(4.26)

Since $v^{\prime}\in(0,v_{1})$, where $v_{1}$ is the constant for which (4.23)
holds, and $N>2$, we have that

$I(v^{\prime})^{\frac{2-N}{N-1}}\leq\left(\vartheta
v^{\prime\frac{N-1}{N}}\right)^{\frac{2-N}{N-1}}=\vartheta^{\frac{2-N}{N-1}}v^{\prime\frac{2-N}{N}}.$
(4.27)

Hence, from (4.26) and (4.27), we have that, since $K\leq 0$,

$-\varphi^{\prime\prime}(v^{\prime})\geq\left(\frac{KN\vartheta^{\frac{2-N}{N-1}}}{N-1}+\frac{2C(N+2)}{N^{2}}\right)v^{\prime\frac{2-N}{N}}=0,$

where the last equality holds by (4.25). Hence
$-\varphi^{\prime\prime}(v^{\prime})\geq 0$, that is the sought conclusion.

Let us deal with the remaining case $N=2$. From Theorem 4.2 we get that the
function $\psi:=I^{2}$ satisfies

$-\psi^{\prime\prime}\geq 2K,$

in the viscosity sense on $(0,\mathscr{H}^{N}(X))$. Hence the function
$\eta(x):=\psi(x)+Kx^{2}$ satisfies $-\eta^{\prime\prime}\geq 0$ in the
viscosity sense on $(0,\mathscr{H}^{N}(X))$. Therefore, in this case we can
take $C:=-K$ on the whole interval $[0,\mathscr{H}^{N}(X)]$ or
$[0,\mathscr{H}^{N}(X))$, depending on whether $\mathscr{H}^{N}(X)$ is finite
or infinite.

In order to prove the last conclusion in the statement of item (1), notice
that $\eta$ is concave on $[0,v_{1}]$ and $\eta(0)=0$, therefore
$v\mapsto\eta(v)/v$ is non-increasing on $[0,v_{1}]$. Hence the function

$[0,\mathscr{H}^{N}(X))\ni
v\mapsto\frac{I^{\frac{N}{N-1}}(v)-Cv^{\frac{2+N}{N}}}{v},$

has limit when $v\to 0$. In particular, also

$[0,\mathscr{H}^{N}(X))\ni v\mapsto\frac{I^{\frac{N}{N-1}}(v)}{v},$

has limit when $v\to 0$, whence the conclusion is obtained by raising to the
power $(N-1)/N$.

Let us now prove item (2). Let $v_{1}=v_{1}(K,N,v_{0})>0$,
$\vartheta=\vartheta(K,N,v_{0})>0$, and $C=C(K,N,v_{0})>0$ be as in subsection
4.2 and item (1) proved above. All the inequalities that will appear during
the proof will be intended in the viscosity sense on $(0,v_{1})$, which is
sufficient since $\tilde{\eta}$ and $\eta(v)\coloneqq
I^{\frac{N}{N-1}}(v)-Cv^{\frac{2+N}{N}}$ are continuous on $(0,v_{1})$. Being
the function $\eta$ concave on $[0,v_{1}]$ by subsection 4.2, we know that
$\big{(}I^{\frac{N}{N-1}}(v)\big{)}^{\prime\prime}-C\frac{2(2+N)}{N^{2}}v^{\frac{2-N}{N}}=\eta^{\prime\prime}\leq
0$. Note that
$I^{\frac{1}{N-1}}(v)\geq\vartheta^{\frac{1}{N-1}}v^{\frac{1}{N}}$ by
subsection 4.2. All in all,

$\big{(}I^{\frac{N}{N-1}}(v)\big{)}^{\prime\prime}\leq
C\frac{2(2+N)}{N^{2}}v^{\frac{2-N}{N}},\qquad
I^{-\frac{1}{N-1}}(v)\leq\vartheta^{-\frac{1}{N-1}}v^{-\frac{1}{N}}.$ (4.28)

Define $\hat{C}\coloneqq C\vartheta^{-\frac{1}{N-1}}\frac{2(2+N)(N-1)}{N^{3}}$
and $\tilde{C}\coloneqq\hat{C}\frac{N^{2}}{1+N}$. Therefore, we can estimate

$\begin{split}I^{\prime\prime}(v)&=\Big{(}\big{(}I^{\frac{N}{N-1}}(v)\big{)}^{\frac{N-1}{N}}\Big{)}^{\prime\prime}=\frac{N-1}{N}\Big{(}\big{(}I^{\frac{N}{N-1}}(v)\big{)}^{-\frac{1}{N}}\big{(}I^{\frac{N}{N-1}}(v)\big{)}^{\prime}\Big{)}^{\prime}\\\
&=\frac{N-1}{N}\big{(}I^{\frac{N}{N-1}}(v)\big{)}^{\prime\prime}I^{-\frac{1}{N-1}}(v)-\frac{N-1}{N^{2}}\big{(}I^{\frac{N}{N-1}}(v)\big{)}^{-\frac{1+N}{N}}\Big{(}\big{(}I^{\frac{N}{N-1}}(v)\big{)}^{\prime}\Big{)}^{2}\\\
&\leq\frac{N-1}{N}\big{(}I^{\frac{N}{N-1}}(v)\big{)}^{\prime\prime}I^{-\frac{1}{N-1}}(v)\overset{\eqref{eq:ViscosityEst}}{\leq}\hat{C}v^{\frac{1-N}{N}}.\end{split}$

