean estimate at the end of the proof above. Moreover the proof of Theorem 6.5
suggests that the infimum of the discrete separating ratio should be realized
by ’slim’ subsets. However notice that a discrete separating set with discrete
width equal to $1$ is not necessarily ‘slim’. For instance one can consider
the graph $V=\mathbb{Z}^{2}$ with $[(v,w)]\in E$ if and only if $v-w\in\\{(\pm
1,0),(0,\pm 1)\\}$. Let $v:=(-5,0)$ and $w:=(5,0)$. Define $A:=\\{(x,y)\in
V:\,|y|\geq|x|\\}$. In such a case, ${\rm disc\text{-}width}_{v,w}(A)=1$ by
simply considering the horizontal path, but the set is ‘thick’.

In Proposition 6.9 we show that the infimum of the discrete separating ratio
can be always computed among slim discrete separating sets, where for us
’slim’ means that every point has position $1$. In other words a discrete
separating set $A$ is slim if for every $z\in A$ there exists a path ${\sf
c}\in\mathcal{P}_{v,w}$ such that $z=({\sf c}\cap A)_{1}$. These sets resemble
the boundaries of separating sets in the continuous case, that should be
thought as model of slim sets. Let us first discuss the definition of slim
sets.

######  Definition 6.7 (Slim separating sets).

Let $({\rm V},E)$ be a graph, $v,w\in{\rm V}$. A subset $A\subseteq{\rm V}$ is
called a slim separating set if

  * (a)

$A\in{\rm disc\text{-}SS}(v,w)$;

  * (b)

for every $z\in A$ there exists a path ${\sf c}\in\mathcal{P}_{v,w}$ such that
$z=({\sf c}\cap A)_{1}$.

Recall that the connected component of a point $v\in{\rm V}$ is the maximal
subset $C$ of ${\rm V}$ satisfying $\mathcal{P}_{v,w}\neq\emptyset$ for every
$w\in C$. A graph is connected if it has only one connected component. Observe
that if $A\in{\rm disc\text{-}SS}(v,w)$ then also its intersection with the
connected component containing $v$ and $w$ is a discrete separating set with
same width and smaller measure. So it is not restrictive to suppose that a
separating set is contained in the connected component of $v$ and $w$. By
definition a slim separating set is contained in the connected component of
$v$ and $w$.

######  Lemma 6.8.

Let $({\rm V},E)$ be a graph, $v,w\in{\rm V}$ and $A\in{\rm
disc\text{-}SS}(v,w)$. Suppose $\mathcal{P}_{v,w}\neq\emptyset$ and that $A$
is contained in the connected component of $v$ and $w$. Then the following
conditions are equivalent:

  * (i)

$A$ is slim;

  * (ii)

the function ${\rm pos}_{A}$ restricted to $A$ is constantly equal to $1$;

  * (iii)

the function ${\rm pos}_{A}$ takes value in $\\{0,1,+\infty\\}$;

  * (iv)

for every point $z$ of the connected component of $v,w$ there exists a path
${\sf c}\in\mathcal{P}_{v,w}$ such that $z\in{\sf c}$ and $\\#({\sf c}\cap
A)\in\\{0,1\\}$.

In particular condition (iv) says that $A$ is slim if and only if every point
of the connected component of $v$ and $w$ lies inside a path joining $v$ to
$w$ that intersects $A$ in exactly one point.

###### Proof.

Suppose $A$ is slim and let $z\in A$. By definition we get ${\rm
pos}_{A}(z)\leq 1$. On the other hand it is also clear from the definition
that ${\rm pos}_{A}(z)\geq 1$ since $z\in A$. Therefore (ii) holds.
Suppose now that (ii) holds and take $z\in{\rm X}$. If $z$ does not belong to
the connected component of $v$ and $w$ then ${\rm pos}_{A}(z)=+\infty$ by
definition. Otherwise there exists a path ${\sf
c}=\\{q_{i}\\}_{i=0}^{N}\in\mathcal{P}_{v,w}$ containing $z$, say
$z=q_{i_{0}}$. Consider the set $\\{q_{i}\\}_{i=0}^{i_{0}}\cap A$. If it is
empty then ${\rm pos}_{A}(z)=0$. Otherwise let $z^{\prime}$ be the biggest
element of $\\{q_{i}\\}_{i=0}^{i_{0}}\cap A$. By (ii) we can find a path ${\sf
c}_{z^{\prime}}\in\mathcal{P}_{v,w}$ such that $({\sf c}_{z^{\prime}}\cap
A)_{1}=z^{\prime}$. Let us consider the concatenation $\tilde{{\sf c}}$ of
${\sf c}_{z^{\prime}}$ up to $z^{\prime}$ and ${\sf c}$ from $z^{\prime}$.
This is a path in $\mathcal{P}_{v,w}$ and, by construction, ${\rm
pos}_{\tilde{{\sf c}},A}(z)=1$, so ${\rm pos}_{A}(z)\leq 1$. This shows (iii).
We notice that (iv) is just a reformulation of (iii). It is also clear that
(iv) implies (i), completing the equivalences. ∎

