# Topological regularity of isoperimetric sets
in PI spaces having a deformation property

Gioacchino<EMAIL_ADDRESS>Courant Institute of Mathematical
Sciences (NYU), 251 Mercer Street, 10012, New York, USA., Enrico
<EMAIL_ADDRESS>Department of Mathematics and
Statistics, P.O. Box 35 (MaD), FI-40014 University of Jyvaskyla, Finland.,
Marco<EMAIL_ADDRESS>Dipartimento di Matematica e
Applicazioni, Università di Napoli Federico II, Via Cintia, Monte S. Angelo,
80126 Napoli, Italy., and Ivan Yuri<EMAIL_ADDRESS>Department of
Mathematics and Statistics, P.O. Box 35 (MaD), FI-40014 University of
Jyvaskyla, Finland.

###### Abstract

We prove topological regularity results for isoperimetric sets in PI spaces
having a suitable deformation property, which prescribes a control on the
increment of the perimeter of sets under perturbations with balls. More
precisely, we prove that isoperimetric sets are open, satisfy boundary density
estimates and, under a uniform lower bound on the volumes of unit balls, are
bounded. Our results apply, in particular, to the class of possibly collapsed
$\mathrm{RCD}(K,N)$ spaces.

As a consequence, the rigidity in the isoperimetric inequality on possibly
collapsed $\mathrm{RCD}(0,N)$ spaces with Euclidean volume growth holds
without the additional assumption on the boundedness of isoperimetric sets.

MSC(2020). Primary: 53C23, 49Q20. Secondary: 26B30, 26A45, 49J40.
Keywords. Isoperimetric set, PI space, deformation property.

##  1 Introduction

In this paper we consider length PI spaces, i.e. metric measure spaces $({\rm
X},{\sf d},\mathfrak{m})$ where $\mathfrak{m}$ is a uniformly locally doubling
Borel measure, there holds a weak local $(1,1)$-Poincaré inequality (see
Definition 2.4), and the distance between any two points $x,y$ is realized as
the infimum of the lengths of curves joining $x$ and $y$. The well-established
theory of $BV$ functions on metric measure spaces [3, 33] allows the treatment
of sets of finite perimeter in this generalized setting. Hence, it makes sense
to consider the classical isoperimetric problem, defined by the following
minimization:

$\inf\big{\\{}P(E)\;\big{|}\;E\subseteq{\rm X}\text{ Borel,
}\mathfrak{m}(E)=v\big{\\}},$

for any assigned volume $v\in(0,\mathfrak{m}({\rm X}))$, where $P(E)$ denotes
the perimeter of $E$. A set $E$ minimizing the previous infimum is called an
_isoperimetric set_ , or an _isoperimetric region_.

One of the fundamental questions about isoperimetric sets addresses their
topological regularity. Namely, one aims at proving that, up to the choice of
a representative, isoperimetric sets are open, bounded and enjoy density
estimates at points of the topological boundary. In the Euclidean space,
topological regularity follows from [24], subsequently generalized in [39]. On
Riemannian manifolds the result is due to [34]. In [6] the result has been
generalized to the setting of _noncollapsed_ $\mathsf{RCD}(K,N)$ spaces $({\rm
X},{\sf d},\mathcal{H}^{N})$, i.e. $N\in\mathbb{N}$ and $\mathfrak{m}$
coincides with the Hausdorff measure $\mathcal{H}^{N}$. We mention also [29],
which addresses the case of quasi-minimal sets in PI spaces.

The purpose of this paper is to prove the topological regularity of
isoperimetric sets in the general setting of length PI spaces that enjoy a so-
called _deformation property_ , which we are going to introduce (we refer to
Definition 3.3 for the precise definition). We say that a metric measure space
$({\rm X},{\sf d},\mathfrak{m})$ has the deformation property provided the
following holds: given a set $E\subseteq{\rm X}$ of finite perimeter and a
point $x\in{\rm X}$, we can find $R,C>0$ such that

$P(E\cup B_{r}(y))\leq C\frac{\mathfrak{m}(B_{r}(y)\setminus
E)}{r}+P(E)\quad\text{ for every }y\in B_{R}(x)\text{ and }r\in(0,R).$ (1.1)

Classes of spaces having the deformation property are collected in Remark 3.4.
Notably, the class includes $\mathsf{RCD}(K,N)$ spaces $({\rm X},{\sf
d},\mathfrak{m})$, thanks to [6, Theorem 1.1]. We shall not introduce
$\mathsf{RCD}$ spaces here, and we refer the reader to the survey [4] and to
the references therein.

The general relation between being PI and having the deformation property
seems worth of further investigations. At the moment, we do not have any
example of a PI space that does not enjoy the deformation property.
Nevertheless, it is simple to construct a metric measure space that is not
doubling and that does not satisfy the deformation property, see Remark 3.5.

Deformation properties for sets of locally finite perimeter are well-known in
the smooth context [32, Lemma 17.21], and they represent a tool of crucial
importance in several classical arguments. We refer, for instance, to [2,
VI.2(3)], to [23, Lemma 4.5] and [35, Lemma 3.6] in the sub-Riemannian
setting, and to [20, 36] which study isoperimetric problems in a weighted
setting.

In fact, it is mostly powerful to couple the topological regularity of an
isoperimetric set, or of a set minimizing some variational problem, with the
deformation property. For instance, knowing that such a set $E$ has an open
representative allows to apply (1.1) centered at points $y$ in the interior,
so that $\mathfrak{m}(E\cup B_{r}(y))>\mathfrak{m}(E)$ only for radii $r$
sufficiently large, and thus (1.1) implies that one can increase the volume of
$E$ controlling the perimeter of the deformed set $E\cup B_{r}(y)$ _linearly_
with respect to the increase of mass $\mathfrak{m}(B_{r}(y)\setminus E)$. An
analogous observation holds applying (1.1) to the complement, in case the
complement of the considered set has an open representative.

We can now state our main result, which yields the topological regularity at
the more general level of _volume-constrained minimizers_ of the perimeter,
i.e. sets which minimize the perimeter with respect to any bounded variation
that locally preserves the measure, see Definition 3.1. We will denote by
$E^{(1)}$, $E^{(0)}$, and $\partial^{e}E$ the essential interior, the
essential exterior, and the essential boundary, respectively, of a Borel set
$E\subseteq{\rm X}$; see Section 2.2 for their definitions.

######  Theorem 1.1 (Topological regularity of volume-constrained minimizers).

Let $({\rm X},{\sf d},\mathfrak{m})$ be a length PI space having the
deformation property. Let $E\subseteq{\rm X}$ be a volume-constrained
minimizer of the perimeter. Then $E^{(1)}={\rm int}(E^{(1)})$ and
$E^{(0)}={\rm int}(E^{(0)})$. In particular, it holds that $E^{(1)}$,
$E^{(0)}$ are open sets and $\partial E^{(1)}=\partial E^{(0)}=\partial^{e}E$.

The previous theorem implies density estimates on the volume and on the
perimeter measure of a volume-constrained minimizer at points of the essential
boundary, see Theorem 3.8. For an isoperimetric set, we can additionally prove
its boundedness. Namely:

######  Theorem 1.2.

Let $({\rm X},{\sf d},\mathfrak{m})$ be a length PI space having the
deformation property. Suppose that $\inf_{x\in{\rm
X}}\mathfrak{m}(B_{1}(x))>0$. Let $E\subseteq{\rm X}$ be an isoperimetric set.
Then $E^{(1)}$ is bounded. In particular, every isoperimetric set in ${\rm X}$
has a bounded representative.

Since $\mathsf{RCD}(K,N)$ spaces with $N<\infty$ are length PI spaces (see
[37, 38] and [31]), and as recalled above they have the deformation property,
putting together Theorem 1.1 and Theorem 1.2 we obtain the following.

######  Corollary 1.3.

Let $({\rm X},{\sf d},\mathfrak{m})$ be an $\mathsf{RCD}(K,N)$ space with
$N<\infty$. Let $E\subseteq{\rm X}$ be an isoperimetric set. Then the sets
$E^{(1)},E^{(0)}$ are open and $\partial^{e}E=\partial E^{(1)}=\partial
E^{(0)}$. Moreover, if in addition $\inf_{x\in{\rm
X}}\mathfrak{m}(B_{1}(x))>0$, then $E^{(1)}$ is bounded.

In the case of noncollapsed $\mathsf{RCD}(K,N)$ spaces, the above result has
been previously proved in [6, Theorem 1.4].

As an application of Corollary 1.3, we can refine the rigidity part in the
sharp isoperimetric inequality on $\mathsf{RCD}(0,N)$ spaces $({\rm X},{\sf
d},\mathfrak{m})$ with Euclidean volume growth. We recall that “Euclidean
volume growth” means that the _asymptotic volume ratio_

${\rm AVR}({\rm X},{\sf
d},\mathfrak{m})\coloneqq\lim_{R\to\infty}\frac{\mathfrak{m}(B_{R}(p))}{\omega_{N}R^{N}},\quad\text{
for some }p\in{\rm X},$

of the space is strictly positive. Recall that the existence of the above
limit is guaranteed by the monotonicity of $(0,+\infty)\ni
r\mapsto\mathfrak{m}(B_{r}(p))/\omega_{N}r^{N}$, which in turn follows from
the Bishop–Gromov inequality (see e.g. [38]). Observe that the condition ${\rm
AVR}({\rm X},{\sf d},\mathfrak{m})>0$ implies that $\inf_{x\in{\rm
X}}\mathfrak{m}(B_{1}(x))>0$. The sharp isoperimetric inequality on these
spaces, see (1.2) below, was obtained at different levels of generality in [1,
13, 9, 7, 15, 16]. In [7] the rigidity for the isoperimetric inequality was
proved for noncollapsed $\mathsf{RCD}(0,N)$ spaces. In the recent [16, Theorem
1.5], the authors prove the rigidity for the inequality in all
$\mathsf{RCD}(0,N)$ spaces with Euclidean volume growth under the additional
assumption that the set achieving the equality is bounded. An application of
our Corollary 1.3 allows to drop the previous boundedness requirement, thus
obtaining the following unconditional rigidity statement.