Then the function $\tilde{\eta}(v)\coloneqq I(v)-\tilde{C}v^{\frac{1+N}{N}}$
satisfies
$\tilde{\eta}^{\prime\prime}(v)=\big{(}I^{\prime}(v)-\tilde{C}\frac{1+N}{N}v^{\frac{1}{N}}\big{)}^{\prime}=I^{\prime\prime}(v)-\hat{C}v^{\frac{1-N}{N}}\leq
0$. This implies that $\tilde{\eta}$ is concave on $[0,v_{1}]$, thus reaching
the sought conclusion. ∎

######  Remark .

Item (1) of subsection 4.2 answers in the affirmative to Questions 2 and 3 in
[78] in the more general setting of $N$-dimensional $\operatorname{RCD}(K,N)$
spaces. As a consequence, it is possible to drop the additional hypothesis (H)
in [78, Lemma 4.9].

Let us now derive some further consequences from the concavity properties
above.

######  Proposition .

Let $K\leq 0$ and $N\geq 2$. Let $(X,\mathsf{d},\mathscr{H}^{N})$ be an
$\operatorname{RCD}(K,N)$ space such that there exists $v_{0}\geq 0$ such that
$\mathscr{H}^{N}(B_{1}(x))\geq v_{0}$ for every $x\in X$. Let us denote
$\vartheta:=\lim_{v\to 0}I(v)/v^{\frac{N-1}{N}}>0$, which exists due to item
(1) of subsection 4.2. Hence the following hold.

  1. (1)

There holds

$\lim_{v\to
0}\frac{I^{\prime}_{+}(v)}{v^{-\frac{1}{N}}}=\frac{N-1}{N}\vartheta.$

  2. (2)

Let $\alpha>N$. Hence there exists
$\varepsilon=\varepsilon(K,N,v_{0},\alpha)>0$ such that
$I^{\frac{\alpha}{\alpha-1}}$ is concave on $(0,\varepsilon)$. As a
consequence, $I$ is strictly subadditive on $(0,\varepsilon)$.

###### Proof.

We know from subsection 4.2 that there exist
$\tilde{C}=\tilde{C}(K,N,v_{0})>0$ and $v_{1}=v_{1}(K,N,v_{0})$ such that the
function $\tilde{\eta}(v)\vcentcolon=I(v)-\tilde{C}v^{1+\frac{1}{N}}$ is
concave on $[0,v_{1}]$. For $v,\delta>0$ such that $(1+\delta)v<v_{1}$ we can
estimate

$\begin{split}\frac{\tilde{\eta}((1+\delta)v)-\tilde{\eta}(v)}{((1+\delta)v)^{\frac{N-1}{N}}-v^{\frac{N-1}{N}}}&=\frac{\int_{v}^{(1+\delta)v}\tilde{\eta}^{\prime}_{+}(t)\,\mathrm{d}t}{\frac{N-1}{N}\int_{v}^{(1+\delta)v}t^{-\frac{1}{N}}\,\mathrm{d}t}\\\
&\leq\frac{\delta
v\tilde{\eta}^{\prime}_{+}(v)}{\frac{N-1}{N}((1+\delta)v)^{-\frac{1}{N}}\delta
v}=\frac{N(1+\delta)^{\frac{1}{N}}}{N-1}\frac{\tilde{\eta}^{\prime}_{+}(v)}{v^{-\frac{1}{N}}}.\end{split}$
(4.29)

Moreover

$\begin{split}\left|\frac{\tilde{\eta}((1+\delta)v)-\tilde{\eta}(v)}{((1+\delta)v)^{\frac{N-1}{N}}-v^{\frac{N-1}{N}}}-\vartheta\right|&=\bigg{|}\frac{\tilde{\eta}((1+\delta)v)}{((1+\delta)^{\frac{N-1}{N}}-1)v^{\frac{N-1}{N}}}-\frac{(1+\delta)^{\frac{N-1}{N}}}{(1+\delta)^{\frac{N-1}{N}}-1}\vartheta+\\\
&\phantom{=}-\frac{\tilde{\eta}(v)}{((1+\delta)^{\frac{N-1}{N}}-1)v^{\frac{N-1}{N}}}+\frac{1}{(1+\delta)^{\frac{N-1}{N}}-1}\vartheta\bigg{|}\\\
&\leq\frac{(1+\delta)^{\frac{N-1}{N}}}{(1+\delta)^{\frac{N-1}{N}}-1}\left|\frac{\tilde{\eta}((1+\delta)v)}{((1+\delta)v)^{\frac{N-1}{N}}}-\vartheta\right|\\\
&\phantom{=}+\frac{1}{(1+\delta)^{\frac{N-1}{N}}-1}\left|\frac{\tilde{\eta}(v)}{v^{\frac{N-1}{N}}}-\vartheta\right|.\end{split}$
(4.30)

