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 [Reference Ambrosio3, Reference Miranda35] 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:
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 [Reference Gonzalez, Massari and Tamanini25], subsequently generalized in [Reference Xia41]. The proof in the Euclidean setting can be further simplified, see [Reference Maggi34, Example 21.3, Theorem 21.11]. On Riemannian manifolds the result is due to [Reference Morgan36]. In [Reference Antonelli, Pasqualetto and Pozzetta6] 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 [Reference Kinnunen, Korte, Lorent and Shanmugalingam31], 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
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 [Reference Antonelli, Pasqualetto and Pozzetta6, Theorem 1.1]. We shall not introduce $\mathsf {RCD}$
spaces here, and we refer the reader to the survey [Reference Ambrosio4] and the references therein.
We point out that being a PI space does not imply that the deformation property holds, see the examples in remarks 3.5 and Reference Galli and Ritoré3.6. Anyway, we are not aware of any example of a PI space where the deformation property fails when tested on an isoperimetric set $E$
, nor of an example of a PI space where the essential interior of an isoperimetric set is not topologically open.
Deformation properties for sets of locally finite perimeter are well-known in the smooth context [Reference Maggi34, Lemma 17.21], and they represent a tool of crucial importance in several classical arguments. We refer, for instance, to [Reference Almgren2, VI.2(3)], [Reference Galli and Ritoré24, Lemma 4.5], and [Reference Pozuelo37, Lemma 3.6] in the sub-Riemannian setting, and to [Reference Cinti and Pratelli21, Reference Pratelli and Saracco38] 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) centred 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. Observe that the previous improved deformation property with linear control follows from (1.1) only after topological regularity of the set has been established. This is in contrast to the Euclidean setting, where the stronger form of deformation property is always available [Reference Maggi34, Lemma 17.21]. The latter result follows by deforming sets of finite perimeter by flows of vector fields, an argument out of reach in the metric setting. Hence, the simplest Euclidean proof for the topological regularity of isoperimetric sets [Reference Maggi34, Example 21.3] has no hope of being performed in our framework, and we must look for an alternative argument.
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 § 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 3.9. 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 [Reference Rajala39, Reference Sturm40] and [Reference Lott and Villani33]), and as recalled above they have the deformation property, putting together theorems 1.1 and 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 [Reference Antonelli, Pasqualetto and Pozzetta6, 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
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. [Reference Sturm40]). 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), was obtained at different levels of generality in [Reference Agostiniani, Fogagnolo and Mazzieri1, Reference Antonelli, Pasqualetto, Pozzetta and Semola7, Reference Balogh and Kristály9, Reference Brendle13, Reference Cavalletti and Manini16, Reference Cavalletti and Manini17]. In [Reference Antonelli, Pasqualetto, Pozzetta and Semola7] the rigidity for the isoperimetric inequality was proved for noncollapsed $\mathsf {RCD}(0,N)$
spaces. Recently in [Reference Cavalletti and Manini17, 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
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 centred at one of the tips of ${\rm X}$
.
In the previous theorem, when we say that ${\rm X}$
is a Euclidean metric measure cone over an ${\sf RCD}(N-2,N-1)$
space we mean that there is a compact ${\sf RCD}(N-2,N-1)$
metric measure space $(Z,{\sf d}_Z,\mathfrak {m}_Z)$
such that $({\rm X},{\sf d},\mathfrak {m})$
is isomorphic, as a metric measure space, to the metric measure cone $(C(Z),{\sf d}_{c},t^{N-1}\mathrm {d}t\otimes {m}_Z)$
, where ${\sf d}_c$
is the cone metric built using ${\sf d}_Z$
. In the case of $1< N<2$
, it is understood that in the rigidity part of the previous statement, the space $X$
is either a weighted Euclidean half-line or a weighted Euclidean line.
We stress that theorem 1.4 is not a straightforward consequence of the results in [Reference Antonelli, Pasqualetto, Pozzetta and Semola7], according to which the same result holds in the class of noncollapsed spaces. Indeed, an $\mathsf {RCD}(0,N)$
space with $1< N<\infty$
and with Euclidean volume growth might not be noncollapsed. A simple example is given by the weighted Euclidean half-line $([0,+\infty ),{\sf d}_{\mathrm {eu}},t^{N-1}\mathrm {d}t)$
, with $N>1$
.