######  Theorem 1.4 (Sharp and rigid isoperimetric inequality on ${\sf
RCD}(0,N)$ spaces with Euclidean volume growth).

Let $({\rm X},{\sf d},\mathfrak{m})$ be an ${\sf RCD}(0,N)$ space with
$1<N<\infty$ having Euclidean volume growth. Then for every set of finite
perimeter $E\subseteq{\rm X}$ with $\mathfrak{m}(E)<+\infty$ it holds that

$P(E)\geq N\omega_{N}^{\frac{1}{N}}{\rm AVR}({\rm X},{\sf
d},\mathfrak{m})^{\frac{1}{N}}\mathfrak{m}(E)^{\frac{N-1}{N}}.$ (1.2)

Moreover, the equality in (1.2) holds for some set of finite perimeter
$E\subseteq{\rm X}$ with $\mathfrak{m}(E)\in(0,+\infty)$ if and only if ${\rm
X}$ is isometric to a Euclidean metric measure cone over an ${\sf
RCD}(N-2,N-1)$ space and $E$ is isometric, up to negligible sets, to a ball
centered at one of the tips of ${\rm X}$.

We now briefly discuss our strategy for the proof of Theorem 1.1. As in [24,
39], we gain information on a volume-constrained minimizer by comparison with
suitable competitors exploiting the deformation property, but our argument is
different, more direct, and much shorter. The classical strategy of [24, 39]
consists in proving first that $E$ has an interior and an exterior point, i.e.
${\rm int}(E^{(1)})\neq\varnothing$ and ${\rm int}(E^{(0)})\neq\varnothing$
(see [24, Theorem 1]), then one deduces that $E$ is a
$(\Lambda,r_{0})$-perimeter minimizer, and thus finally that $E$ is open.
Instead, we prove directly that if $x\in E^{(0)}$ and $y\in E^{(1)}$ are
arbitrary points, then $x\in{\rm int}(E^{(0)})$ and $y\in{\rm int}(E^{(1)})$.
To do so we avoid deriving quantitative estimates on the decay of
$\mathfrak{m}(B_{r}(y)\setminus E)$ as in [24, 39, 6], and we rather adopt a
more qualitative approach. More precisely, the key point is to show (see the
key Lemma 3.6) that if the function $v(r)\coloneqq\mathfrak{m}(B_{r}(x)\cap
E)$ vanishes, as $r\to 0^{+}$, slower than the function
$w(r)\coloneqq\mathfrak{m}(B_{r}(y)\setminus E)$, then $x\in{\rm
int}(E^{(0)})$ (and vice versa for $y\in{\rm int}(E^{(1)})$). By “slower” we
mean, roughly speaking, that $v(r)\geq w(r)$ for many $r>0$ in a measure-
theoretic sense (see Lemma 3.6 for the precise statement). However, up to
exchanging $E$ with its complement ${\rm X}\setminus E$, we can always ensure
that $v(r)$ vanishes slower than $w(r)$, thus deducing that $x\in{\rm
int}(E^{(0)})$. By symmetry, we get $y\in{\rm int}(E^{(1)})$ as well.

We conclude the introduction by pointing out that the strategy of [24, 39]
does not seem to generalize to our setting, unless we require additional
assumptions – such as Ahlfors regularity – which we do not want to make (in
order to obtain a result which applies to the whole class of collapsed ${\sf
RCD}(K,N)$ spaces). This motivated us to look for an alternative proof of the
topological result, which – we believe – is of interest even in the Euclidean
setting.

Acknowledgements. Part of this research has been carried out at the Fields
Institute (Toronto) in November 2022, during the Thematic Program on Nonsmooth
Riemannian and Lorentzian Geometry. The authors gratefully acknowledge the
warm hospitality and the stimulating atmosphere. The third author is partially
supported by the INdAM - GNAMPA Project 2022 CUP_ E55F22000270001
“Isoperimetric problems: variational and geometric aspects”. The fourth author
is supported by the Academy of Finland project _Incidences on Fractals_ ,
Grant No. 321896. The authors thank Camillo Brena, Vesa Julin, Tapio Rajala,
and Daniele Semola for fruitful discussions on the topic of the paper.

##  2 Preliminaries

Given a metric space $({\rm X},{\sf d})$, we denote by ${\rm
LIP}_{\mathrm{loc}}({\rm X})$ the space of all locally Lipschitz functions
from ${\rm X}$ to $\mathbb{R}$. The _slope_ ${\rm lip}(f)\colon{\rm
X}\to[0,+\infty)$ of a function $f\in{\rm LIP}_{\mathrm{loc}}({\rm X})$ is
defined as ${\rm lip}(f)(x)\coloneqq 0$ if $x\in{\rm X}$ is an isolated point
and

${\rm lip}(f)(x)\coloneqq\varlimsup_{y\to x}\frac{|f(x)-f(y)|}{{\sf
d}(x,y)}\quad\text{ if }x\in{\rm X}\text{ is an accumulation point.}$

The topological interior and the topological boundary of a set $E\subseteq{\rm
X}$ are denoted by ${\rm int}(E)$ and $\partial E$, respectively.

###  2.1 Sets of finite perimeter in metric measure spaces

In this paper, by a _metric measure space_ $({\rm X},{\sf d},\mathfrak{m})$ we
mean a complete separable metric space $({\rm X},{\sf d})$ together with a
boundedly-finite Borel measure $\mathfrak{m}\geq 0$ on ${\rm X}$. Following
[33], we define the _total variation_ $|{\bf D}f|(B)\in[0,+\infty]$ of a given
function $f\in L^{1}_{\mathrm{loc}}({\rm X})$ in a Borel set $B\subseteq{\rm
X}$ as

$|{\bf D}f|(B)\coloneqq\inf_{B\subseteq\Omega\text{
open}}\inf\bigg{\\{}\varliminf_{n\to\infty}\int_{\Omega}{\rm
lip}(f_{n})\,{\mathrm{d}}\mathfrak{m}\;\bigg{|}\;(f_{n})_{n\in\mathbb{N}}\subseteq{\rm
LIP}_{\mathrm{loc}}(\Omega),\,f_{n}\to f\text{ in
}L^{1}_{\mathrm{loc}}(\Omega)\bigg{\\}}.$

If for some open cover $(\Omega_{n})_{n\in\mathbb{N}}$ of ${\rm X}$ we have
that $|{\bf D}f|(\Omega_{n})<+\infty$ holds for every $n\in\mathbb{N}$, then
$|{\bf D}f|$ is a locally-finite Borel measure on ${\rm X}$. We say that a
Borel set $E\subseteq{\rm X}$ is _of locally finite perimeter_ if
$P(E,\cdot)\coloneqq|{\bf D}{\raise 1.29167pt\hbox{$\chi$}}_{E}|$ is a
locally-finite measure, called the _perimeter measure_ of $E$. When
$P(E)\coloneqq P(E,{\rm X})<+\infty$, we say that $E$ is _of finite
perimeter_.

######  Remark 2.1.

If $E\subseteq{\rm X}$ is a set of locally finite perimeter and $x\in{\rm X}$
is a given point, then $P(E,\partial B_{r}(x))=0$ for all but countably many
radii $r>0$. This is due to the fact that $\partial B_{r}(x)\cap\partial
B_{s}(x)=\varnothing$ whenever $0<r<s$.  $\blacksquare$

Given any function $f\in{\rm LIP}_{\mathrm{loc}}({\rm X})$, it holds that
$|{\bf D}f|$ is a locally-finite measure and $|{\bf D}f|\leq{\rm
lip}(f)\mathfrak{m}$.

######  Theorem 2.2 (Coarea formula [33, Proposition 4.2]).

Let $({\rm X},{\sf d},\mathfrak{m})$ be a metric measure space. Fix any $f\in
L^{1}_{\mathrm{loc}}({\rm X})$ such that $|{\bf D}f|$ is a locally-finite
measure. Fix a Borel set $E\subseteq{\rm X}$. Then $\mathbb{R}\ni t\mapsto
P(\\{f<t\\},E)\in[0,+\infty]$ is a Borel measurable function and it holds that

$|{\bf D}f|(E)=\int_{\mathbb{R}}P(\\{f<t\\},E)\,{\mathrm{d}}t.$

######  Corollary 2.3.

Let $({\rm X},{\sf d},\mathfrak{m})$ be a metric measure space. Fix $x\in{\rm
X}$ and a Borel set $E\subseteq{\rm X}$. Define
$f\colon(0,+\infty)\to\mathbb{R}$ as $f(r)\coloneqq|{\bf D}{\sf d}_{x}|(E\cap
B_{r}(x))$ for every $r>0$, where we denote ${\sf d}_{x}\coloneqq{\sf
d}(\cdot,x)\in{\rm LIP}_{\mathrm{loc}}({\rm X})$. Then the function $f$ is
locally absolutely continuous and it holds that $f^{\prime}(r)=P(B_{r}(x),E)$
for $\mathcal{L}^{1}$-a.e. $r>0$.

###### Proof.

By virtue of the coarea formula, we obtain that
$f(r)=\int_{\mathbb{R}}P\big{(}\\{{\sf d}_{x}<s\\},E\cap
B_{r}(x)\big{)}\,{\mathrm{d}}r=\int_{0}^{r}P(B_{s}(x),E)\,{\mathrm{d}}r$ for
every $r>0$, whence it follows that
$f(r)-f(\tilde{r})=\int_{\tilde{r}}^{r}P(B_{s}(x),E)\,{\mathrm{d}}s$ for every
$r>\tilde{r}>0$. Hence, $f$ is locally absolutely continuous and
$f^{\prime}(r)=P(B_{r}(x),E)$ for every Lebesgue point $r$ of $s\mapsto
P(B_{s}(x),E)$, thus for $\mathcal{L}^{1}$-a.e. $r>0$. ∎

###  2.2 PI spaces

Even though the general theory of sets of finite perimeter is meaningful in
any metric measure space, a much more refined calculus is available in the
class of doubling spaces supporting a weak form of $(1,1)$-Poincaré
inequality, which we refer to as _PI spaces_. Below we recall the definition
of PI space we adopt in this paper, referring e.g. to [12, 27] for a thorough
account of this topic. We will also recall some key features of sets of finite
perimeter in PI spaces.