######  Proposition 6.9.

Let $({\rm V},E,\mu)$ be a measure graph and let $v,w\in{\rm V}$. Suppose
$\mathcal{P}_{v,w}\neq\emptyset$. Then:

$\inf_{A\subseteq{\rm V}}{\rm disc\text{-}SR}_{v,w}(A)=\inf_{A\textup{
slim}}{\rm disc\text{-}SR}_{v,w}(A).$

###### Proof.

It suffices to prove the inequality $\geq$. We recall that for sets $A\in{\rm
disc\text{-}SS}(v,w)$ with ${\rm disc\text{-}width}_{v,w}(A)=1$ we have ${\rm
disc\text{-}SR}(A)=\mu(A)$. Because of the proof of Theorem 6.5 and our
assumption $\mathcal{P}_{v,w}\neq\emptyset$, it suffices to prove that for
every $A\in{\rm disc\text{-}SS}(v,w)$ with ${\rm
disc\text{-}width}_{v,w}(A)=1$, there exists $A^{\prime}$ slim with
$\mu(A^{\prime})\leq\mu(A)$. Let ${\rm pos}_{A}$ be the position function
associated to $A$ and take $A^{\prime}:=\\{z\in{\rm V}\,:\,{\rm
pos}_{A}(z)=1\\}$. Then $A^{\prime}$ belongs to ${\rm disc\text{-}SS}(v,w)$ as
we already showed in the proof of Theorem 6.5. It is also contained in the
connected component of $v$ and $w$ since the position function is finite on
$A^{\prime}$. It is also slim since ${\rm pos}_{A^{\prime}}(z)\leq{\rm
pos}_{A}(z)=1$ for every $z\in A^{\prime}$, while on the other hand we always
have ${\rm pos}_{A^{\prime}}(z)\geq 1$ for $z\in A^{\prime}$. So ${\rm
pos}_{A^{\prime}}$ is constantly equal to $1$ on $A^{\prime}$ and we can apply
Lemma 6.8. Since it is a subset of $A$ it is clear that
$\mu(A^{\prime})\leq\mu(A)$. ∎

######  Remark 6.10.

Let $({\rm X},{\sf d},\mathfrak{m})$ be a doubling metric measure space. Let
$x,y\in{\rm X}$ and let $(V_{r},E_{r},\mu_{r})$ be the graph at scale $r>0$
introduced in Remark 6.4 with the requirement that $x,y\in{\rm V}_{r}$. It
would be interesting to study the $\Gamma$-convergence of the functionals on
graphs to the functionals in the continuous case. The goal would be to prove
the following relations:

$\lim_{r\to 0}\inf_{A\subseteq{\rm V}_{r}}\frac{{\rm
disc\text{-}SR}_{x,y}(A)}{r}\approx\inf_{A\subseteq{\rm
X}\,\textup{closed}}{\rm SR}_{x,y}(A)$

and

$\lim_{r\to 0}\inf_{A\in{\rm disc\text{-}SS}(x,y,{\rm
V}_{r})}\frac{\mu_{r}(A)}{r}\approx\inf_{\Omega\in\SS_{\textup{top}}(x,y)}\mathfrak{m}^{+}(A).$

We do not know what are the conditions on $({\rm X},{\sf d},\mathfrak{m})$ to
ensure this kind of estimates and we do not insist in this direction.