Since $\lim_{v\to 0}\tilde{\eta}(v)/v^{\frac{N-1}{N}}=\vartheta>0$ and
$\tilde{\eta}^{\prime}_{+}(v)=I^{\prime}_{+}(v)-\tilde{C}(1+\tfrac{1}{N})v^{\frac{1}{N}}$,
combining (4.29) and (4.30) we get

$\begin{split}\liminf_{v\to
0}\frac{I^{\prime}_{+}(v)}{v^{-\frac{1}{N}}}&\geq\liminf_{v\to
0}\left(\frac{\tilde{\eta}((1+\delta)v)-\tilde{\eta}(v)}{((1+\delta)v)^{\frac{N-1}{N}}-v^{\frac{N-1}{N}}}\frac{N-1}{N(1+\delta)^{\frac{1}{N}}}+\tilde{C}\left(1+\frac{1}{N}\right)v^{\frac{2}{N}}\right)\\\
&=\frac{N-1}{N(1+\delta)^{\frac{1}{N}}}\vartheta.\end{split}$ (4.31)

Letting $\delta\to 0$ we obtain that $\liminf_{v\to
0}\tfrac{I^{\prime}_{+}(v)}{v^{-\frac{1}{N}}}\geq\tfrac{N-1}{N}\vartheta$. An
analogous argument shows the opposite estimate on $\limsup_{v\to
0}\tfrac{I^{\prime}_{+}(v)}{v^{-\frac{1}{N}}}$. Hence the first claim is
proved.

The proof of the second claim now easily follows from the first one employing
(4.6). In order to show that $\varepsilon$ only depends on $K,N,v_{0},\alpha$,
we observe that, similarly as before,

$\begin{split}\frac{\tilde{\eta}((1+\delta)v)-\tilde{\eta}(v)}{((1+\delta)v)^{\frac{N-1}{N}}-v^{\frac{N-1}{N}}}&=\frac{(1+\delta)^{\frac{N-1}{N}}}{[(1+\delta)^{\frac{N-1}{N}}-1]}\frac{\tilde{\eta}((1+\delta)v)}{((1+\delta)v)^{\frac{N-1}{N}}}-\frac{1}{[(1+\delta)^{\frac{N-1}{N}}-1]}\frac{\tilde{\eta}(v)}{v^{\frac{N-1}{N}}}\\\
&\geq\frac{(1+\delta)^{\frac{N-1}{N}}}{[(1+\delta)^{\frac{N-1}{N}}-1]}\left(\theta-\tilde{C}(1+\delta)^{\frac{2}{N}}v^{\frac{2}{N}}\right)\\\
&\phantom{=}-\frac{1}{[(1+\delta)^{\frac{N-1}{N}}-1]}\left(\overline{C}_{K,N}-\tilde{C}v^{\frac{2}{N}}\right),\end{split}$

for every $\delta,v$ such that $(1+\delta)v\leq V_{1}=V_{1}(K,N,v_{0})<v_{1}$,
where we used that $I((1+\delta)v)\geq\theta((1+\delta)v)^{\frac{N-1}{N}}$ for
some $\theta:=\theta(N,K,v_{0})$ by subsection 4.2, and the fact that
$I(v)\leq\overline{C}_{K,N}v^{\frac{N-1}{N}}$ for small volumes, by subsection
4.2. Hence we can first fix $\delta=\bar{\delta}=\bar{\delta}(K,N,v_{0})$
sufficiently large to get

$\frac{\tilde{\eta}((1+\bar{\delta})v)-\tilde{\eta}(v)}{((1+\bar{\delta})v)^{\frac{N-1}{N}}-v^{\frac{N-1}{N}}}\geq\frac{\theta}{2}-\frac{(1+\bar{\delta})^{\frac{N+1}{N}}-1}{[(1+\bar{\delta})^{\frac{N-1}{N}}-1]}\tilde{C}v^{\frac{2}{N}},$

for any $v\leq V_{1}/(1+\bar{\delta})$, and then take $v\leq V_{2}(K,N,v_{0})$
sufficiently small to deduce that

$\frac{\tilde{\eta}((1+\bar{\delta})v)-\tilde{\eta}(v)}{((1+\bar{\delta})v)^{\frac{N-1}{N}}-v^{\frac{N-1}{N}}}\geq\frac{\theta}{4},$