######  Definition 2.4 (PI space).

Let $({\rm X},{\sf d},\mathfrak{m})$ be a metric measure space. Then:

  * •

We say that $({\rm X},{\sf d},\mathfrak{m})$ is _uniformly locally doubling_
if there exists a function $C_{D}\colon(0,+\infty)\to(0,+\infty)$ such that

$\mathfrak{m}\big{(}B_{2r}(x)\big{)}\leq
C_{D}(R)\,\mathfrak{m}\big{(}B_{r}(x)\big{)}\quad\text{ for every }0<r<R\text{
and }x\in{\rm X}.$

  * •

We say that $({\rm X},{\sf d},\mathfrak{m})$ supports a _weak local
$(1,1)$-Poincaré inequality_ if there exist a constant $\lambda\geq 1$ and a
function $C_{P}\colon(0,+\infty)\to(0,+\infty)$ such that for any function
$f\in{\rm LIP}_{\mathrm{loc}}({\rm X})$ it holds that

$\fint_{B_{r}(x)}\bigg{|}f-\fint_{B_{r}(x)}f\,{\mathrm{d}}\mathfrak{m}\bigg{|}\,{\mathrm{d}}\mathfrak{m}\leq
C_{P}(R)\,r\fint_{B_{\lambda r}(x)}{\rm
lip}(f)\,{\mathrm{d}}\mathfrak{m}\quad\text{ for every }0<r<R\text{ and
}x\in{\rm X}.$

  * •

$({\rm X},{\sf d},\mathfrak{m})$ is a _PI space_ if it is uniformly locally
doubling and it supports a weak local $(1,1)$-Poincaré inequality.

######  Remark 2.5.

Let $({\rm X},{\sf d},\mathfrak{m})$ be a PI space such that $({\rm X},{\sf
d})$ is a _length space_ , i.e. the distance between any two points in ${\rm
X}$ is the infimum of the lengths of rectifiable curves joining them. Then the
weak local $(1,1)$-Poincaré inequality is in fact _strong_ , namely it holds
with $\lambda=1$; see for example [25, Corollary 9.5 and Theorem 9.7].
$\blacksquare$

Given a Borel set $E\subseteq{\rm X}$ in a PI space $({\rm X},{\sf
d},\mathfrak{m})$, we define its _essential interior_ and _essential exterior_
as

$E^{(1)}\coloneqq\bigg{\\{}x\in{\rm X}\;\bigg{|}\;\lim_{r\to
0}\frac{\mathfrak{m}(E\cap
B_{r}(x))}{\mathfrak{m}(B_{r}(x))}=1\bigg{\\}},\qquad
E^{(0)}\coloneqq\bigg{\\{}x\in{\rm X}\;\bigg{|}\;\lim_{r\to
0}\frac{\mathfrak{m}(E\cap B_{r}(x))}{\mathfrak{m}(B_{r}(x))}=0\bigg{\\}},$

respectively. The _essential boundary_ of $E$ is defined as
$\partial^{e}E\coloneqq{\rm X}\setminus(E^{(1)}\cup E^{(0)})$. Notice that
$E^{(1)}$, $E^{(0)}$, and $\partial^{e}E$ are Borel sets and that
$\partial^{e}E\subseteq\partial E$. It follows from the Lebesgue
differentiation theorem (which holds on every uniformly locally doubling
metric measure space, see e.g. [26, Theorem 1.8]) that
$\mathfrak{m}(E^{(1)}\Delta E)=0$ and $\mathfrak{m}(E^{(0)}\Delta({\rm
X}\setminus E))=0$. Moreover, if $E$ is a set of finite perimeter, then we
know from [3, Theorem 5.3] that $P(E,\cdot)$ is concentrated on
$\partial^{e}E$.

######  Proposition 2.6.

Let $({\rm X},{\sf d},\mathfrak{m})$ be a PI space. Let $E,F\subseteq{\rm X}$
be sets of locally finite perimeter with $P(E,\partial^{e}F)=0$. Then

$P(E\cap F,\cdot)\leq P(E,\cdot)\mathop{\hbox{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits
F^{(1)}+P(F,\cdot)\mathop{\hbox{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits E^{(1)}.$

###### Proof.

We know from [3, Theorem 5.3] that the perimeter measure $P(G,\cdot)$ of a set
$G\subseteq{\rm X}$ of locally finite perimeter can be written as
$P(G,\cdot)=\theta_{G}\mathcal{H}^{h}|_{\partial^{e}G}$ for some Borel
function $\theta_{G}\colon{\rm X}\to(0,+\infty)$, where $\mathcal{H}^{h}$
stands for the codimension-one Hausdorff measure (see [3, Section 5]). Since
$P(E\cap F,\cdot)\leq P(E,\cdot)+P(F,\cdot)$ and $P(E,\cdot)|_{{\rm
X}\setminus\partial^{e}E}=P(F,\cdot)|_{{\rm X}\setminus\partial^{e}F}=0$, we
deduce that $\theta_{E\cap F}\leq\theta_{E}$ and $\theta_{E\cap
F}\leq\theta_{F}$ hold $\mathcal{H}^{h}$-a.e. in
$\partial^{e}E\setminus\partial^{e}F$ and
$\partial^{e}F\setminus\partial^{e}E$, respectively. Moreover, we deduce from
$\int_{\partial^{e}F}\theta_{E}\,{\mathrm{d}}\mathcal{H}^{h}|_{\partial^{e}E}=P(E,\partial^{e}F)=0$
that $\mathcal{H}^{h}(\partial^{e}E\cap\partial^{e}F)=0$. Given that
$\partial^{e}(E\cap F)=(\partial^{e}E\cap F^{(1)})\sqcup(\partial^{e}F\cap
E^{(1)})$ up to an $\mathcal{H}^{h}$-negligible set, which is shown e.g. in
the proof of [6, Lemma 2.5], we conclude that

$\begin{split}P(E\cap F,\cdot)&=\theta_{E\cap
F}\mathcal{H}^{h}|_{\partial^{e}(E\cap F)}=\theta_{E\cap
F}\mathcal{H}^{h}|_{\partial^{e}E\cap F^{(1)}}+\theta_{E\cap
F}\mathcal{H}^{h}|_{\partial^{e}F\cap
E^{(1)}}\leq\theta_{E}\mathcal{H}^{h}|_{\partial^{e}E\cap
F^{(1)}}+\theta_{F}\mathcal{H}^{h}|_{\partial^{e}F\cap E^{(1)}},\end{split}$

which yields the statement. ∎

The following is a direct consequence of the study in [3], taking Remark 2.5
into account.

######  Theorem 2.7 (Relative isoperimetric inequality [3, Remark 4.4]).

Let $({\rm X},{\sf d},\mathfrak{m})$ be a length PI space. Then there exists a
function
$C_{I}=C_{I}(C_{D},C_{P})\colon(1,+\infty)\times(0,+\infty)\to(0,+\infty)$
such that the following property holds: given a set $E\subseteq{\rm X}$ of
finite perimeter, a radius $R>0$, and an exponent
$\alpha>\max\\{\log_{2}(C_{D}(R)),1\\}$, we have that

$\min\big{\\{}\mathfrak{m}(B_{r}(x)\cap E),\mathfrak{m}(B_{r}(x)\setminus
E)\big{\\}}\leq
C_{I}(\alpha,R)\bigg{(}\frac{r^{\alpha}}{\mathfrak{m}\big{(}B_{r}(x)\big{)}}\bigg{)}^{\frac{1}{\alpha-1}}P\big{(}E,B_{r}(x)\big{)}^{\frac{\alpha}{\alpha-1}},$

for every $x\in{\rm X}$ and $r\in(0,R)$.

In the next proposition we recall the well-known fact that in the class of PI
spaces where unit balls have measure uniformly bounded away from zero, there
holds an isoperimetric inequality for sets of small volume. Such a result is
essentially due to [22], after [14, 28, 19]. For a proof, we refer the reader
to the argument in [18, Lemma V.2.1].

######  Proposition 2.8 (Isoperimetric inequality for small volumes).

Let $({\rm X},{\sf d},\mathfrak{m})$ be a length PI space. Then there exist
constants $\alpha>1$, $C>0$ such that the following holds. If
$v_{0}\coloneqq\inf_{x\in{\rm X}}\mathfrak{m}(B_{1}(x))>0$, then for all Borel
sets $E\subseteq{\rm X}$ with $\mathfrak{m}(E)<v_{0}/2$ it holds that

$P(E)\geq Cv_{0}^{\frac{1}{\alpha}}\mathfrak{m}(E)^{\frac{\alpha-1}{\alpha}}.$

######  Remark 2.9.

Let $({\rm X},{\sf d},\mathfrak{m})$ be a length PI space and $E\subseteq{\rm
X}$ a set of finite perimeter such that $\mathfrak{m}(E),\mathfrak{m}({\rm
X}\setminus E)>0$. Then the relative isoperimetric inequality ensures that
$P(E)\neq 0$. In order to prove it, fix any $x\in{\rm X}$ and notice that we
have $\mathfrak{m}(B_{R}(x)\cap E),\mathfrak{m}(B_{R}(x)\setminus E)>0$ for
some $R>0$ sufficiently large, thus $P(E)\geq P(E,B_{R}(x))>0$.
$\blacksquare$

######  Lemma 2.10.

Let $({\rm X},{\sf d},\mathfrak{m})$ be a length PI space. Then there exists
$c=c(\inf C_{D},\inf C_{P})\in(0,1)$ such that

$c\,\mathfrak{m}\leq|{\bf D}{\sf d}_{x}|\leq\mathfrak{m}\quad\text{ for every
}x\in{\rm X},$ (2.1)

where we denote ${\sf d}_{x}\coloneqq{\sf d}(\cdot,x)\in{\rm
LIP}_{\mathrm{loc}}({\rm X})$. In particular, it holds that

$\mathfrak{m}\big{(}\partial B_{r}(x)\big{)}=0\quad\text{ for every }x\in{\rm
X}\text{ and }r>0.$ (2.2)

###### Proof.

Recall that $|{\bf D}{\sf d}_{x}|\leq{\rm lip}({\sf d}_{x})\mathfrak{m}$.
Moreover, we deduce from [5, Equation (4.5)] that there exists a constant
$c=c(\inf C_{D},\inf C_{P})\in(0,1)$ such that $c\,{\rm lip}({\sf
d}_{x})\mathfrak{m}\leq|{\bf D}{\sf d}_{x}|$. To obtain (2.1), observe that
${\rm lip}({\sf d}_{x})\equiv 1$: the inequality ${\rm lip}({\sf d}_{x})\leq
1$ holds in any metric space, while the converse inequality readily follows
from the assumption that $({\rm X},{\sf d})$ is a length space. Finally, (2.2)
can be proved by combining (2.1) with the coarea formula: we can estimate

$\mathfrak{m}\big{(}\partial B_{r}(x)\big{)}\leq\frac{1}{c}|{\bf D}{\sf
d}_{x}|\big{(}\partial
B_{r}(x)\big{)}=\frac{1}{c}\int_{\mathbb{R}}P\big{(}B_{s}(x),\partial
B_{r}(x)\big{)}\,{\mathrm{d}}s=0,$

where the last identity follows from the fact that $P(B_{s}(x),\cdot)$ is
concentrated on $\partial^{e}B_{s}(x)\subseteq\partial B_{s}(x)$. ∎

##  3 Topological regularity

Let us begin with the definition of a volume-constrained minimizer of the
perimeter.