We now briefly discuss our strategy for the proof of theorem 1.1. As mentioned above, the Euclidean proof [Reference Maggi34, Example 21.3] cannot be adapted to our setting. As in the classical [Reference Gonzalez, Massari and Tamanini25, Reference Xia41], 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 strategy of [Reference Gonzalez, Massari and Tamanini25, Reference Xia41] 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 [Reference Gonzalez, Massari and Tamanini25, 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 [Reference Antonelli, Pasqualetto and Pozzetta6, Reference Gonzalez, Massari and Tamanini25, Reference Xia41], and we rather adopt a more qualitative approach. More precisely, the key point is to show (see the key lemma 3.7) 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 $v(r)\ge w(r)$
for many $r>0$
in a measure-theoretic sense (see lemma 3.7 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 point out that the strategy of [Reference Gonzalez, Massari and Tamanini25, Reference Xia41] 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, since it brings simplifications to the classical arguments in [Reference Gonzalez, Massari and Tamanini25], still (necessarily) avoiding the use of the smooth structure of the ambient.
We conclude the introduction by explicitly recording the following open problem.
Question 1.5 Let $({\rm X},{\sf d},\mathfrak {m})$
be a length PI space and let $E\subseteq {\rm X}$
be a volume-constrained minimizer of the perimeter. Is it true that $E^{(1)}$
is open?
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 $\mathbb {R}$
, i.e. of those functions $f\colon {\rm X}\to \mathbb {R}$
such that for any $x\in {\rm X}$
there exists $r_x>0$
for which $f$
is Lipschitz on $B_{r_x}(x)$
. 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
The topological interior and the topological boundary of a set $E\subseteq {\rm X}$
are denoted by ${\rm int}(E)$
and $\partial E$
, respectively. A Borel measure $\mu \geq 0$
on ${\rm X}$
is locally finite if for any $x\in {\rm X}$
there exists $r_x>0$
such that $\mu (B_{r_x}(x))<+\infty$
, while we say that $\mu$
is boundedly finite if $\mu (B)<+\infty$
whenever $B\subseteq {\rm X}$
is bounded Borel. Trivially, each boundedly finite measure is locally finite, while the converse holds e.g. if $({\rm X},{\sf d})$
is proper, i.e. bounded closed subsets of ${\rm X}$
are compact. Notice that locally finite Borel measures on a complete separable metric space are $\sigma$
-finite.
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 [Reference Miranda35], 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
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 $P(E,\cdot )\coloneqq |{\bf D}{\raise.3ex\hbox{$\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 $0< r< s$
and to the $\sigma$
-finiteness of $P(E,\cdot )$
. $\blacksquare$
Given any $f\in {\rm LIP}_{\mathrm {loc}}({\rm X})$
, 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 [Reference Miranda35, 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
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 ${\sf d}_x\coloneqq {\sf d}(\cdot,x)\in {\rm LIP}({\rm X})$
. Then, the function $f$
is locally absolutely continuous and it holds that $f'(r)=P(B_r(x),E)$
for $\mathcal {L}^1$
-a.e. $r>0$
.
Proof. By virtue of the coarea formula, we obtain that $f(r)=\int _\mathbb {R} P\big (\{{\sf d}_x< s\},E\cap B_r(x)\big )\,{\mathrm d} s=\int _0^r P(B_s(x),E)\,{\mathrm d} s$
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 $f'(r)=P(B_r(x),E)$
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 [Reference Björn and Björn12, Reference Heinonen, Koskela, Shanmugalingam and Tyson29] 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 is 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
\begin{align*} & \fint_{B_r(x)}\bigg|f-\fint_{B_r(x)}f\,{\mathrm d}\mathfrak{m}\bigg|\,{\mathrm d}\mathfrak{m}\\& \quad \leq C_P(R)\,r\fint_{B_{\lambda r}(x)}{\rm lip}(f)\,{\mathrm d}\mathfrak{m} \quad\text{ for all }0< r< R\text{ and }x\in{\rm X}. \end{align*}
• $({\rm X},{\sf d},\mathfrak {m})$
is a PI space if it is uniformly locally doubling and it supports a weak local $(1,1)$-Poincaré inequality.
We point out that if $({\rm X},{\sf d},\mathfrak {m})$
is a uniformly locally doubling space, then $({\rm X},{\sf d})$
is proper, so $({\rm X},{\sf d})$
is locally compact, and locally finite Borel measures on $({\rm X},{\sf d})$
are boundedly finite.