Combining with (4.29) we obtain

$\frac{I^{\prime}_{+}(v)}{v^{-\frac{1}{N}}}\geq\frac{N-1}{N(1+\bar{\delta})^{\frac{1}{N}}}\frac{\theta}{4},$

for any $v\leq V_{2}(K,N,v_{0})$. ∎

######  Remark .

Under the same assumptions of subsection 4.2, we prove in [20] that the
isoperimetric profile satisfies the asymptotic behavior for small volume given
by

$\lim_{v\to
0}\frac{I_{X}(v)}{v^{\frac{N-1}{N}}}=N\left(\omega_{N}\vartheta_{\infty,\mathrm{min}}\right)^{\frac{1}{N}}\,,$
(4.32)

where, being $v(N,K/(N-1),r)$ the volume of the ball of radius $r$ in the
simply connected model space with constant sectional curvature $K/(N-1)$ and
dimension $N$, we have that

$\vartheta_{\infty,\mathrm{min}}\vcentcolon=\lim_{r\to 0}\inf_{x\in
X}\frac{\mathscr{H}^{N}(B_{r}(x))}{v(N,K/(N-1),r)}\,>\,0\,$

is the minimum of all the possible densities at any point in $X$ or in any
pmGH limit at infinity of $X$. The limit in (4.32) yields an answer to
Questions 4 in [78].
It follows that the limit $\vartheta$ in subsection 4.2 is now known to be
equal to $N(\omega_{N}\vartheta_{\infty,\min})^{\frac{1}{N}}$. Hence

$\lim_{v\to
0}\frac{I^{\prime}_{+}(v)}{v^{-\frac{1}{N}}}=(N-1)(\omega_{N}\vartheta_{\infty,\min})^{\frac{1}{N}}\,.$

Exploiting subsection 4.2, we can derive local lower bounds on the
isoperimetric profile and local upper bounds on the Lipschitz constant of the
isoperimetric profile which are _independent_ of the ambient space. Such
bounds were unknown also in the case of smooth manifolds.

######  Corollary .

Let $0<V_{1}<V_{2}<V_{3}$ and let $K\in\mathbb{R},N\in\mathbb{N}_{\geq
2},v_{0}>0$. Then there exists
$\mathscr{I}=\mathscr{I}(K,N,v_{0},V_{1},V_{2},V_{3})>0$ such that the
following holds. If $(X,\mathsf{d},\mathscr{H}^{N})$ is an
$\operatorname{RCD}(K,N)$ space with $\inf_{x\in
X}\mathscr{H}^{N}(B_{1}(x))\geq v_{0}>0$ and $\mathscr{H}^{N}(X)\geq V_{3}$,
then

$I_{X}(v)\geq\mathscr{I}\qquad\forall\,v\in[V_{1},V_{2}]\,.$ (4.33)

###### Proof.

Suppose by contradiction that there exist sequences $v_{i}\in[V_{1},V_{2}]$
with $\lim v_{i}=v_{\infty}$ and $\operatorname{RCD}(K,N)$ spaces
$(X_{i},\mathsf{d}_{i},\mathscr{H}^{N})$ with $\inf_{x\in
X_{i}}\mathscr{H}^{N}(B_{1}(x))\geq v_{0}>0$ and $\mathscr{H}^{N}(X_{i})\geq
V_{3}$ such that $I_{X_{i}}(v_{i})\to 0$.

By Theorem 4.1 we know that for any $i$ there exist finitely many
isoperimetric regions $Z_{i}^{1},\ldots,Z_{i}^{\overline{N}_{i}}$ contained in
pmGH limits along sequences of points in $X_{i}$ such that

$\sum_{j=1}^{\overline{N}_{i}}\mathscr{H}^{N}(Z_{i}^{j})=v_{i}\,,\qquad\sum_{j=1}^{\overline{N}_{i}}\operatorname{Per}(Z_{i}^{j})=I_{X_{i}}(v_{i})\,.$
(4.34)

Without loss of generality we may assume that
$\mathscr{H}^{N}(Z_{i}^{j})\geq\mathscr{H}^{N}(Z_{i}^{j+1})$ for any $i$ and
$j=1,\ldots,\overline{N}_{i}-1$. Recalling subsection 4.2, there exists
$\varepsilon>0$ independent of $i$ such that $I_{X_{i}}$ is strictly
subadditive on $(0,\varepsilon)$. Without loss of generality, assume that
$\varepsilon<V_{1}/3$. Therefore, if
$\mathscr{H}^{N}(Z_{i}^{1})<\varepsilon/2$ for large $i$ (up to subsequence),
then $\overline{N}_{i}\geq 3$ and