##  Appendix A Revision of conditions equivalent to the Poincaré inequality

There are many equivalent characterizations of the Poincaré inequality in
terms of other geometric or analytical quantities, all involving curves
connecting two points. However there is an asymmetry in these conditions: some
of them are expressed in terms of $\Gamma_{x,y}$, while some of them in terms
of $\Gamma_{x,y}^{L}$ for a suitable $L<\infty$. The scope of this appendix is
twofold: first we want to show that there is no difference in considering
$\Gamma_{x,y}^{L}$ or $\Gamma_{x,y}$ in all of these conditions and secondly
we will see that the equivalence is still valid for the pointwise version of
the conditions, without requiring them _for every couple of points_ , assuming
some regularity of the space. Furthermore we show that, given
$p\in[1,\infty)$, the $p$-Poincaré inequality is equivalent to the existence
of a $p$-pencil of curves, a result which is new for $p>1$ (thus generalizing
the previous works [7] and [12]).
For the scope of this appendix the pointwise $p$-Poincaré inequality at
$x,y\in{\rm X}$ is the following condition: there exist $C>0$ and $L\geq 1$
such that for every Borel function $u$ and every $g\in{\rm UG}(u)$ it holds

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

###  A.1. Keith’s module characterization

The first characterization is due to Keith (see [17]). In order to state it we
need to recall the notion of $p$-modulus of a family of curves. Let $({\rm
X},{\sf d},\mathfrak{m})$ be a metric measure space and let $p\geq 1$. Let
$\Gamma$ be a family of rectifiable curves of ${\rm X}$. The set of admissible
densities for $\Gamma$ is

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

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

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

Keith’s characterization reads as follows.

######  Proposition A.1.

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

  * (i)

(41) holds;

  * (ii)

there exist $c>0$, $L\geq 1$ such that
$\textup{Mod}_{p}(\Gamma_{x,y},\mathfrak{m}_{x,y}^{L})\geq c{\sf
d}(x,y)^{1-p}$ for all $x,y\in{\rm X}$;

  * (iii)

there exist $c>0$, $L\geq 1$ such that
$\textup{Mod}_{p}(\Gamma_{x,y}^{L},\mathfrak{m}_{x,y}^{L})\geq c{\sf
d}(x,y)^{1-p}$ for all $x,y\in{\rm X}$.

The equivalence between (i) and (ii) is essentially proved in [17], we will
write it in order to show that the pointwise version still works. The
equivalence between (ii) and (iii) is consequence of the next lemma (see also
the discussion at the beginning of the proof of [7, Theorem 3.7]).

######  Lemma A.2.

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

$\textup{Mod}_{p}(\Gamma_{x,y}\setminus\Gamma_{x,y}^{\Lambda},\mathfrak{m}_{x,y}^{L})\leq\frac{8}{\Lambda^{p}}C_{D}L{\sf
d}(x,y)^{1-p}.$

###### Proof.

The function $\rho=\frac{1}{\Lambda{\sf d}(x,y)}$ is admissible for the family
$\Gamma_{x,y}\setminus\Gamma_{x,y}^{\Lambda}$, i.e. $\rho\in{\rm
Adm}(\Gamma_{x,y}\setminus\Gamma_{x,y}^{\Lambda})$. So

$\textup{Mod}_{p}(\Gamma_{x,y}\setminus\Gamma_{x,y}^{\Lambda},\mathfrak{m}_{x,y}^{L})\leq\int\rho^{p}{\mathrm{d}}\mathfrak{m}_{x,y}^{L}\leq\frac{1}{\Lambda^{p}}\frac{1}{{\sf
d}(x,y)^{p}}\mathfrak{m}_{x,y}^{L}({\rm X})\leq\frac{8}{\Lambda^{p}}C_{D}L{\sf
d}(x,y)^{1-p}.$

The last inequality follows from Lemma 2.3. ∎

###### Proof of Proposition A.1.

It is known that in every metric measure space the $p$-modulus of the family
$\Gamma_{x,y}$ coincides with the $p$-capacity of the sets $\\{x\\}$ and
$\\{y\\}$ (see [15, Theorem 7.31]). Namely

${\rm
Mod}_{p}(\Gamma_{x,y},\mathfrak{m}_{x,y}^{L})=\inf\left\\{\int\rho^{p}\,{\mathrm{d}}\mathfrak{m}_{x,y}^{L}\,:\,\rho\in{\rm
UG}(u),\,u\colon{\rm X}\to\mathbb{R},u(x)=0,u(y)=1\right\\}.$

For every $u\colon{\rm X}\to\mathbb{R}$ such that $u(x)=0$ and $u(y)=1$ we
get, by (41),

$\int\rho^{p}\,{\mathrm{d}}\mathfrak{m}_{x,y}^{L}\geq\frac{1}{C}{\sf
d}(x,y)^{1-p}.$