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 [Reference Hajłasz and Koskela26, Corollary 9.5 and Theorem 9.7]. Moreover, the completeness and the local compactness of $({\rm X},{\sf d})$
ensure that $({\rm X},{\sf d})$
is also geodesic. $\blacksquare$
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
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. [Reference Heinonen27, 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 [Reference Ambrosio3, 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,
Proof. We know from [Reference Ambrosio3, 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 [Reference Ambrosio3, 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 [Reference Antonelli, Pasqualetto and Pozzetta6, Lemma 2.5], we conclude that
which yields the statement.
The following is a direct consequence of the study in [Reference Ambrosio3], taking remark 2.5 into account.
Theorem 2.7 (Relative isoperimetric inequality [Reference Ambrosio3, 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
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 [Reference Coulhon and Saloff-Coste23], after [Reference Buser15, Reference Chavel and Feldman20, Reference Kanai30]. For a proof, we refer the reader to the argument in [Reference Chavel19, 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
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
where we denote ${\sf d}_x\coloneqq {\sf d}(\cdot,x)\in {\rm LIP}_{\mathrm {loc}}({\rm X})$
. In particular, it holds that
Proof. Recall that $|{\bf D}{\sf d}_x|\leq {\rm lip}({\sf d}_x)\mathfrak {m}$
. Moreover, we deduce from [Reference Ambrosio, Pinamonti and Speight5, 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 fact that $({\rm X},{\sf d})$
is geodesic. Finally, (2.2) can be proved by combining (2.1) with the coarea formula: we can estimate
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 $P(E, K)\leq P(F, K)$
.
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 with $0<\mathfrak {m}(E)<+\infty$
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 with $({\rm X},{\sf d})$
proper. 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 $C\ge 0$
such that
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 $C\ge 0$
and we define $C_x(E)\ge 0$
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 $R$
. We also observe that if $E\subseteq {\rm X}$
is a given set of finite perimeter (resp. of locally finite perimeter), then (3.1a) is equivalent to asking that $P(E\setminus B_r(y),S)\leq C({\mathfrak {m}(B_r(y)\cap E)}/{r})+P(E,S)$
holds for every $(y,r)\in B_R(x)\times (0,R)$
and every Borel set (resp. bounded Borel set) $S\subseteq {\rm X}$
with $B_{2R}(x)\subseteq S$
. Similarly for (3.1b). We will often make use of this observation without further notice. Also,
Indeed, the compactness of the closure of $B$
ensures that $B\subseteq \bigcup _{i=1}^n B_{R_{x_i}(E)/2}(x_i)$
for some $x_1,\ldots,x_n\in B$
, which gives $R_x(E)\geq \delta \coloneqq \min \big \{R_{x_i}(E)/2\,:\,i=1,\ldots,n\big \}>0$
for every $x\in B$
. The same argument shows also that (3.1a) and (3.1b) hold for every $x\in B$
for some $R$
and $C$
that depend only on $B$
and $E$
, e.g. by taking $R\coloneqq \delta$
and $C\coloneqq \max \big \{C_{x_i}(E)\,:\,i=1,\ldots,n\big \}$
.
Remark 3.4 (Spaces having the deformation property)
These are some spaces with the deformation property:
(i) Euclidean spaces (see e.g. [Reference Gonzalez, Massari and Tamanini25] and the references therein).
(ii) Riemannian manifolds (this can be proved e.g. by following the proof of [Reference Antonelli, Pasqualetto and Pozzetta6, Theorem 1.1] and using the fact that the Ricci curvature is locally bounded from below).
(iii) ${\sf RCD}(K,N)$
spaces with $K\in \mathbb {R}$ and $N\in [1,\infty )$ (proved in [Reference Antonelli, Pasqualetto and Pozzetta6, Theorem 1.1] building upon the Gauss–Green formula in [Reference Bruè, Pasqualetto and Semola14, Theorem 2.4]).