$\begin{split}I_{X_{i}}(v_{i})&=\sum_{j=1}^{\overline{N}_{i}}\operatorname{Per}(Z_{i}^{j})\geq
I_{X_{i}}(\mathscr{H}^{N}(Z_{i}^{1}))+I_{X_{i}}(\mathscr{H}^{N}(Z_{i}^{2}))+\sum_{j=3}^{\overline{N}_{i}}\operatorname{Per}(Z_{i}^{j})\\\
&>I_{X_{i}}(\mathscr{H}^{N}(Z_{i}^{1})+\mathscr{H}^{N}(Z_{i}^{2}))+\sum_{j=3}^{\overline{N}_{i}}\operatorname{Per}(Z_{i}^{j})\geq
I_{X_{i}}\left(\sum_{j=1}^{\overline{N}_{i}}\mathscr{H}^{N}(Z_{i}^{j})\right)=I_{X_{i}}(v_{i})\,.\end{split}$

Hence $\mathscr{H}^{N}(Z_{i}^{1})\geq\varepsilon/2$ for any large $i$. We can
apply the Generalized Compactness result in [18, Theorem 1.2] on the sequence
$Z_{i}^{1}$ to get the existence of at most countably many sets $F_{k}$ in
$\operatorname{RCD}(K,N)$ spaces $(Y_{k},\mathsf{d}_{k},\mathscr{H}^{N})$ with
$\inf_{y\in Y_{k}}\mathscr{H}^{N}(B_{1}(y))\geq v_{0}>0$ and
$\mathscr{H}^{N}(Y_{k})\geq V_{3}$ for any $k$ such that, up to subsequence,

$\frac{\varepsilon}{2}\leq\lim_{i}\mathscr{H}^{N}(Z_{i}^{1})=\sum_{k}\mathscr{H}^{N}(F_{k})\,,\qquad\sum_{k}\operatorname{Per}(F_{k})\leq\liminf_{i}\operatorname{Per}(Z_{i}^{1})=0\,.$
(4.35)

Since $\varepsilon>0$ and $\mathscr{H}^{N}(Y_{k})\geq
V_{3}>\sum_{k}\mathscr{H}^{N}(F_{k})$ for any $k$, by relative isoperimetric
inequality there is a set $F_{k_{0}}$ with strictly positive perimeter, and
then (4.35) yields a contradiction. ∎

######  Remark .

The argument in the proof of subsection 4.2 can be employed in order to prove
an upper bound on the number $\overline{N}$ in Theorem 4.1 in terms of the
volume $V$ of the minimizing sequence. More precisely, in the hypotheses and
notation of Theorem 4.1, one can show that there exists
$\varepsilon=\varepsilon(K,N,v_{0})>0$ such that $\overline{N}\leq
1+V/\varepsilon$, see [18].

######  Corollary .

Let $0<V_{1}<V_{2}$ and let $K\leq 0,N\in\mathbb{N}_{\geq 2},v_{0}>0$. Then
there exist $\mathscr{C},\mathscr{L}>0$ depending on $K,N,v_{0},V_{1},V_{2}$
such that the following holds. If $(X,\mathsf{d},\mathscr{H}^{N})$ is
$\operatorname{RCD}(K,N)$ with $\inf_{x\in X}\mathscr{H}^{N}(B_{1}(x))\geq
v_{0}>0$ and $\mathscr{H}^{N}(X)\geq V_{2}$, then

$\begin{split}v\mapsto
I^{\frac{N}{N-1}}(v)-\mathscr{C}v^{\frac{2+N}{N}}&\quad\text{is concave on
$[0,V_{1}]$}\,,\\\ v\mapsto I^{\frac{N}{N-1}}(v)&\quad\text{is
$\mathscr{L}$-Lipschitz on $[0,V_{1}]$}\,.\end{split}$ (4.36)

In particular, for any $V\in(0,V_{1})$ there is $L>0$ depending on
$K,N,v_{0},V_{1},V_{2},V$ such that

$v\mapsto\text{$I(v)$ is $L$-Lipschitz on $[V,V_{1}]$}\,.$ (4.37)

###### Proof.

We already know from subsection 4.2 that there exist $v_{1},C>0$ depending on
$K,N,v_{0}$ such that $I^{\frac{N}{N-1}}(v)-Cv^{\frac{2+N}{N}}$ is concave on
$[0,v_{1}]$. Also assume without loss of generality that $v_{1}<V_{1}$. By the
argument in the proof of subsection 4.2, we know that
$I^{\frac{N}{N-1}}(v)-Dv^{2}$ is concave on an interval
$J\Subset(0,\mathscr{H}^{N}(X))$, provided