######  Definition 3.1 (Volume-constrained minimizer).

Let $({\rm X},{\sf d},\mathfrak{m})$ be a metric measure space. Then a set
$E\subseteq{\rm X}$ of locally finite perimeter is said to be a _volume-
constrained minimizer_ of the perimeter if the following property is verified:
given a Borel set $F\subseteq{\rm X}$ and a compact set $K\subseteq{\rm X}$
satisfying $\mathfrak{m}((E\Delta F)\setminus K)=0$ and $\mathfrak{m}(E\cap
K)=\mathfrak{m}(F\cap K)$, it holds that $P(E,K)\leq P(F,K)$.

Observe that $E$ is a volume-constrained minimizer if and only if ${\rm
X}\setminus E$ is a volume-constrained minimizer.

######  Remark 3.2.

An _isoperimetric set_ , i.e. a set $E\subseteq{\rm X}$ of finite perimeter
such that $0<\mathfrak{m}(E)<+\infty$ and such that $P(E)\leq P(F)$ for any
Borel set $F\subseteq{\rm X}$ with $\mathfrak{m}(F)=\mathfrak{m}(E)$, is a
volume-constrained minimizer of the perimeter.  $\blacksquare$

Next we introduce our definition of a metric measure space having the
_deformation property_ , which will be our standing assumption throughout the
rest of the paper.

######  Definition 3.3 (Deformation property).

Let $({\rm X},{\sf d},\mathfrak{m})$ be a metric measure space. Then we say
that $({\rm X},{\sf d},\mathfrak{m})$ has the _deformation property_ if the
following property holds: for every set of locally finite perimeter
$E\subseteq{\rm X}$ and any point $x\in{\rm X}$, there exist constants
$R\in(0,1]$ and $C\geq 0$ such that

$\displaystyle P(E\setminus B_{r}(y),K)\leq C\frac{\mathfrak{m}(B_{r}(y)\cap
E)}{r}+P(E,K)$ $\displaystyle\quad\text{ for every }y\in B_{R}(x)\text{ and
}r\in(0,R),$ (3.1a) $\displaystyle P(E\cup B_{r}(y),K)\leq
C\frac{\mathfrak{m}(B_{r}(y)\setminus E)}{r}+P(E,K)$ $\displaystyle\quad\text{
for every }y\in B_{R}(x)\text{ and }r\in(0,R),$ (3.1b)

for any bounded closed set $K\subseteq{\rm X}$ such that $B_{2R}(x)\subseteq
K$. For convenience, we define from now on $R_{x}(E)\in(0,1]$ to be the
maximal $R\in(0,1]$ such that the above holds for some $C\geq 0$ and we define
$C_{x}(E)\geq 0$ to be the minimal constant such that (3.1a) and (3.1b) hold
with $R=R_{x}(E).$ Note that, by symmetry, we have that $R_{x}(E)=R_{x}({\rm
X}\setminus E)$ and $C_{x}(E)=C_{x}({\rm X}\setminus E)$; this is the reason
why in Definition 3.3 we require the validity of both (3.1a) and (3.1b) with
the same constants $C$ and $R$.

######  Remark 3.4 (Spaces having the deformation property).

These are some spaces with the deformation property:

  * $\rm i)$

Euclidean spaces (see e.g. [24] and the references therein).

  * $\rm ii)$

Riemannian manifolds (this can be proved e.g. by following the proof of [6,
Theorem 1.1] and using the fact that the Ricci curvature is locally bounded
from below).

  * $\rm iii)$

${\sf RCD}(K,N)$ spaces with $K\in\mathbb{R}$ and $N\in[1,\infty)$ (see [6,
Theorem 1.1]).

We point out that in the above cases a stronger version of the deformation
property holds, since, given an arbitrary $R>0$, the constants $C_{x}(E)$ for
which the deformation property holds at every point $x\in{\rm X}$ and for
every $0<r<R$, can be chosen to be independent of $E,x$, and to be dependent
only on $K,N,R$.

It would be interesting to study whether there are other distinguished
examples of PI spaces having the deformation property. One natural class to
investigate is the one of sub-Riemannian manifolds, or, more specifically, the
one of Carnot groups. For example, in the first Heisenberg group one has a
sub-Laplacian comparison theorem. Being $r$ the Carnot–Carathéodory distance
from the origin, we have that $\Delta_{\mathrm{H}}r\leq 4/r$ holds in the
distributional sense, where $\Delta_{\mathrm{H}}$ is the horizontal Laplacian.
See [11] for the study of sub-Laplacian comparison theorems in more general
sub-Riemannian structures, and [17, Corollary 4.19] for the Laplacian
comparison theorem in arbitrary essentially non-branching $\sf MCP$ spaces.
Then, coupling this with the Gauss–Green formulae for Carnot groups in [21],
one could argue following the lines of [6, Theorem 2.32 and Theorem 1.1] to
obtain that at least $\mathbb{H}^{1}$, and more in general all the groups that
are essentially non-branching ${\sf MCP}(K,N)$ spaces, with $K\in\mathbb{R}$
and $N\in(1,\infty)$ (cf. [10], [8]), have the deformation property. Since
this is out of the scope of the present note, and since there are also some
regularity issues of the distance function to deal with, we do not treat these
examples here, but we leave it to possible future investigations.

We mention that, on the other hand, the topological regularity of
isoperimetric sets is already proved in [30] in the setting of Carnot groups
and in [23] on a certain class of sub-Riemannian manifolds.  $\blacksquare$

######  Remark 3.5.

There exist metric measure spaces where the deformation property fails. For
example, fix a continuous function $\varphi\colon[0,+\infty)\to[0,+\infty)$
with $\varphi(0)=0$ and $\varphi(t)>0$ for every $t>0$. Define the density
function $\rho\colon\mathbb{R}^{n}\to\mathbb{R}$ as
$\rho(x)\coloneqq\varphi(|x|)$ for every $x\in\mathbb{R}^{n}$. Now consider
the metric measure space $(\mathbb{R}^{n},|\cdot|,\mathfrak{m})$, where we set
$\mathfrak{m}\coloneqq\rho\mathcal{L}^{n}$. It is easy to check that
$\mathfrak{m}(B_{r}(0))=\int_{0}^{r}\varphi(t)t^{n-1}\,{\mathrm{d}}t$ and
$P(B_{r}(0))=\frac{{\mathrm{d}}}{{\mathrm{d}}r}\mathfrak{m}(B_{r}(0))=\varphi(r)r^{n-1}$
for every $r>0$. Therefore, if we choose $\varphi$ so that the function
$\psi(r)\coloneqq\int_{0}^{r}\varphi(t)t^{n-1}\,{\mathrm{d}}t$ satisfies
$r\psi^{\prime}(r)/\psi(r)\to+\infty$ as $r\to\infty$, then

$\frac{rP(B_{r}(0))}{\mathfrak{m}(B_{r}(0))}\to+\infty\quad\text{ as }r\to 0,$
(3.2)

showing that (3.1b) fails for every $C>0$ (when $E=\varnothing$ and $y=0$).
Notice that, for example, $\varphi(t)=e^{-1/t}/t^{n+1}$ does the job. However,
integrating (3.2) over suitably small intervals $(r,2r)$ readily implies that
$\mathfrak{m}$ is not doubling. In fact, it is an open problem to find an
example of a PI space not satisfying the deformation property.  $\blacksquare$

Given a metric measure space $({\rm X},{\sf d},\mathfrak{m})$, a point
$x\in{\rm X}$, and a Borel set $E\subseteq{\rm X}$, we introduce the notation

$v_{E,x}(r)\coloneqq\mathfrak{m}(B_{r}(x)\cap E),\quad
w_{E,x}(r)\coloneqq\mathfrak{m}(B_{r}(x)\setminus E)\quad\text{ for every
}r>0.$ (3.3)

The core of the proof of our main Theorem 1.1 is contained in the following
technical result.

######  Lemma 3.6.

Let $({\rm X},{\sf d},\mathfrak{m})$ be a length PI space having the
deformation property. Let $E\subseteq{\rm X}$ be a volume-constrained
minimizer of the perimeter. Fix any $x\in E^{(0)}$ and $y\in E^{(1)}$. Define
the functions $v_{E,x},w_{E,y}\colon(0,+\infty)\to[0,+\infty)$ as in (3.3).
Fix a sequence $(r_{n})_{n}\subseteq(0,1)$ such that $r_{n}\to 0$. For any
$n\in\mathbb{N}$, we define the Borel set
$A_{E,r_{n}}^{x,y}\subseteq(0,r_{n})$ as

$A_{E,r_{n}}^{x,y}\coloneqq\big{\\{}r\in(0,r_{n})\;\big{|}\;v_{E,x}(r)\geq
w_{E,y}(r)\big{\\}}.$ (3.4)

Suppose the following conditions are verified:

  * $\rm i)$

There exists $\delta\in(0,R_{y}(E))$ such that
$\bar{B}_{\delta}(x)\cap\bar{B}_{\delta}(y)=\varnothing$, $v_{E,x}(\delta)>0$,
and $w_{E,y}(\delta)>0$.

  * $\rm ii)$

The inequality $\mathcal{L}^{1}(A_{E,r_{n}}^{x,y})\geq r_{n}/2$ holds for
infinitely many $n\in\mathbb{N}$.

Then it holds that $x\in{\rm int}(E^{(0)})$.