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 [Reference Baudoin, Grong, Kuwada and Thalmaier11] for the study of sub-Laplacian comparison theorems in more general sub-Riemannian structures, and [Reference Cavalletti and Mondino18, 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 [Reference Comi and Magnani22], one could argue following the lines of [Reference Antonelli, Pasqualetto and Pozzetta6, Theorems 2.32 and 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. [Reference Barilari and Rizzi10], [Reference Badreddine and Rifford8]), have the deformation property. Since this is out of the scope of the present paper, 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 contrary, the topological regularity of isoperimetric sets is already proved in [Reference Leonardi and Rigot32] in the setting of Carnot groups and in [Reference Galli and Ritoré24] on a certain class of sub-Riemannian manifolds. $\blacksquare$
Remark 3.5 There exist PI spaces where the deformation property fails. For example, fix a sequence of pairwise well-separated non-empty balls $B_n\coloneqq B_{r_n}(x_n)$
in $\mathbb {R}^2$
such that $x_n\to 0$
and $\sum _n r_n<+\infty$
. Now, consider the density function $\rho \colon \mathbb {R}^2\to [1,2]$
given by $\rho \coloneqq {\raise.3ex\hbox{$\chi$}}_E+2{\raise.3ex\hbox{$\chi$}}_{\mathbb {R}^2\setminus E}$
, where $E\coloneqq \bigcup _n B_n$
. Letting $\mathfrak {m}\coloneqq \rho \mathcal {L}^2$
we have $\mathcal {L}^2\leq \rho \mathcal {L}^2\leq 2\mathcal {L}^2$
, so that $(\mathbb {R}^2,|\cdot |,\mathfrak {m})$
is an Ahlfors regular geodesic PI space. We claim that the deformation property is not valid for the set of finite perimeter $E$
at the origin $0$
. To check it, notice that for any $n\in \mathbb {N}$
it holds $P(B_n)=2\pi r_n$
, while $P(B_{r_n+\varepsilon }(x_n))=4\pi (r_n+\varepsilon )$
and $\mathfrak {m}(B_{r_n+\varepsilon }(x_n)\setminus B_n)=2\pi (2r_n\varepsilon +\varepsilon ^2)$
for any $\varepsilon \in (0,\varepsilon _n)$
for some $\varepsilon _n>0$
sufficiently small. Therefore,
which shows that the deformation property fails at the origin. However, we are not aware of any example of a PI space where the deformation property fails when tested on an isoperimetric set, nor of an example of a PI space where the essential interior of some isoperimetric set is not topologically open. $\blacksquare$
Remark 3.6 The validity of the deformation property on a metric measure space $({\rm X}, {\sf d},\mathfrak {m})$
entails a growth condition: given $x\in {\rm X}$
, there exist $C_x,r_x>0$
such that
Equation (3.3) follows just by taking $E\coloneqq \varnothing$
in the deformation property. We have that (3.3) is not equivalent to the deformation property (e.g. in the example in remark 3.5 property (3.3) is satisfied). However, there are examples of PI spaces where also (3.3) fails. The example we are going to describe has been pointed out to the authors by Panu Lahti. Consider the measure $\mathfrak {m}\coloneqq |x|^{-1/2}{\mathrm d} x$
in $\mathbb {R}$
. Since the function $|x|^{-1/2}$
is an $A_1$
-Muckenhoupt weight, we know that $(\mathbb {R},|\cdot |,\mathfrak {m})$
is a PI space (see e.g. [Reference Heinonen, Kilpeläinen and Martio28]). Using that ${\mathfrak {m}(B_r(0))}/{r}=2\fint _0^r({1}/{\sqrt x})\,{\mathrm d} x={4}/{\sqrt r}\to +\infty$
as $r\searrow 0$
, one can easily check that the codimension-one Hausdorff measure of the singleton $\{0\}$
diverges, i.e. $\mathcal {H}^h(\{0\})=+\infty$
. It follows from [Reference Ambrosio3, Theorem 5.3] that $B_{|y|}(y)$
is not a set of locally finite perimeter when $y\in (0,+\infty )$
. Hence, (3.3) fails for $x=0$
. $\blacksquare$
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
The core of the proof of our main theorem 1.1 is contained in the following technical result.