This shows that (i) implies (ii).
We now show the converse. Let $u\colon{\rm X}\to\mathbb{R}$ and let
$\rho\in{\rm UG}(u)$. If $u(x)=u(y)$ there is nothing to prove. Otherwise we
define the function

$\bar{u}(z)=\left|\frac{u(z)-u(x)}{u(z)-u(y)}\right|$

and $\bar{\rho}(z)=\frac{\rho(z)}{|u(y)-u(x)|}$. By triangle inequality of the
absolute value we obtain that $\bar{\rho}\in{\rm UG}(\bar{u})$. Since
$\bar{u}(x)=0$ and $\bar{u}(y)=1$ we can use $\bar{\rho}$ in the estimate of
the $p$-capacity. Namely we have

$c\,{\sf d}(x,y)^{1-p}\leq{\rm
Mod}_{p}(\Gamma_{x,y},\mathfrak{m}_{x,y}^{L})\leq\int\bar{\rho}^{p}\,{\mathrm{d}}\mathfrak{m}_{x,y}^{L}=\frac{1}{|u(y)-u(x)|^{p}}\int\rho^{p}\,{\mathrm{d}}\mathfrak{m}_{x,y}^{L},$

which is (41).
Finally, since it is clear that (iii) implies (ii), it is sufficient to prove
that (ii) implies (iii). Suppose (ii) holds with constants $c,L$. Fix
$\Lambda\geq 1$ such that $\frac{8}{\Lambda^{p}}C_{D}L\leq\frac{c}{2}$. Notice
that $\Lambda$ depends only on $C_{D},L,p$ and $c$. By Lemma A.2 we get

$\textup{Mod}_{p}(\Gamma_{x,y}^{\Lambda},\mathfrak{m}_{x,y}^{L})\geq\frac{c}{2}{\sf
d}(x,y)^{1-p}.$

Setting $c^{\prime}=\frac{c}{2}$ and $L^{\prime}=\max\\{\Lambda,L\\}$ we get
the thesis. Indeed

$\textup{Mod}_{p}(\Gamma_{x,y}^{L^{\prime}},\mathfrak{m}_{x,y}^{L^{\prime}})\geq\textup{Mod}_{p}(\Gamma_{x,y}^{\Lambda},\mathfrak{m}_{x,y}^{L})\geq
c^{\prime}{\sf d}(x,y)^{1-p}.$

∎

###  A.2. Pencil and $A_{p}$-connectedness characterizations

We now move to other characterizations that are known in the literature with
the name of $A_{p}$-connectedness (see [9]) and $p$-pencil (see for instance
[12] or [7] for the case $p=1$). We recall that given a complete and separable
metric space $({\rm Y},{\sf d}_{\rm Y})$ we denote by $\mathscr{P}({\rm Y})$
the set of Borel probability measures on ${\rm Y}$. We will always consider
the topology induced by the weak convergence on $\mathscr{P}({\rm Y})$. We
recall that a sequence of probability measures $\alpha_{n}\in\mathscr{P}({\rm
Y})$ converges weakly to a probability measure $\alpha\in\mathscr{P}({\rm Y})$
if $\int u\,{\mathrm{d}}\alpha_{n}\to\int u\,{\mathrm{d}}\alpha$ for every
bounded, continuous function $u\colon{\rm Y}\to\mathbb{R}$. This is the
sequential convergence induced by a topology on $\mathscr{P}({\rm Y})$ that
makes it a Polish space. Moreover $\mathscr{P}({\rm Y})$ is compact if ${\rm
Y}$ is. We recall also that every family of continuous curves with values in a
metric space ${\rm X}$ is endowed with the supremum distance. In particular if
$({\rm X},{\sf d})$ is a complete metric space and $x,y\in{\rm X}$ are two
points then the sets $\Gamma_{x,y}^{L}$ are compact for every $L\geq 1$ and
$\Gamma_{x,y}$ is closed. Finally we denote by ${\rm Lip}_{1}({\rm X})$ the
set of $1$-Lipschitz maps $u\colon{\rm X}\to\mathbb{R}$.

######  Theorem A.3.

Let $({\rm X},{\sf d},\mathfrak{m})$ be a path connected, pointwise
quasiconvex, doubling metric measure space. Let $x,y\in{\rm X}$. Then the
following conditions are quantitatively equivalent.