###### Proof.

We argue by contradiction: suppose that $x\notin{\rm int}(E^{(0)})$. Recalling
that $\mathfrak{m}(B_{\delta}(y)\setminus E)=w_{E,y}(\delta)>0$ and noticing
that $\mathfrak{m}(B_{r}(x)\cap E)\to 0$ as $r\to 0$, we can extract a (not
relabeled) subsequence of $(r_{n})_{n}$ for which

$r_{n}<\delta,\qquad\mathfrak{m}(B_{r_{n}}(x)\cap
E)<\mathfrak{m}(B_{\delta}(y)\setminus
E),\qquad\mathcal{L}^{1}(A_{E,r_{n}}^{x,y})\geq\frac{r_{n}}{2},$ (3.5)

for every $n\in\mathbb{N}$. Now let $n\in\mathbb{N}$ be fixed. We claim that
for any $r\in A_{n}\coloneqq A_{E,r_{n}}^{x,y}$ there exists
$s(r)\in[r,\delta)$ such that

$v_{E,x}(r)=\mathfrak{m}(B_{r}(x)\cap E)=\mathfrak{m}(B_{s(r)}(y)\setminus
E)=w_{E,y}(s(r)).$

Indeed, if $w_{E,y}(r)=v_{E,x}(r)$, then we can take $s(r)\coloneqq r$. If
$w_{E,y}(r)\neq v_{E,x}(r)$, then $v_{E,x}(r)>w_{E,y}(r)$ by definition of
$A_{n}$, thus the continuity of $w_{E,y}$ (which follows from (2.2)) ensures
that $w_{E,y}(s(r))=v_{E,x}(r)$ for some $s(r)>0$. Since $w_{E,y}$ is non-
decreasing, we infer that $s(r)\geq r$. Moreover, the second inequality in
(3.5) implies that $s(r)<\delta$.

Given any $r\in A_{n}$, we define the Borel set $E_{r}\subseteq{\rm X}$ as
$E_{r}\coloneqq(E\setminus B_{r}(x))\cup B_{s(r)}(y)$. The first inequality in
(3.5) ensures that $\bar{B}_{r}(x)\cap\bar{B}_{s(r)}(y)=\varnothing$, whence
it follows that
$\mathfrak{m}\big{(}E_{r}\cap(\bar{B}_{r}(x)\cup\bar{B}_{s(r)}(y))\big{)}=\mathfrak{m}\big{(}E\cap(\bar{B}_{r}(x)\cup\bar{B}_{s(r)}(y))\big{)}$.
Let $K\subseteq{\rm X}$ be a compact set such that
$\bar{B}_{2\delta}(x)\cup\bar{B}_{2\delta}(y)\subseteq K$. The assumption that
$E$ is a volume-constrained minimizer of the perimeter then implies that
$P(E,K)\leq P(E_{r},K)$. For ease of notation from now on we will denote
$C_{y}(E)$ simply by $C_{y}$. Thanks to Proposition 2.6, Remark 2.1, the
deformation property, and $s(r)\geq r$, we deduce that for
$\mathcal{L}^{1}$-a.e. $r\in A_{n}$ one has

$\begin{split}P(E,K)&\leq P(E_{r},K)\leq P\big{(}E\cup
B_{s(r)}(y),B_{r}(x)^{(0)}\cap K\big{)}+P\big{(}B_{r}(x),(E\cup
B_{s(r)}(y))^{(1)}\cap K\big{)}\\\ &=P(E\cup B_{s(r)}(y),K)-P\big{(}E\cup
B_{s(r)}(y),\partial B_{r}(x)\cup B_{r}(x)^{(1)}\big{)}+P(B_{r}(x),E^{(1)})\\\
&\leq P(E\cup B_{s(r)}(y),K)-P(E,B_{r}(x))+P(B_{r}(x),E^{(1)})\\\ &\leq
P(E,K)+C_{y}\frac{\mathfrak{m}(B_{s(r)}(y)\setminus
E)}{s(r)}-P(E,B_{r}(x))+P(B_{r}(x),E^{(1)})\\\
&=P(E,K)+C_{y}\frac{\mathfrak{m}(B_{r}(x)\cap
E)}{s(r)}-P(E,B_{r}(x))+P(B_{r}(x),E^{(1)})\\\ &\leq
P(E,K)+C_{y}\frac{\mathfrak{m}(B_{r}(x)\cap
E)}{r}-P(E,B_{r}(x))+P(B_{r}(x),E^{(1)}).\end{split}$

Notice that the constant $C_{y}$ depends on $y$ and $E$, but neither on $n$
nor on $r$. Therefore, we have shown that

$P(E,B_{r}(x))\leq C_{y}\frac{\mathfrak{m}(B_{r}(x)\cap
E)}{r}+P(B_{r}(x),E^{(1)})\quad\text{ for every }n\in\mathbb{N}\text{ and
}\mathcal{L}^{1}\text{-a.e.\ }r\in A_{n}.$ (3.6)

Now fix any $\alpha>\max\\{\log_{2}(C_{D}(\delta)),1\\}$. We know from the
relative isoperimetric inequality, i.e. Theorem 2.7, that

$P(E,B_{r}(x))\geq
2\tilde{C}\min\\{v_{E,x}(r),w_{E,x}(r)\big{\\}}^{1-\frac{1}{\alpha}}\frac{\mathfrak{m}(B_{r}(x))^{\frac{1}{\alpha}}}{r}\quad\text{
for every }n\in\mathbb{N}\text{ and }r\in A_{n},$ (3.7)

where we define $\tilde{C}\coloneqq
1/2C_{I}(\alpha,\delta)^{(\alpha-1)/\alpha}$. Exploiting the fact that $x\in
E^{(0)}$, we can find $\bar{n}\in\mathbb{N}$ such that

$v_{E,x}(r)<w_{E,x}(r),\qquad C_{y}\bigg{(}\frac{\mathfrak{m}(B_{r}(x)\cap
E)}{\mathfrak{m}(B_{r}(x))}\bigg{)}^{\frac{1}{\alpha}}\leq\tilde{C}\quad\text{
for every }n\geq\bar{n}\text{ and }r\in A_{n}.$ (3.8)

By combining (3.6), (3.7), and (3.8), we deduce that for every $n\geq\bar{n}$
and $\mathcal{L}^{1}$-a.e. $r\in A_{n}$ it holds that

$\begin{split}2\tilde{C}\mathfrak{m}(B_{r}(x)\cap
E)^{1-\frac{1}{\alpha}}\frac{\mathfrak{m}(B_{r}(x))^{\frac{1}{\alpha}}}{r}&\leq
P(B_{r}(x),E^{(1)})+\mathfrak{m}(B_{r}(x)\cap
E)^{1-\frac{1}{\alpha}}C_{y}\bigg{(}\frac{\mathfrak{m}(B_{r}(x)\cap
E)}{\mathfrak{m}(B_{r}(x))}\bigg{)}^{\frac{1}{\alpha}}\frac{\mathfrak{m}(B_{r}(x))^{\frac{1}{\alpha}}}{r}\\\
&\leq P(B_{r}(x),E^{(1)})+\tilde{C}\mathfrak{m}(B_{r}(x)\cap
E)^{1-\frac{1}{\alpha}}\frac{\mathfrak{m}(B_{r}(x))^{\frac{1}{\alpha}}}{r}.\end{split}$

Rearranging the terms, we infer that

$\tilde{C}\frac{\mathfrak{m}(B_{r}(x))^{\frac{1}{\alpha}}}{r}\mathfrak{m}(B_{r}(x)\cap
E)^{1-\frac{1}{\alpha}}\leq P(B_{r}(x),E^{(1)})\quad\text{ for every
}n\geq\bar{n}\text{ and }\mathcal{L}^{1}\text{-a.e.\ }r\in A_{n}.$ (3.9)

Now define the function $f\colon(0,+\infty)\to\mathbb{R}$ as
$f(r)\coloneqq|{\bf D}{\sf d}_{x}|(B_{r}(x)\cap E^{(1)})$ for every $r>0$.
Corollary 2.3 tells that $f$ is locally absolutely continuous and
$f^{\prime}(r)=P(B_{r}(x),E^{(1)})$ for $\mathcal{L}^{1}$-a.e. $r>0$.
Moreover, Lemma 2.10 gives $f(r)\leq\mathfrak{m}(B_{r}(x)\cap E)$ for every
$r>0$. Consequently, it follows from (3.9) that

$\tilde{C}\frac{\mathfrak{m}(B_{r}(x))^{\frac{1}{\alpha}}}{r}f(r)^{1-\frac{1}{\alpha}}\leq
f^{\prime}(r)\quad\text{ for every }n\geq\bar{n}\text{ and
}\mathcal{L}^{1}\text{-a.e.\ }r\in A_{n}.$ (3.10)

Using again Lemma 2.10 and the fact that $x\notin{\rm int}(E^{(0)})$, we see
that $f(r)\geq c\,\mathfrak{m}(B_{r}(x)\cap E)>0$ for every $r>0$, thus we can
divide both sides of (3.10) by $\alpha f(r)^{1-\frac{1}{\alpha}}$, obtaining
that

$\frac{\tilde{C}}{\alpha}\frac{\mathfrak{m}(B_{r}(x))^{\frac{1}{\alpha}}}{r}\leq\frac{f^{\prime}(r)}{\alpha
f(r)^{1-\frac{1}{\alpha}}}=(f^{\frac{1}{\alpha}})^{\prime}(r)\quad\text{ for
every }n\geq\bar{n}\text{ and }\mathcal{L}^{1}\text{-a.e.\ }r\in A_{n}.$
(3.11)

The third inequality in (3.5) implies that
$\mathcal{L}^{1}([r_{n}/4,r_{n}]\cap A_{n})\geq r_{n}/4$ for every
$n\in\mathbb{N}$, thus integrating (3.11) (and taking into account that
$(f^{1/\alpha})^{\prime}(r)\geq 0$ holds for $\mathcal{L}^{1}$-a.e. $r>0$) we
get that

$\begin{split}\frac{\tilde{C}}{4\alpha}\mathfrak{m}(B_{r_{n}/4}(x))^{\frac{1}{\alpha}}&\leq\frac{\tilde{C}}{\alpha}\frac{\mathfrak{m}(B_{r_{n}/4}(x))^{\frac{1}{\alpha}}}{r_{n}}\mathcal{L}^{1}([r_{n}/4,r_{n}]\cap
A_{n})\leq\frac{\tilde{C}}{\alpha}\int_{[r_{n}/4,r_{n}]\cap
A_{n}}\frac{\mathfrak{m}(B_{r}(x))^{\frac{1}{\alpha}}}{r}\,{\mathrm{d}}r\\\
&\leq\int_{[r_{n}/4,r_{n}]\cap
A_{n}}(f^{\frac{1}{\alpha}})^{\prime}(r)\,{\mathrm{d}}r\leq\int_{0}^{r_{n}}(f^{\frac{1}{\alpha}})^{\prime}(r)\,{\mathrm{d}}r=f(r_{n})^{\frac{1}{\alpha}}\leq\mathfrak{m}(B_{r_{n}}(x)\cap
E)^{\frac{1}{\alpha}}\end{split}$

for every $n\geq\bar{n}$. Letting
$C\coloneqq\frac{1}{C_{D}(\delta)^{2}}\big{(}\frac{\tilde{C}}{4\alpha}\big{)}^{\alpha}$,
we can conclude that $\mathfrak{m}(B_{r_{n}}(x)\cap E)\geq
C\mathfrak{m}(B_{r_{n}}(x))$ for every $n\geq\bar{n}$. This leads to a
contradiction with the fact that $x\in E^{(0)}$. Therefore, the proof of the
statement is achieved. ∎

Having Lemma 3.6 at our disposal, we can now easily prove Theorem 1.1.