$D\geq-\frac{KN}{2(N-1)}I^{\frac{2-N}{N-1}}(v),$

for any $v\in J$. Exploiting subsection 4.2, we get that

$-\frac{2(N-1)}{KN}I^{\frac{N-2}{N-1}}(v)\geq-\frac{2(N-1)}{KN}\mathscr{I}^{\frac{N-2}{N-1}}\qquad\forall\,v\in\left[v_{1},\frac{V_{1}+V_{2}}{2}\right],$

for some constant $\mathscr{I}>0$ depending on $K,N,v_{0},v_{1},V_{1},V_{2}$.
Hence, letting
$D^{-1}\vcentcolon=-\tfrac{2(N-1)}{KN}\mathscr{I}^{\frac{N-2}{N-1}}$, we have
that $I^{\frac{N}{N-1}}(v)-Dv^{2}$ is concave on
$[v_{1},\tfrac{V_{1}+V_{2}}{2}]$. Hence there exists
$\mathscr{C}=\mathscr{C}(C,D,v_{1},V_{1},V_{2})>0$ such that
$I^{\frac{N}{N-1}}(v)-\mathscr{C}v^{\frac{2+N}{N}}$ is concave on
$[0,\tfrac{V_{1}+V_{2}}{2}]$, and thus on $[0,V_{1}]$.

Let us denote
$f(v)\vcentcolon=I^{\frac{N}{N-1}}(v)-\mathscr{C}v^{\frac{2+N}{N}}$. In order
to show the Lipschitzianity of $I^{\frac{N}{N-1}}$, we first observe that by
concavity, subsection 4.2, subsection 4.2, and subsection 4.2, we have

$\begin{split}f^{\prime}_{+}(v)&\leq\lim_{v\to 0}f^{\prime}_{+}(v)=\lim_{v\to
0}\frac{N}{N-1}I^{\frac{1}{N-1}}I^{\prime}_{+}(v)=\lim_{v\to
0}\frac{N}{N-1}\frac{I^{\frac{1}{N-1}}}{v^{\frac{1}{N}}}\frac{I^{\prime}_{+}(v)}{v^{-\frac{1}{N}}}\\\
&=\left(\lim_{v\to 0}\frac{I}{v^{\frac{N-1}{N}}}\right)^{\frac{N}{N-1}}\leq
C_{1}(K,N,v_{0}),\end{split}$ (4.38)

for any $v\in[0,V_{1}]$. On the other hand, by subsection 4.2, we get that

$f(v)\geq\widetilde{\mathscr{I}}^{\frac{N}{N-1}}-\mathscr{C}\left(\frac{V_{1}+V_{2}}{2}\right)^{\frac{2+N}{N}}\qquad\forall\,v\in\left[\frac{V_{1}}{2},\frac{V_{1}+V_{2}}{2}\right]\,,$
(4.39)

where $\widetilde{\mathscr{I}}$ depends on $K,N,v_{0},V_{1},V_{2}$. Since $f$
is concave on $[0,\tfrac{V_{1}+V_{2}}{2}]$, then there exists a constant
$C_{2}=C_{2}(K,N,v_{0},V_{1},V_{2})>0$ such that $f^{\prime}_{+}(V_{1})\geq-
C_{2}$. All in all we obtained that

$-C_{2}\leq f^{\prime}_{+}(v)\leq C_{1}\qquad\forall\,v\in[0,V_{1}]\,,$

and then $I^{\frac{N}{N-1}}$ is $\mathscr{L}$-Lipschitz on $[0,V_{1}]$ for
some constant $\mathscr{L}$ depending on $K,N,v_{0},V_{1},V_{2}$. ∎

The above uniform bounds on the isoperimetric profile allow to derive the
following uniform regularity properties on isoperimetric regions, that is, we
prove that isoperimetric sets satisfy almost-minimality properties and density
estimates with constants _independent_ of the specific ambient space. This is
new even in the case of isoperimetric sets in smooth Riemannian manifolds.

######  Corollary .

Let $0<V_{1}<V_{2}<V_{3}$ and let $K\in\mathbb{R},N\in\mathbb{N}_{\geq
2},v_{0}>0$. Then there exist $\Lambda,R>0$ depending on
$K,N,v_{0},V_{1},V_{2},V_{3}$ such that the following holds.

If $(X,\mathsf{d},\mathscr{H}^{N})$ is an $\operatorname{RCD}(K,N)$ space with
$\inf_{x\in X}\mathscr{H}^{N}(B_{1}(x))\geq v_{0}>0$, $\mathscr{H}^{N}(X)\geq
V_{3}$, and $E\subset X$ is an isoperimetric region with
$\mathscr{H}^{N}(E)\in[V_{1},V_{2}]$, then $E$ is a $(\Lambda,R)$-minimizer,
i.e., for any $F\subset X$ such that $F\Delta E\subset B_{R}(x)$ for some
$x\in X$, then

$\operatorname{Per}(E)\leq\operatorname{Per}(F)+\Lambda\mathscr{H}^{N}(F\Delta
E)\,.$

In particular there exist $R^{\prime}>0$, $C_{1}\in(0,1)$ and $C_{2},C_{3}>0$
depending on $K,N,v_{0},V_{1},V_{2},V_{3}$ such that