  * (i)

(41) holds;

  * (ii)

there exist $C>0$ and $L\geq 1$ such that for every $g\geq 0$ Borel there
exists $\gamma\in\Gamma_{x,y}^{L}$ such that

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

  * (iii)

there exist $C>0$ and $L\geq 1$ such that for every $g\geq 0$ bounded,
$g\in{\rm Lip}_{1}({\rm X})$, there exists $\gamma\in\Gamma_{x,y}$ such that

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

  * (iv)

there exist $C>0$, $L\geq 1$ and $\alpha\in\mathscr{P}(\Gamma_{x,y})$ such
that

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

for all $g\in{\rm Lip}_{1}({\rm X})$, $g\geq 0$ and bounded;

  * (v)

there exist $C>0$, $L\geq 1$ and $\alpha\in\mathscr{P}(\Gamma_{x,y}^{L})$ such
that

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

for all $g\geq 0$ Borel.

If moreover $({\rm X},{\sf d})$ is locally $\Lambda$-quasiconvex then the
conditions above are also equivalent to:

  * (vi)

there exist $C>0,L\geq 1$ such that

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

for every $u\in{\rm Lip}({\rm X})$.

Some connectivity assumption on ${\rm X}$ is necessary since the nice
behaviour of every functions at just $x$ and $y$ as expressed by (41) is not
enough to guarantee the existence of many curves in the space.
Condition (ii) is very similar to the $A_{p}$-connectedness condition of [9].
Of course we just ask it for a fixed couple of points. Condition (iii) is a
weakening of (ii) since it requires the same condition for less functions and
it allows to use more curves.
Conditions (iv) and (v) are closely related, at least in case $p=1$, to the so
called pencil of curves. The formulation and the proof in case $p>1$ is new.
The difference between these two conditions is similar to the difference
between (ii) and (iii). Finally observe that in (ii) and (iii) it is required
that for every function $g$ in a certain class there exists a curve, depending
on the function $g$, for which the estimate holds. In conditions (iv) and (v)
we ask for a class of curves on which the same condition holds _in average for
every function $g$_ in the class. This step is made, as in [7], with the help
of the following well known min-max theorem.

######  Proposition A.4 ([19, Thm. 9.4.2] (original proof in [20])).

Let ${\rm X}$ be a vector space and ${\rm Y}$ be a topological vector space.
Let $G\subseteq{\rm X}$ and $K\subseteq{\rm Y}$ be convex subsets, with $K$
compact. Let $F\colon G\times K\to\mathbb{R}$ be such that

  * a)

$F(\cdot,y)$ is convex on $G$ for every $y\in K$;

  * b)

$F(x,\cdot)$ is concave and upper semicontinuous in $K$ for every $x\in G$.

Then

(43) $\max_{y\in K}\inf_{x\in G}F(x,y)=\inf_{x\in G}\max_{y\in K}F(x,y).$

###### Proof of Theorem A.3.

We start assuming (i) and we show (iii). We tailor the argument in [9, Thm.
1.5] to our situation. First of all we recall that ${\rm X}$ is rectifiable
path connected by Lemma 2.2. For every $g\in{\rm Lip}_{1}({\rm X})$, $g\geq 0$
bounded, we define the map
$u(z):=\inf_{\gamma\in\Gamma_{x,z}}\int_{\gamma}g\,{\mathrm{d}}s$. Observe
that $u(z)<+\infty$ for every $z\in{\rm X}$ since ${\rm X}$ is rectifiable
path connected. We claim that $u$ is continuous, hence Borel. Indeed let
$z_{n}\in{\rm X}$ be points converging to $z$. Let $\varepsilon>0$ and let
$\gamma_{n}\in\Gamma_{x,z_{n}}$ be a curve such that
$\int_{\gamma_{n}}g\,{\mathrm{d}}s\leq u(z_{n})+\varepsilon$. Let also
$\Lambda_{z},r_{z}$ be the constants of pointwise quasiconvexity at $z$. Since
${\sf d}(z_{n},z)<r_{z}$ for $n$ big enough, then we can find curves
$\eta_{n}\in\Gamma_{z_{n},z}^{\Lambda_{n}}$ for every such $n$. The curve
$\gamma_{n}\star\eta_{n}\in\Gamma_{x,z}$ satisfies

$u(z)\leq\int_{\gamma_{n}\star\eta_{n}}g\,{\mathrm{d}}s\leq
u(z_{n})+\varepsilon+\Lambda_{z}{\sf d}(z_{n},z)\|g\|_{\infty}.$