Lemma 3.7 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 (3.4). 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
Suppose the following conditions are verified:
(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 $w_{E,y}(\delta )>0$.(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 relabelled) subsequence of $(r_n)_n$
for which
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
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 (3.6) 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 (3.6) ensures that $\bar B_r(x)\cap \bar B_{s(r)}(y)=\varnothing$
, whence it follows that $\mathfrak {m}\big (E_r\cap (\bar B_r(x)\cup \bar B_{s(r)}(y))\big )=\mathfrak {m}\big (E\cap (\bar B_r(x)\cup \bar B_{s(r)}(y))\big )$
. Denote $K\coloneqq \bar B_{2\delta }(x) \cup \bar B_{2\delta }(y)$
for brevity. 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
Notice that the constant $C_y$
depends on $y$
and $E$
, but neither on $n$
nor on $r$
. Therefore, we have shown that
Now, fix any $\alpha >\max \{\log _2(C_D(\delta )),1\}$
. We know from the relative isoperimetric inequality, i.e. theorem 2.7, that
where we define $\tilde C\coloneqq 1/2 C_I(\alpha,\delta )^{(\alpha -1)/\alpha }$
. Exploiting the fact that $x\in E^{(0)}$
, we can find $\bar n\in \mathbb {N}$
such that
By combining (3.7), (3.8), and (3.9), we deduce that for every $n\geq \bar n$
and $\mathcal {L}^1$
-a.e. $r\in A_n$
it holds that
Rearranging the terms, we infer that
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 $f'(r)=P(B_r(x),E^{(1)})$
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 (3.10) that
Using that $x\notin {\rm int}(E^{(0)})$
, which is the contradiction assumption, and lemma 2.10 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 (3.11) by $\alpha f(r)^{1-{1}/{\alpha }}$
, obtaining that
The third inequality in (3.6) 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 (3.12) (and taking into account that $(f^{1/\alpha })'(r)\geq 0$
holds for $\mathcal {L}^1$
-a.e. $r>0$
) we get that
for every $n\geq \bar n$
. Letting $C\coloneqq ({1}/{C_D(\delta )^2})({\tilde C}/{4\alpha })^\alpha$
, 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 3.7 at our disposal, we can now easily prove theorem 1.1.
Proof 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 $\delta \in (0,R_x(E))\cap (0,2 R_z(E)/3)\cap (0,{\sf d}(x,z)/3)$
. 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 (3.4), 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 $2 R_z(E)/3\le R_y(E)$
, 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 3.7 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 3.7 fails. Letting $A_{E,r_n}^{x,y}$
be as in (3.5), we get that
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 3.7 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 (3.13). This proves that $E^{(0)}={\rm int}(E^{(0)})$
.
Remark 3.8 (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.
(i) By inspecting the proof of lemma 3.7, 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.(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 $\partial E^{(1)}\cap \Omega =\partial E^{(0)}\cap \Omega =\partial ^e E\cap \Omega$.(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 $\sigma =\sigma (U)\in \big (1-({1}/{\max \{1,\log _2(\inf C_D)\}}),1\big ]$ 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 3.7.
Theorem 3.9 (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
for every $x\in \partial ^e E\cap B$
and $r\in (0,\bar r)$
.
In particular, there exists a constant $\tilde C = \tilde C(C,C_D(\bar r/2)) \geq 1$
such that
Proof. If $\partial ^e E$
contains only one point, the first one in (3.14) follows by the definition $\partial ^e E$
, while the second follows from [Reference Ambrosio3, 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 $\rho \coloneqq \min \big \{R_z(E),R_{\tilde z}(E),({1}/{5}){\sf d}(z,\tilde z)\big \}$
, 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
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
Thanks to (3.2), 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 relabelling $y$
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 (3.16), (3.17), 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 3.7 we obtain
for any $x\in \partial ^e E\cap B$
and $\mathcal {L}^1\text{-a.e.}\;r\in (0,\bar r_0)$
. 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 (3.18) and using lemma 2.10, we deduce that
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
then we infer from (3.19) that
This also proves (by considering ${\rm X}\setminus E$
instead of $E$
) that, up to shrinking $\bar r>0$
, it holds that
Let us now define the function $f_x\colon (0,+\infty )\to \mathbb {R}$
as
Corollary 2.3 ensures that $f_x$
is locally absolutely continuous and
Observe that $A_x(E)\cup A_x({\rm X}\setminus E)=(0,+\infty )$
. Arguing as in lemma 3.7, we deduce from (3.20) and (3.21) that
Given any $r\in (0,\bar r)$
, we can integrate the inequality in (3.22) over the interval $[r/2,r]$
, thus obtaining that
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 $C_1\coloneqq C_D(\bar r/2)\big ({2\alpha }/{C_0}\big )^\alpha$
.