$\operatorname{Per}(E,B_{r}(x))\leq(1+C_{3}r)\operatorname{Per}(F,B_{r}(x)),$
(4.40)

for any $x\in X$, $r\in(0,R^{\prime}]$, and any $F$ such that $E\Delta
F\Subset B_{r}(x)$. Moreover

$C_{1}\leq\frac{\mathscr{H}^{N}(B_{r}(x)\cap
E)}{\mathscr{H}^{N}(B_{r}(x))}\leq 1-C_{1}\,,\qquad
C_{2}^{-1}\leq\frac{\operatorname{Per}(E,B_{r}(x))}{r^{N-1}}\leq C_{2}\,,$

for any $x\in\partial E$ and any $r\in(0,R^{\prime}]$.

###### Proof.

Let $R>0$ be a radius such that
$\mathscr{H}^{N}(B_{R}(x))\leq\min\\{V_{1}/2,(V_{3}-V_{2})/2\\}$ for any $x\in
X$. Let also $L>0$ be the Lipschitz constant of $I$ on
$[V_{1}/2,(V_{2}+V_{3})/2]$ given by subsection 4.2. Then for any $F\subset X$
with $F\Delta E\subset B_{R}(x)$ it holds

$\operatorname{Per}(F)\geq I(\mathscr{H}^{N}(F))\geq
I(\mathscr{H}^{N}(E))-L|\mathscr{H}^{N}(E)-\mathscr{H}^{N}(F)|\geq\operatorname{Per}(E)-L\mathscr{H}^{N}(F\Delta
E).$

For the second part of the claim let us exploit [19, Equation (3.51) in Remark
3.25]. According to the latter, one can find $R^{\prime}>0$ possibly smaller
than $R$ and only depending on $K,N,v_{0},V_{1},V_{2},V_{3}$ such that,
calling $v(N,K/(N-1),r)$ the volume of the geodesic ball of radius $r$ in the
model of constant sectional curvature $K/(N-1)$ and dimension $N$, one has,
for some constant $\widetilde{C}_{1}$ only depending on $N,K,v_{0}$, that

$\operatorname{Per}(E,B_{r}(x))\leq\frac{1+\Lambda\widetilde{C}_{1}v(N,K/(N-1),r)^{1/N}}{1-\Lambda\widetilde{C}_{1}v(N,K/(N-1),r)^{1/N}}\operatorname{Per}(F,B_{r}(x)),$

for any $x\in X$, $r\in(0,R^{\prime}]$, and any $F$ such that $E\Delta
F\Subset B_{r}(x)$. Hence, by taking $R^{\prime}$ smaller if needed, and by
using that there exists $\widetilde{C}_{2}$ only depending on $N,K$ such that
$v(N,K/(N-1),r)\leq\widetilde{C}_{2}r^{N}$ for every $r\leq 1$, we get the
conclusion in (4.40).

The last part of the claim follows as in the proof of [19, Proposition 3.27].
∎

###  4.3. Consequences

From the previous results on the concavity of the isoperimetric profile one
can prove that in the $N$-dimensional $\operatorname{RCD}(K,N)$ spaces,
isoperimetric regions of sufficiently small volume are connected. If $K=0$,
the conclusion holds for all volumes.

######  Corollary .

Let $(X,\mathsf{d},\mathscr{H}^{N})$ be an $\operatorname{RCD}(K,N)$ space
with $N\geq 2$. Let us assume that $\inf_{x\in X}\mathscr{H}^{N}(B_{1}(x))\geq
v_{0}>0$. Let $\varepsilon$ be such that the isoperimetric profile $I$ is
strictly subadditive on $(0,\varepsilon)$. Such an $\varepsilon>0$ exists
thanks to item (2) of subsection 4.2, and, if $K=0$, one can take
$\varepsilon=\mathscr{H}^{N}(X)$.
Then, if $E$ is an isoperimetric region in $X$ with
$\mathscr{H}^{N}(E)<\varepsilon$, $E$ is connected. If in addition
$\mathscr{H}^{N}$ is finite, then $E$ is simple (i.e. $E$ and $X\setminus E$
are indecomposable) and $E^{(0)}$ is connected.

###### Proof.

From item (2) of subsection 4.2 one gets that there exists $\varepsilon>0$
such that $I$ is strictly subadditive on $(0,\varepsilon)$. Notice that if
$K=0$, we have that $I^{N/(N-1)}$ is concave as a consequence of item (2) of
Theorem 4.2, hence in this case $I$ is striclty subadditive on
$(0,\mathscr{H}^{N}(X))$.