For $n\to+\infty$ and $\varepsilon\to 0$ we get
$u(z)\leq\varliminf_{n\to+\infty}u(z_{n})$. Arguing similarly we get also the
other inequality, namely $u(z)\geq\varlimsup_{n\to+\infty}u(z_{n})$. Thus $u$
is Borel and $g$ is clearly an upper gradient of $u$. By (i) and the
definition of $u$ we find $\gamma\in\Gamma_{x,y}$ such that

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

The next step is to prove that (iii) implies (v). We fix $n\in\mathbb{N}$ and
we observe that ${\rm Lip}_{1}({\rm X},[0,n])$ is a convex subset of the
vector space $(C^{0}({\rm X},\mathbb{R}),\|\cdot\|_{\infty})$. Moreover
$\mathscr{P}(\Gamma_{x,y}^{\Lambda})$ is a compact, convex subset of the
vector space of all measures of $\Gamma_{x,y}^{\Lambda}$ endowed with the weak
convergence. For every $\Lambda\geq 1$ we consider the functional:

$F_{\Lambda}^{n}\colon\mathcal{P}(\Gamma_{x,y}^{\Lambda})\times{\rm
Lip}_{1}\left({\rm X},\left[0,n\right]\right),\quad(\alpha,g)\mapsto
C^{1/p}{\sf d}(x,y)^{\frac{p-1}{p}}\left(\int
g^{p}{\mathrm{d}}\mathfrak{m}_{x,y}^{L}\right)^{1/p}-\int\int_{\gamma}g{\mathrm{d}}\alpha.$

Observe that the functional has always the same expression, the only
dependence on $\Lambda$ and $n$ is in the domain of definition. We notice also
that $F_{\Lambda}^{n}(\cdot,g)$ is concave and upper semicontinuous because it
is linear, while $F_{\Lambda}^{n}(\alpha,\cdot)$ is convex because it is a
$L^{p}$-norm plus a linear term. By applying Proposition A.4 we have

$M_{\Lambda}^{n}:=\max_{\alpha\in\mathscr{P}(\Gamma_{x,y}^{\Lambda})}\inf_{g\in{\rm
Lip}_{1}({\rm X},[0,n])}F_{\Lambda}^{n}(\alpha,g)=\inf_{g\in{\rm Lip}_{1}({\rm
X},[0,n])}\max_{\alpha\in\mathscr{P}(\Gamma_{x,y}^{\Lambda})}F_{\Lambda}^{n}(\alpha,g)$

for every $\Lambda\geq 1$. We claim that
$\varliminf_{\Lambda\to+\infty}M_{\Lambda}^{n}\geq 0$, for $n$ fixed. Suppose
it is not the case. Then we can find $\varepsilon>0$, a sequence
$\Lambda_{j}\to+\infty$ and a sequence of functions $g_{j}\in{\rm
Lip}_{1}({\rm X},[0,n])$ such that for every
$\alpha\in\mathscr{P}(\Gamma_{x,y}^{\Lambda_{j}})$ it holds
$F_{\Lambda_{j}}^{n}(\alpha,g_{j})\leq-\varepsilon<0$. By Ascolì-Arzela we can
suppose that $g_{j}$ converges uniformly to a limit function
$g_{\infty}\in{\rm Lip}_{1}({\rm X},[0,n])$. By assumption (iii) we can find
$\gamma\in\Gamma_{x,y}$ such that