###### Proof of Theorem 1.1.

Since $E^{(1)}=({\rm X}\setminus E)^{(0)}$, it is sufficient to check that
$E^{(0)}={\rm int}(E^{(0)})$. To prove it, we argue by contradiction: suppose
there exists a point $x\in E^{(0)}\setminus{\rm int}(E^{(0)})$. This implies
that both $\mathfrak{m}(E)>0$ (otherwise $E^{(0)}={\rm X}={\rm int}(E^{(0)})$)
and $\mathfrak{m}({\rm X}\setminus E)>0$ (otherwise $E^{(0)}=\varnothing$),
thus we know from Remark 2.9 that $P(E)\neq 0$. Since $P(E,\cdot)$ is
concentrated on $\partial^{e}E$, we can find a point $z\in\partial^{e}E$.
Notice that $\mathfrak{m}(B_{r}(z)\cap E)>0$ and
$\mathfrak{m}(B_{r}(z)\setminus E)>0$ for all $r>0$. Since $z\neq x$, we can
fix some radius $\delta\in(0,R_{x}(E))\cap(0,2R_{z}(E)/3)\cap(0,{\sf
d}(x,z)/3)$. Thanks to the fact that $\mathfrak{m}(B_{\delta/2}(z)\cap E)>0$,
we can find a point $y\in E^{(1)}\cap B_{\delta/2}(z)$. Notice that
$B_{\delta/2}(z)\setminus E\subseteq B_{\delta}(y)\setminus E$, so that
$\mathfrak{m}(B_{\delta}(y)\setminus
E)\geq\mathfrak{m}(B_{\delta/2}(z)\setminus E)>0$. The fact that $x\notin{\rm
int}(E^{(0)})$ implies that also $\mathfrak{m}(B_{\delta}(x)\cap E)>0$. Hence,
letting $v_{E,x}$, $w_{E,y}$ be defined in (3.3), we have proved that
$v_{E,x}(\delta)>0$ and $w_{E,y}(\delta)>0$. By our construction and by the
definition of $R_{z}(E),R_{y}(E)$ it holds $2R_{z}(E)/3\leq R_{y}(E)$, hence
we have $\delta\in(0,R_{y}(E))$. Moreover, the inequality $\delta<{\sf
d}(x,z)/3$ implies that ${\sf d}(x,y)>2{\sf d}(x,z)/3>2\delta$, which means
that $\bar{B}_{\delta}(x)\cap\bar{B}_{\delta}(y)=\varnothing$. All in all, we
showed that item i) of Lemma 3.6 holds. Hence, fixed any sequence
$(r_{n})_{n}\subseteq(0,1)$ with $r_{n}\to 0$, we deduce from the assumption
$x\notin{\rm int}(E^{(0)})$ that item ii) of Lemma 3.6 fails. Letting
$A_{E,r_{n}}^{x,y}$ be as in (3.4), we get that

$\mathcal{L}^{1}(A_{E,r_{n}}^{x,y})\geq\frac{r_{n}}{2}\quad\text{ holds only
for finitely many }n\in\mathbb{N}.$ (3.12)

Since $A_{E,r_{n}}^{x,y}\cup A_{{\rm X}\setminus E,r_{n}}^{y,x}=(0,r_{n})$ for
every $n\in\mathbb{N}$, we infer that $\mathcal{L}^{1}(A_{{\rm X}\setminus
E,r_{n}}^{y,x})\geq r_{n}/2$ for infinitely many $n\in\mathbb{N}$. Given that
$v_{{\rm X}\setminus E,y}(\delta)=w_{E,y}(\delta)>0$ and $w_{{\rm X}\setminus
E,x}(\delta)=v_{E,x}(\delta)>0$, we are in a position to apply Lemma 3.6
again, obtaining that $y\in{\rm int}(({\rm X}\setminus E)^{(0)})={\rm
int}(E^{(1)})$. This gives some $\bar{r}>0$ satisfying $w_{E,y}(r)=0$ for
every $r\in(0,\bar{r})$. On the other hand, we know from $x\notin{\rm
int}(E^{(0)})$ that $v_{E,x}(r)>0$ for all $r\in(0,\bar{r})$. Choosing
$\bar{n}\in\mathbb{N}$ so that $r_{n}<\bar{r}$ for all $n\geq\bar{n}$, we
conclude that $A_{E,r_{n}}^{x,y}=(0,r_{n})$ for every $n\geq\bar{n}$, in
contradiction with (3.12). This proves that $E^{(0)}={\rm int}(E^{(0)})$. ∎

######  Remark 3.7 (Some generalizations of Theorem 1.1).

To keep the presentation of Theorem 1.1 as clear as possible, we decided not
to prove it in its utmost generality. However, below we discuss some
generalizations of our result that can be obtained by slightly adapting our
arguments. The standing assumption is that $({\rm X},{\sf d},\mathfrak{m})$ is
a length PI space.

  * $\rm i)$

By inspecting the proof of Lemma 3.6, one can see that assuming the validity
of a weaker notion of deformation property is sufficient. Namely, one can
allow for the constant $C$ appearing in (3.1a), (3.1b) to depend on $y$ and it
is sufficient to require the deformation property only for volume-constrained
minimizers $E$ of the perimeter.

  * $\rm ii)$

A localized version of Theorem 1.1 holds as well: let $E\subseteq{\rm X}$ be a
volume-constrained minimizer of the perimeter in some open set
$\Omega\subseteq{\rm X}$ (i.e. as in Definition 3.1 but requiring that
$K\subseteq\Omega$ and with $P(\cdot)$ replaced by $P(\cdot,\Omega)$)
satisfying $P(E,\Omega)>0$. Then $E^{(1)}\cap\Omega$, $E^{(0)}\cap\Omega$ are
open sets and $\partial E^{(1)}\cap\Omega=\partial
E^{(0)}\cap\Omega=\partial^{e}E\cap\Omega$.

  * $\rm iii)$

Theorem 1.1 can be generalized to volume-constrained minimizers of a suitable
class of _quasi-perimeters_. Fix an open set $\Omega\subseteq{\rm X}$ and a
functional $G\colon\mathscr{B}(\Omega)\to\mathbb{R}\cup\\{+\infty\\}$ with
$G(\varnothing)<+\infty$ having the following property: for any
$U\Subset\Omega$ open, there exist constants $C=C(U)>0$ and
$\sigma=\sigma(U)\in\big{(}1-\frac{1}{\max\\{1,\log_{2}(\inf
C_{D})\\}},1\big{]}$ such that

$G(E)\leq G(F)+C\mathfrak{m}(E\Delta F)^{\sigma}\quad\text{ whenever
}E,F\in\mathscr{B}(\Omega)\text{ satisfy }E\Delta F\subseteq U.$

We then define the quasi-perimeter $\mathscr{P}_{G}$ restricted to $\Omega$ as
$\mathscr{P}_{G}(E,\Omega)\coloneqq P(E,\Omega)+G(E\cap\Omega)$ for every
$E\in\mathscr{B}(\Omega)$. Then an adaption of the previous arguments yields
the validity of the following statement: if $E\subseteq{\rm X}$ is a volume-
constrained minimizer of the quasi-perimeter $\mathscr{P}_{G}$ in $\Omega$
(i.e. as in Definition 3.1 but requiring that $K\subseteq\Omega$, and with
$P(\cdot)$ replaced by $\mathscr{P}_{G}(\cdot,\Omega)$) satisfying
$P(E,\Omega)>0$, then $E^{(1)}\cap\Omega$ and $E^{(0)}\cap\Omega$ are open
sets, and it holds that $\partial E^{(1)}\cap\Omega=\partial
E^{(0)}\cap\Omega=\partial^{e}E\cap\Omega$.  $\blacksquare$

Once we know that volume-constrained minimizers of the perimeter have an open
representative, we can obtain the following expected boundary density
estimates by suitably adapting the arguments in the proof of Lemma 3.6.

######  Theorem 3.8 (Boundary density estimates).

Let $({\rm X},{\sf d},\mathfrak{m})$ be a length PI space having the
deformation property. Let $E\subseteq{\rm X}$ be a volume-constrained
minimizer of the perimeter. Let $B\subseteq{\rm X}$ be a given bounded set.
Then there exist constants $\bar{r}=\bar{r}(E,B,C_{D},C_{I})>0$ and
$C=C(E,B,C_{D},C_{I})>1$ such that

$\frac{1}{C}\leq\frac{\mathfrak{m}(B_{r}(x)\cap
E)}{\mathfrak{m}(B_{r}(x))}\leq
1-\frac{1}{C},\qquad\frac{1}{C}\leq\frac{rP(E,B_{r}(x))}{\mathfrak{m}(B_{r}(x))}\leq
C\quad\text{ for every }x\in\partial^{e}E\cap B\text{ and }r\in(0,\bar{r}).$
(3.13)

In particular, there exists a constant
$\tilde{C}=\tilde{C}(C,C_{D}(\bar{r}/2))\geq 1$ such that

$P(E,B_{2r}(x))\leq\tilde{C}\,P(E,B_{r}(x))\quad\text{ for every
}x\in\partial^{e}E\cap B\text{ and }r\in(0,\bar{r}/2).$ (3.14)

###### Proof.

If $\partial^{e}E$ contains only one point, the first one in (3.13) follows by
the definition $\partial^{e}E$, while the second follows from [3, Theorem
5.4]. Thus we can assume that $\partial^{e}E$ contains at least two distinct
points $z$ and $\tilde{z}$, otherwise there is nothing to prove. In
particular, letting
$\rho\coloneqq\min\big{\\{}R_{z}(E),R_{\tilde{z}}(E),\frac{1}{5}{\sf
d}(z,\tilde{z})\big{\\}}$, we can find two points $y\in B_{\rho/2}(z)\cap
E^{(1)}$ and $\tilde{y}\in B_{\rho/2}(\tilde{z})\cap E^{(1)}$. In fact,
Theorem 1.1 ensures that $y,\tilde{y}\in{\rm int}(E^{(1)})$, so that there
exists $r_{0}\in(0,\rho)$ such that