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 (3.23), by using the relative isoperimetric inequality, and recalling that $2C_0=1/C_I(\alpha,\rho )^{(\alpha -1)/\alpha }$
, we get that
On the contrary, up to shrinking $\bar r$
(depending only on $B$
), we can find a constant $C_2>0$
(depending only on $B$
) such that $P(B_{\tilde r}(x))\leq C_2{\mathfrak {m}(B_{\tilde r}(x))}/{\tilde r}$
for every $\tilde r\in (0,\bar r)$
; recall the discussion after (3.2). Then,
for $\mathcal {L}^1$
-a.e. $\tilde r\in (r,\bar r)$
, thanks to (3.18) and to the deformation property. Hence, ${r P(E,B_r(x))}/{\mathfrak {m}(B_r(x))}\leq \max \{C_z,C_{\tilde z}\}+C_2$
for all $x\in \partial ^e E\cap B$
and $r\in (0,\bar r)$
. Taking
we conclude that (3.14) holds. Finally, applying (3.14) we conclude that for every $x\in \partial ^e E\cap B$
and $r\in (0,\bar r/2)$
it holds that
which proves the validity of (3.15). 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. [Reference Antonelli, Pasqualetto and Pozzetta6, Remark 3.23, Theorem 3.24], exploiting theorem 1.1 and the deformation property.
Remark 3.10 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 3.9, 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 $P(E,B_r(x)) \le P(F,B_r(x)) + \Lambda \,\mathfrak {m}(E\Delta F)$
.
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 $P(E,B_r(x)) \le L\, P(F,B_r(x))$
. The class of quasi-minimal sets has been studied e.g. in [Reference Kinnunen, Korte, Lorent and Shanmugalingam31].
It is worth pointing out that, once we know that volume-constrained minimizers of the perimeter are $(L,r_0)$
-quasi minimal sets, theorem 3.9 follows directly from [Reference Kinnunen, Korte, Lorent and Shanmugalingam31, Theorem 4.2 and Lemma 5.1]. Nevertheless, we opted for a self-contained proof of theorem 3.9, which takes advantage of the openness of volume-constrained minimizers. $\blacksquare$
4. Boundedness of isoperimetric sets
In this final 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 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
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 $\mathfrak {m}(B_{r_0}(y)\setminus E)=0.$
We consider the function $f: (0,+\infty ) \to \mathbb {R}$
defined by
and observe that $|{\bf D} {\sf d}_y|(E^{(1)})\le {\rm lip}({\sf d}_y) \mathfrak {m}(E^{(1)})=\mathfrak {m}(E)<+\infty.$
By corollary 2.3 the function $f$
is locally absolutely continuous and satisfies $f'(r)=-P(B_R(x),E^{(1)})$
for $\mathcal {L}^1$
-a.e. $R>0.$
Thanks to lemma 2.10 and since $\mathfrak {m}(E^{(1)}\Delta E)=0$
, there also exists a constant $c>0$
such that
Observe that, since $\mathfrak {m}(E)<+\infty$
, it holds $\mathfrak {m}(E\setminus B_R(y))\to 0$
as $R\to +\infty$
. Hence, $f(R)\to 0$
as $R\to +\infty$
and so we can find $R_0>\rho$
such that $f(R)<\min \{\delta,v_0/2\}$
for all $R\ge R_0.$
By continuity, for every $R\ge R_0$
there exists $r(R)\in (0,\rho )$
such that
For every $R\ge R_0$
we define the set $F_R\coloneqq (E\cup B_{r(R)}(y))\cap B_{R}(y)$
, which satisfies $\mathfrak {m}(F_R)=\mathfrak {m}(E)$
thanks to (4.2) and $r(R)< R$
. Hence, by minimality, $P(E)\le P(F_R)$
for every $R\ge R_0$
. Moreover, using proposition 2.6 and the deformation property, for $\mathcal {L}^1$
-a.e. $R\ge R_0$
we have
with $C>0, \alpha >1$
constants independent of $R,$
where in the fifth 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 $\mathfrak {m}( E \setminus B_{R}(y))< v_0/2$
). This combined with (4.1) shows that
for some constant $R_1\ge R_0$
big enough and where $C_1>0$
is a constant independent of $R$
. Note that in the last inequality we used that $f(R)\to 0$
as $R\to +\infty$
and $\alpha >1.$
Since $f(R)>0$
for all $R>0$
, this shows that
which contradicts the fact that $f(R)$
is strictly positive for any $R>0$
.
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 authors thank Panu Lahti for pointing out the example in remark 3.6. The authors also thank Camillo Brena, Vesa Julin, Tapio Rajala, and Daniele Semola for fruitful discussions on the topic of the paper. The authors also thank the reviewer for the careful reading and for pointing out an inaccuracy in a preliminary version of the paper.