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

By definition we have that $\gamma\in\Gamma_{x,y}^{\Lambda_{j}}$ if $j\geq
j_{0}$, for some $j_{0}$. Therefore, denoting by
$\delta_{\gamma}\in\mathscr{P}(\Gamma_{x,y})$ the atomic measure concentrated
on $\gamma$, we get $F_{\Lambda_{j_{0}}}^{n}(\delta_{\gamma},g_{\infty})\geq
0$. Since $\gamma$ has finite length and $\mathfrak{m}_{x,y}^{L}$ is finite
and since $g_{j}$ converges uniformly on compact subsets to $g_{\infty}$, we
have that
$F_{\Lambda_{j}}^{n}(\delta_{\gamma},g_{j})=F_{\Lambda_{j_{0}}}^{n}(\delta_{\gamma},g_{j})$
converges to $F_{\Lambda_{j_{0}}}^{n}(\delta_{\gamma},g_{\infty})$, which is a
contradiction. Therefore $\varliminf_{\Lambda\to+\infty}M_{\Lambda}^{n}\geq
0$. This means that for all $\varepsilon>0$ there exist
$\Lambda_{\varepsilon}^{n}\geq 1$ and
$\alpha_{\varepsilon}^{n}\in\mathscr{P}(\Gamma_{x,y}^{\Lambda_{\varepsilon}^{n}})$
such that
$F_{\Lambda_{\varepsilon}^{n}}^{n}(\alpha_{\varepsilon}^{n},g)\geq-\varepsilon$
for all $g\in{\rm Lip}_{1}({\rm X},[0,n])$. We consider only
$\varepsilon<(8CC_{D}L)^{\frac{1}{p}}{\sf d}(x,y)$. We claim that if
$\Lambda_{0}=4(8CC_{D}L)^{\frac{1}{p}}$ then
$\alpha_{\varepsilon}^{n}(\Gamma_{x,y}^{\Lambda_{0}})\geq\frac{1}{2}$. Indeed
if not then
$\alpha(\Gamma_{x,y}\setminus\Gamma_{x,y}^{\Lambda_{0}})>\frac{1}{2}$. We
apply the condition above to $g=1$ getting
$F^{n}_{\Lambda^{n}_{\varepsilon}}(\alpha^{n}_{\varepsilon},1)\geq-\varepsilon$,
which means

$C^{\frac{1}{p}}{\sf d}(x,y)^{\frac{p-1}{p}}\mathfrak{m}_{x,y}^{L}({\rm
X})^{\frac{1}{p}}-\int\ell(\gamma){\mathrm{d}}\alpha_{\varepsilon}^{n}(\gamma)\geq-\varepsilon.$

This implies

$\frac{\Lambda_{0}}{2}{\sf
d}(x,y)<\int\ell(\gamma){\mathrm{d}}\alpha_{\varepsilon}^{n}(\gamma)\leq
C^{\frac{1}{p}}{\sf d}(x,y)^{\frac{p-1}{p}}\mathfrak{m}_{x,y}^{L}({\rm
X})^{\frac{1}{p}}+\varepsilon\leq 2(8CC_{D}L)^{\frac{1}{p}}{\sf d}(x,y),$

which is a contradiction. In the last inequality we used Lemma 2.3.
Renormalizing the restriction of $\alpha_{\varepsilon}^{n}$ to
$\Gamma_{x,y}^{\Lambda_{0}}$ we obtain the following. For every
$\varepsilon>0$ define
$\beta_{\varepsilon}^{n}:=\alpha_{\varepsilon}^{n}(\Gamma_{x,y}^{\Lambda_{0}})^{-1}\,\alpha_{\varepsilon}^{n}\lower
3.0pt\hbox{$|_{\Gamma_{x,y}^{\Lambda_{0}}}$}\in\mathscr{P}(\Gamma_{x,y}^{\Lambda_{0}})$
and

$\left(\int\int_{\gamma}g\,{\mathrm{d}}s\,{\mathrm{d}}\beta_{\varepsilon}^{n}\right)^{p}\leq
2^{p}C{\sf d}(x,y)^{p-1}\int
g^{p}{\mathrm{d}}\mathfrak{m}_{x,y}^{L}+\varepsilon$

for all $g\in{\rm Lip}_{1}({\rm X},[0,n])$. The family
$\\{\beta_{\varepsilon}^{n}\\}_{\varepsilon>0}$ is relatively compact. Hence
we take a limit point $\beta^{n}$ of the sequence $\beta_{\varepsilon}^{n}$
for $\varepsilon$ going to zero. We have that
$\beta^{n}\in\mathscr{P}(\Gamma_{x,y}^{\Lambda_{0}})$ and

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

for all $g\in{\rm Lip}_{1}({\rm X},[0,n])$, by the properties of weak
convergence, since the map
$\Gamma_{x,y}^{\Lambda_{0}}\ni\gamma\mapsto\int_{\gamma}g\,{\mathrm{d}}s\in[0,\infty)$
is bounded and lower semicontinuous. We now take a limit point
$\beta\in\mathscr{P}(\Gamma_{x,y}^{\Lambda_{0}})$ of the sequence $\beta^{n}$
for $n$ going to $+\infty$. Then we have that