$\mathfrak{m}(B_{r_{0}}(y)\setminus
E)=\mathfrak{m}(B_{r_{0}}(\tilde{y})\setminus E)=0.$ (3.15)

Notice that $\mathfrak{m}(B_{\rho}(y)\setminus
E)\geq\mathfrak{m}(B_{\rho/2}(z)\setminus E)>0$ and similarly
$\mathfrak{m}(B_{\rho}(\tilde{y})\setminus E)>0$. The doubling assumption
ensures that the closure $K$ of $B$ is compact, thus an application of Dini’s
theorem yields the existence of $r_{1}>0$ such that

$\mathfrak{m}(B_{r}(x)\cap E)<\min\big{\\{}\mathfrak{m}(B_{\rho}(y)\setminus
E),\mathfrak{m}(B_{\rho}(\tilde{y})\setminus E)\big{\\}}\quad\text{ for every
}x\in K\text{ and }r\in(0,r_{1}).$ (3.16)

Again by compactness of $K$, we can also find $r_{2}>0$ such that
$r_{2}<R_{y}(E)$, $r_{2}<R_{\tilde{y}}(E)$, and $r_{2}<R_{x}(\varnothing)$
hold for every $x\in K$. Now define
$\bar{r}_{0}\coloneqq\min\\{r_{0},r_{1},r_{2}\\}>0$. Let
$x\in\partial^{e}E\cap B$ be fixed. Our choice of $\rho$ ensures that
$\bar{B}_{\rho}(x)$ is disjoint from at least one between $\bar{B}_{\rho}(y)$
and $\bar{B}_{\rho}(\tilde{y})$. Up to relabeling $y$ and $\tilde{y}$, say
that $\bar{B}_{\rho}(x)\cap\bar{B}_{\rho}(y)=\varnothing$. Given any
$r\in(0,\bar{r}_{0})$, we deduce from (3.15), (3.16), and the continuity of
$s\mapsto\mathfrak{m}(B_{s}(y)\setminus E)$ that there exists
$s(r)\in(\bar{r}_{0},\rho)$ such that $\mathfrak{m}(B_{r}(x)\cap
E)=\mathfrak{m}(B_{s(r)}(y)\setminus E)$. Define the Borel set
$E_{r}\subseteq{\rm X}$ as $E_{r}\coloneqq(E\setminus B_{r}(x))\cup
B_{s(r)}(y)$. By the minimality assumption on $E$, arguing as we did in the
proof of Lemma 3.6 we obtain that

$P(E,B_{r}(x))\leq\max\\{C_{z},C_{\tilde{z}}\\}\frac{\mathfrak{m}(B_{r}(x)\cap
E)}{\bar{r}_{0}}+P(B_{r}(x),E^{(1)})\quad\text{ for every
}x\in\partial^{e}E\cap B\text{ and }\mathcal{L}^{1}\text{-a.e.\
}r\in(0,\bar{r}_{0}).$ (3.17)

For any $x\in\partial^{e}E\cap B$, define
$A_{x}(E)\coloneqq\big{\\{}r>0\,:\,|{\bf D}{\sf d}_{x}|(B_{r}(x)\cap
E)\leq|{\bf D}{\sf d}_{x}|(B_{r}(x)\setminus E)\big{\\}}$. Fix
$\alpha>\max\\{\log_{2}(C_{D}(\rho)),1\\}$. Applying the relative
isoperimetric inequality to the left-hand side of (3.17) and using Lemma 2.10,
we deduce that

$\begin{split}2C_{0}\frac{\mathfrak{m}(B_{r}(x))^{\frac{1}{\alpha}}}{r}|{\bf
D}{\sf d}_{x}|(B_{r}(x)\cap
E)^{1-\frac{1}{\alpha}}&=2C_{0}\frac{\mathfrak{m}(B_{r}(x))^{\frac{1}{\alpha}}}{r}\min\big{\\{}|{\bf
D}{\sf d}_{x}|(B_{r}(x)\cap E),|{\bf D}{\sf d}_{x}|(B_{r}(x)\setminus
E)\big{\\}}^{1-\frac{1}{\alpha}}\\\ &\leq
2C_{0}\frac{\mathfrak{m}(B_{r}(x))^{\frac{1}{\alpha}}}{r}\min\big{\\{}\mathfrak{m}(B_{r}(x)\cap
E),\mathfrak{m}(B_{r}(x)\setminus E)\big{\\}}^{1-\frac{1}{\alpha}}\\\ &\leq
P(B_{r}(x),E^{(1)})+\frac{\max\\{C_{z},C_{\tilde{z}}\\}}{c^{1-\frac{1}{\alpha}}}\frac{r}{\bar{r}_{0}}\frac{\mathfrak{m}(B_{r}(x))^{\frac{1}{\alpha}}}{r}|{\bf
D}{\sf d}_{x}|(B_{r}(x)\cap E)^{1-\frac{1}{\alpha}}\end{split}$ (3.18)

holds for $\mathcal{L}^{1}$-a.e. $r\in(0,\bar{r}_{0})\cap A_{x}(E)$, where we
set $C_{0}\coloneqq 1/\big{(}2\,C_{I}(\alpha,\rho)^{(\alpha-1)/\alpha}\big{)}$
for brevity. Therefore, if we let

$\bar{r}\coloneqq\min\bigg{\\{}\frac{c^{1-\frac{1}{\alpha}}C_{0}\bar{r}_{0}}{\max\\{C_{z},C_{\tilde{z}}\\}},\bar{r}_{0}\bigg{\\}}\in(0,\bar{r}_{0}],$

then we infer from (3.18) that

$C_{0}\frac{\mathfrak{m}(B_{r}(x))^{\frac{1}{\alpha}}}{r}|{\bf D}{\sf
d}_{x}|(B_{r}(x)\cap E)^{1-\frac{1}{\alpha}}\leq
P(B_{r}(x),E^{(1)})\quad\text{ for }\mathcal{L}^{1}\text{-a.e.\
}r\in(0,\bar{r})\cap A_{x}(E).$ (3.19)

This also proves (by considering ${\rm X}\setminus E$ instead of $E$) that, up
to shrinking $\bar{r}>0$, it holds that

$C_{0}\frac{\mathfrak{m}(B_{r}(x))^{\frac{1}{\alpha}}}{r}|{\bf D}{\sf
d}_{x}|(B_{r}(x)\setminus E)^{1-\frac{1}{\alpha}}\leq
P(B_{r}(x),E^{(0)})\quad\text{ for }\mathcal{L}^{1}\text{-a.e.\
}r\in(0,\bar{r})\cap A_{x}({\rm X}\setminus E).$ (3.20)

Let us now define the function $f_{x}\colon(0,+\infty)\to\mathbb{R}$ as

$f_{x}(r)\coloneqq\min\big{\\{}|{\bf D}{\sf d}_{x}|(B_{r}(x)\cap
E^{(1)}),|{\bf D}{\sf d}_{x}|(B_{r}(x)\cap E^{(0)})\big{\\}}\quad\text{ for
every }r>0.$

Corollary 2.3 ensures that $f_{x}$ is locally absolutely continuous and

$f^{\prime}_{x}(r)=\left\\{\begin{array}[]{ll}P(B_{r}(x),E^{(1)})\\\
P(B_{r}(x),E^{(0)})\end{array}\quad\begin{array}[]{ll}\text{ for
}\mathcal{L}^{1}\text{-a.e.\ }r\in A_{x}(E),\\\ \text{ for
}\mathcal{L}^{1}\text{-a.e.\ }r\in A_{x}({\rm X}\setminus
E).\end{array}\right.$

Observe that $A_{x}(E)\cup A_{x}({\rm X}\setminus E)=(0,+\infty)$. Arguing as
in Lemma 3.6, we deduce from (3.19) and (3.20) that

$\frac{C_{0}}{\alpha}\frac{\mathfrak{m}(B_{r}(x))^{\frac{1}{\alpha}}}{r}\leq(f_{x}^{\frac{1}{\alpha}})^{\prime}(r)\quad\text{
for }\mathcal{L}^{1}\text{-a.e.\ }r\in(0,\bar{r}).$ (3.21)

Given any $r\in(0,\bar{r})$, we can integrate the inequality in (3.21) over
the interval $[r/2,r]$, thus obtaining that

$\begin{split}\frac{C_{0}}{2\alpha\,(C_{D}(\bar{r}/2))^{\frac{1}{\alpha}}}\mathfrak{m}(B_{r}(x))^{\frac{1}{\alpha}}&\leq\frac{C_{0}}{\alpha}\frac{\mathfrak{m}(B_{r/2}(x))^{\frac{1}{\alpha}}}{r}\frac{r}{2}\leq\frac{C_{0}}{\alpha}\int_{r/2}^{r}\frac{\mathfrak{m}(B_{s}(x))^{\frac{1}{\alpha}}}{s}\,{\mathrm{d}}s\leq\int_{0}^{r}(f_{x}^{\frac{1}{\alpha}})^{\prime}(s)\,{\mathrm{d}}s=f_{x}(r)^{\frac{1}{\alpha}}\\\
&\leq\min\big{\\{}\mathfrak{m}(B_{r}(x)\cap E),\mathfrak{m}(B_{r}(x)\setminus
E)\big{\\}}^{\frac{1}{\alpha}}.\end{split}$ (3.22)

It follows that $\mathfrak{m}(B_{r}(x))\leq C_{1}\mathfrak{m}(B_{r}(x)\cap E)$
for every $x\in\partial^{e}E\cap B$ and $r\in(0,\bar{r})$, where we define
$C_{1}\coloneqq C_{D}(\bar{r}/2)\big{(}\frac{2\alpha}{C_{0}}\big{)}^{\alpha}$.

Let $x\in\partial^{e}E\cap B$ and $r\in(0,\bar{r})$ be fixed. Since
$\mathfrak{m}(B_{r}(x))\leq C_{1}\min\big{\\{}\mathfrak{m}(B_{r}(x)\cap
E),\mathfrak{m}(B_{r}(x)\setminus E)\big{\\}}$ by (3.22), by using the
relative isoperimetric inequality (recall that
$2C_{0}=1/C_{I}(\alpha,\rho)^{(\alpha-1)/\alpha}$) we get that

$\frac{2C_{0}}{C_{1}^{1-\frac{1}{\alpha}}}\frac{\mathfrak{m}(B_{r}(x))}{r}\leq
2C_{0}\frac{\mathfrak{m}(B_{r}(x))^{\frac{1}{\alpha}}}{r}\min\big{\\{}\mathfrak{m}(B_{r}(x)\cap
E),\mathfrak{m}(B_{r}(x)\setminus E)\big{\\}}^{1-\frac{1}{\alpha}}\leq
P(E,B_{r}(x)).$

On the other hand, since we can assume that $C_{2}\coloneqq\sup_{x\in
B}C_{x}(\varnothing)<+\infty$ by the compactness of $K$, we obtain that

$\begin{split}P(E,B_{r}(x))&\leq
P(E,B_{\tilde{r}}(x))\leq\max\\{C_{z},C_{\tilde{z}}\\}\frac{\mathfrak{m}(B_{\tilde{r}}(x)\cap
E)}{\tilde{r}}+P(B_{\tilde{r}}(x),E^{(1)})\leq\max\\{C_{z},C_{\tilde{z}}\\}\frac{\mathfrak{m}(B_{\tilde{r}}(x))}{\tilde{r}}+P(B_{\tilde{r}}(x))\\\
&\leq\big{(}\max\\{C_{z},C_{\tilde{z}}\\}+C_{2}\big{)}\frac{\mathfrak{m}(B_{\tilde{r}}(x))}{\tilde{r}}\end{split}$

for $\mathcal{L}^{1}$-a.e. $\tilde{r}\in(r,\bar{r})$, thanks to (3.17) and to
the deformation property. Hence,
$\frac{rP(E,B_{r}(x))}{\mathfrak{m}(B_{r}(x))}\leq\max\\{C_{z},C_{\tilde{z}}\\}+C_{2}$
for all $x\in\partial^{e}E\cap B$ and $r\in(0,\bar{r})$. Picking
$C\coloneqq\max\big{\\{}C_{1},C_{1}^{(\alpha-1)/\alpha}/(2C_{0}),\max\\{C_{z},C_{\tilde{z}}\\}+C_{2}\big{\\}}$,
we conclude that (3.13) holds. Finally, applying (3.13) we conclude that for
every $x\in\partial^{e}E\cap B$ and $r\in(0,\bar{r}/2)$ it holds that

$\frac{P(E,B_{2r}(x))}{P(E,B_{r}(x))}\leq\frac{C\mathfrak{m}(B_{2r}(x))}{2r}\frac{Cr}{\mathfrak{m}(B_{r}(x))}\leq\frac{C^{2}C_{D}(\bar{r}/2)}{2},$

which proves the validity of (3.14). Consequently, the statement is achieved.
∎

We conclude with a final comment on further minimality properties satisfied by
volume-constrained minimizers. Such properties can be derived by reproducing
well-known arguments, see, e.g. [6, Remark 3.23, Theorem 3.24], exploiting
Theorem 1.1 and the deformation property.

######  Remark 3.9.

Let $({\rm X},{\sf d},\mathfrak{m})$ be a length PI space having the
deformation property. Let $E\subseteq{\rm X}$ be a volume-constrained
minimizer of the perimeter. Using Theorem 1.1 and with arguments similar to
those in the proof of Theorem 3.8, it is possible to prove that for any
compact set $K\subseteq{\rm X}$ there exist $\Lambda,r_{0}>0$ such that $E$ is
a $(\Lambda,r_{0})$-perimeter minimizer on $K$, i.e. whenever $F\Delta
E\subseteq B_{r}(x)$ for some $x\in K$ and $r<r_{0}$ it holds
$P(E,B_{r}(x))\leq P(F,B_{r}(x))+\Lambda\,\mathfrak{m}(E\Delta F)$.

Moreover, for any given compact set $K\subseteq{\rm X}$ there exist constants
$L,r_{0}>0$ such that $E$ is $(L,r_{0})$-quasi minimal on $K$, i.e. whenever
$F\Delta E\subseteq B_{r}(x)$ for some $x\in K$ and $r<r_{0}$ it holds that
$P(E,B_{r}(x))\leq L\,P(F,B_{r}(x))$. The class of quasi-minimal sets has been
studied e.g. in [29].

It is worth pointing out that, once we know that volume-constrained minimizers
of the perimeter are $(L,r_{0})$-quasi minimal sets, Theorem 3.8 follows
directly from [29, Theorem 4.2 and Lemma 5.1]. Nevertheless, we opted for a
self-contained proof of Theorem 3.8, which takes advantage of the openness of
volume-constrained minimizers.  $\blacksquare$

##  4 Boundedness of isoperimetric sets

In this last section, we prove the boundedness of isoperimetric sets in length
PI spaces satisfying the deformation property and with a uniform lower bound
on the volume of unit balls (Theorem 1.2). The argument makes use of the
topological regularity given by our main result Theorem 1.1.

###### Proof of Theorem 1.2.

Suppose by contradiction that $E$ has no bounded representatives, i.e.
$\mathfrak{m}(E\setminus B_{R}(x))>0$ for all $R>0$ and $x\in{\rm X}.$ In
particular ${\rm X}$ is unbounded and, since $v_{0}\coloneqq\inf_{x\in{\rm
X}}\mathfrak{m}(B_{1}(x))>0$, we have $\mathfrak{m}({\rm X})=\infty$ and
$\mathfrak{m}({\rm X}\setminus E)>0$. Then $P(E)>0$ and, arguing as in the
proof of Theorem 1.1, we can find $y\in E^{(1)}$ and $\rho\in(0,R_{y}(E))$
such that $\delta\coloneqq\mathfrak{m}(B_{\rho}(y)\setminus E)>0$. By Theorem
1.1 it holds that $y\in{\rm int}(E^{(1)})$, i.e. there exists $r_{0}>0$ such
that $\mathfrak{m}(B_{r_{0}}(y)\cap E)=0.$ Observe that, since
$\mathfrak{m}(E)<+\infty$, it holds $\mathfrak{m}(E\setminus B_{R}(y))\to 0$
as $R\to+\infty$. Moreover, by the coarea formula and recalling (2.1), we have
that

$\int_{R}^{2R}P(B_{t}(y),E^{(1)})\,{\mathrm{d}}t\leq\int_{\mathbb{R}}P(B_{t}(y),E^{(1)}\setminus
B_{R}(y))\,{\mathrm{d}}t=|{\bf D}{\sf d}_{x}|(E^{(1)}\setminus
B_{R}(y))\leq\mathfrak{m}(E\setminus B_{R}(y))\quad\text{ for every }R>0,$

where in the first inequality we used the fact that the measure
$P(B_{t}(y),\cdot)$ is concentrated on $\partial B_{t}(y)$. In particular, for
all $R>1$ the set $\big{\\{}t\in(R,2R)\,:\,P(B_{t}(y),E^{(1)})\leq
2\,\mathfrak{m}(E\setminus B_{R}(y))\big{\\}}$ has positive measure.
Therefore, there exists a sequence $R_{n}\to+\infty$, with $R_{n}>\rho,$ such
that

$P(B_{R_{n}}(y),E^{(1)})\leq 2\,\mathfrak{m}(E\setminus B_{R_{n}}(y)),\quad
0<\mathfrak{m}(E\setminus B_{R_{n}}(y))<\min\\{\delta,v_{0}/2\\},\quad
P(E,\partial B_{R_{n}}(y))=0\quad\text{ for every }n\in\mathbb{N}.$

By continuity, for every $n\in\mathbb{N}$ there exists $r_{n}\in(r_{0},\rho)$
such that

$\mathfrak{m}(B_{r_{n}}(y)\setminus E)=\mathfrak{m}(E\setminus B_{R_{n}}(y)).$
(4.1)

For every $n\in\mathbb{N}$ we define $F_{n}\coloneqq(E\cup B_{r_{n}}(y))\cap
B_{R_{n}}(y)$, which satisfies $\mathfrak{m}(F_{n})=\mathfrak{m}(E)$ thanks to
(4.1) and $r_{n}<R_{n}$. Hence, by minimality, $P(E)\leq P(F_{n})$ for every
$n\in\mathbb{N}$. Moreover, using Proposition 2.6 and the deformation
property,

$\displaystyle P(E)\leq P(F_{n})$ $\displaystyle=P((E\cup B_{r_{n}}(y))\cap
B_{R_{n}}(y))\leq P(E\cup
B_{r_{n}}(y),B_{R_{n}}(y)^{(1)})+P(B_{R_{n}}(y),(E\cup B_{r_{n}}(y))^{(1)})$
$\displaystyle\leq P(E\cup B_{r_{n}}(y))-P(E\cup
B_{r_{n}}(y),B_{R_{n}}(y)^{(0)})+P(B_{R_{n}}(y),E^{(1)})$ $\displaystyle\leq
P(E)+C_{y}(E)\frac{\mathfrak{m}(B_{r_{n}}(y)\setminus
E)}{r_{0}}-P(E,B_{R_{n}}(y)^{(0)})+P(B_{R_{n}}(y),E^{(1)})$ $\displaystyle\leq
P(E)+C_{y}(E)\frac{\mathfrak{m}(B_{r_{n}}(y)\setminus E)}{r_{0}}-P(E\setminus
B_{R_{n}}(y))+2P(B_{R_{n}}(y),E^{(1)})$ $\displaystyle\leq
P(E)+C_{y}(E)\frac{\mathfrak{m}(E\setminus
B_{R_{n}}(y))}{r_{0}}-Cv_{0}^{\frac{1}{\alpha}}\mathfrak{m}(E\setminus
B_{R_{n}}(y))^{\frac{\alpha-1}{\alpha}}+4\mathfrak{m}(E\setminus
B_{R_{n}}(y)),$

where in the fourth line we used again Proposition 2.6 and in the last line we
used the isoperimetric inequality for small volumes in Proposition 2.8 (recall
that $\mathfrak{m}(E\setminus B_{R_{n}}(y))<v_{0}/2$). Note that $\alpha>1$
and $C>0$ are constants independent of $n.$ Hence, since
$\mathfrak{m}(E\setminus B_{R_{n}}(y))\to 0$ and $\mathfrak{m}(E\setminus
B_{R_{n}}(y))>0$, we obtain that for $n\in\mathbb{N}$ big enough it holds that
$P(E)<P(E),$ which is a contradiction. ∎

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