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

for all $g\in{\rm Lip}_{1}({\rm X},[0,+\infty))$ and all $M\in\mathbb{N}$,
again by weak convergence. The right hand side measure is concentrated on a
compact subset of ${\rm X}$, while the left hand side one is concentrated of
curves of bounded length, so whose image lives in a compact subset of ${\rm
X}$. This means that for a fixed $g\in{\rm Lip}_{1}({\rm X},[0,+\infty))$ we
have $\min\\{g,M\\}=g$ if $M$ is big enough. Therefore we get

(44)
$\left(\int\int_{\gamma}g\,{\mathrm{d}}s\,{\mathrm{d}}\beta\right)^{p}\leq
2^{p}C{\sf d}(x,y)^{p-1}\int g^{p}{\mathrm{d}}\mathfrak{m}_{x,y}^{L}$

for all $g\in{\rm Lip}_{1}({\rm X},[0,+\infty))$. So the same estimate holds
for all $g\in{\rm Lip}({\rm X},[0,+\infty))$. A standard approximation
argument (see [16, Corollary 4.2.3 and the theorem below]) gives that (44)
holds for all $g\geq 0$ Borel. In particular, if we set $C^{\prime}=2^{p}C$
and $L^{\prime}=\Lambda_{0}$ we get

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

for all $g\geq 0$ Borel, and $\beta\in\mathscr{P}(\Gamma_{x,y}^{L^{\prime}})$.
This is property (v).
It is clear that (v) implies (iv), that (v) implies (ii) and that (ii) implies
(iii). Moreover (iv) implies (iii), so (ii)-(v) are equivalent. To conclude
the cycle of equivalences we need to show that (v) implies (i) estimating the
$p$-modulus of $\Gamma_{x,y}$. For a given $\rho$ which is admissible for the
definition of $\textup{Mod}_{p}(\Gamma_{x,y},\mathfrak{m}_{x,y}^{L})$ we get

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

so $\textup{Mod}_{p}(\Gamma_{x,y},\mathfrak{m}_{x,y}^{L})\geq\frac{1}{C}{\sf
d}(x,y)^{1-p}$, which implies (i) by Proposition A.1.
Assume now that ${\rm X}$ is locally quasiconvex. It is always true that if
$u\in{\rm Lip}({\rm X})$ then ${\rm lip}u\in{\rm UG}(u)$. So if (i) holds then
(vi) holds as well. On the other hand we adapt the proof of (i) implies (iii)
to show that (vi) implies (iii). Let $g\in{\rm Lip}_{1}({\rm X})$, $g\geq 0$
bounded and define
$u(z)=\inf_{\gamma\in\Gamma_{x,z}}\int_{\gamma}g\,{\mathrm{d}}s$. Let $r_{z}$
be such that every two points belonging to $B_{r_{z}}(z)$ can be joined by a
$\Lambda$-quasigeodesic. Arguing exactly as above we deduce that $u$ is
$(\Lambda\|g\|_{\infty})$-Lipschitz on $B_{r_{z}}(z)$. Therefore $u\in{\rm
Lip}_{\rm loc}({\rm X})$. By Lemma 2.1 we have that the restriction of $u$ to
$\overline{B}_{x,y}^{L}$ is Lipschitz. By McShane Extension Theorem we can
find a map $\tilde{u}\in{\rm Lip}({\rm X})$ that coincides with $u$ on
$\overline{B}_{x,y}^{L}$. Applying (vi) to $\tilde{u}$ we get

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

where the last equality holds because $\mathfrak{m}_{x,y}^{L}$ is concentrated
on the open set $B_{x,y}^{L}$ where $u$ and $\tilde{u}$ coincide. The missing
step is the following estimate: we claim that ${\rm lip}u\leq\Lambda g$.
Indeed let $z\in{\rm X}$ and take $0<r<r_{z}$. For every $w\in B_{r}(z)$ we
can find a curve $\gamma\in\Gamma_{z,w}^{\Lambda}$. By definition of upper
gradient we have

$|u(z)-u(w)|\leq\int_{\gamma}g\leq\Lambda{\sf d}(z,w)\max_{B_{r}(z)}g.$

Since $g$ is continuous we get

$g(z)=\lim_{r\to 0}\max_{B_{r}(z)}g\geq\frac{1}{\Lambda}\lim_{r\to
0}\sup_{w\in B_{r}(z)}\frac{|u(z)-u(w)|}{{\sf d}(z,w)}={\rm lip}u(z).$

Combining this fact with the estimate above we conclude that

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

Then by definition of $u$ we find $\gamma\in\Gamma_{x,y}$ such that

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

i.e. (iii) holds. ∎

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