Assume $\Omega$ is an isoperimetric region of volume $V<\varepsilon$. We prove
first that $\Omega$ is indecomposable of volume $V$. Suppose by contradiction
it is decomposable. Hence $\Omega=\Omega_{1}\cup\Omega_{2}$ with
$\mathscr{H}^{N}(\Omega_{1})+\mathscr{H}_{N}(\Omega_{2})=V$ and
$\operatorname{Per}(\Omega_{1})+\operatorname{Per}(\Omega_{2})=\operatorname{Per}(\Omega)=I(V)$.
Hence

$I(V)=\operatorname{Per}(\Omega_{1})+\operatorname{Per}(\Omega_{2})\geq
I(\mathscr{H}^{N}(\Omega_{1}))+I(\mathscr{H}^{N}(\Omega_{2}))>I(V),$

where the last inequality is due to the fact that $I$ is strictly subadditive
on $(0,\varepsilon)$. Hence we reach a contradiction.

To prove that $E$ is connected, we argue by contradiction: suppose there exist
non-empty open sets $U,V\subseteq X$ such that $U\cap V=\emptyset$ and $U\cup
V=E$. Being $U$, $V$, $E$ open, we have that $\partial E=\partial
U\cup\partial V$. Since we know that $\partial E=\partial^{e}E$ (recall that
we are always assuming that $E=E^{(1)}$), we deduce that
$\mathscr{H}^{N-1}(\partial U)<+\infty$ and accordingly $U$ is a set of finite
perimeter, see e.g., [65]. Now consider the BV function $f\coloneqq\chi_{U}$.
Again thanks to the fact that $U$ and $V$ are open, we get $|Df|(E)=0$. Given
that $\operatorname{RCD}(K,N)$ spaces have the two-sidedness property in the
sense of [28, Definition 1.28]) (see [28, Example 1.31]) and $E$ is
indecomposable, we deduce from [28, Theorem 2.5] that $f$ is
$\mathscr{H}^{N}$-a.e. constant on $E$, which leads to a contradiction.
Therefore, the set $E$ is connected.

Now assume that $\mathscr{H}^{N}(X)<+\infty$. Then $X\setminus E$ is an
isoperimetric set of its own volume, thus accordingly the last part of the
statement follows from the first one applied to $X\setminus E$. ∎

######  Remark .

Classical examples show that, even for smooth compact Riemannian manifolds,
connectedness of isoperimetric regions might fail for volumes bounded away
from zero if the Ricci curvature is negative somewhere, see for instance [25,
Remark 2.3.12].

From the strictly subadditivity of the isoperimetric profile (for small
volumes, if $K<0$), we infer that there is at most one component in the
asymptotic mass decomposition result in Theorem 4.1 (for small volumes if
$K<0$). This is understood in the following statement.

######  Lemma .

Let $(X,\mathsf{d},\mathscr{H}^{N})$ be an $\operatorname{RCD}(K,N)$ space
with $N\geq 2$. Let us assume that $\inf_{x\in X}\mathscr{H}^{N}(B_{1}(x))\geq
v_{0}>0$. Let $\varepsilon$ be such that the isoperimetric profile $I$ is
strictly subadditive on $(0,\varepsilon)$. Such an $\varepsilon>0$ exists
thanks to item (2) of subsection 4.2 and, if $K=0$, one can take
$\varepsilon=\mathscr{H}^{N}(X)$.

  1. (1)

Let $\\{\Omega_{i}\\}_{i\in\mathbb{N}}$ be a minimizing (for the perimeter)
sequence of bounded finite perimeter sets of volume $v<\varepsilon$ in $X$.
Then, if one applies Theorem 4.1, either $\overline{N}=0$, or $\overline{N}=1$
and $\mathscr{H}^{N}(\Omega)=0$.

  2. (2)

Let $X_{1},\ldots,X_{\overline{N}}$ be pmGH limits of $X$ along sequences of
points $\\{p_{i,j}\\}_{i\in\mathbb{N}}$, for
$j=1,\ldots,\overline{N}\in\mathbb{N}\cup\\{+\infty\\}$. Let
$\Omega=E\cup\bigcup_{j=1}^{\overline{N}}E_{j}$, with $E\subset X,E_{j}\subset
X_{j}$ be a set achieving the infimum in (4.2) for some $v<\varepsilon$. Then
exactly one component among $E,E_{1},\ldots,E_{\overline{N}}$ is nonempty.

In particular, for any $v<\varepsilon$ there is an $\operatorname{RCD}(K,N)$
space $(Y,\mathsf{d},\mathscr{H}^{N})$ which is either $X$ or a pmGH limit of
$X$ along a sequence $\\{p_{i}\\}_{i}\subset X$, and a set $E\subset Y$ such
that $\mathscr{H}^{N}(E)=v$ and $I_{X}(v)=\operatorname{Per}(E)$.

###### Proof.

Let us prove the first item, and to this aim we adopt the notation of Theorem
4.1. Let us assume that the assertion is not true. Hence there are $j\geq 2$
nonempty sets $E_{1},\dots,E_{j}$ among $\Omega,Z_{1},\dots,Z_{\overline{N}}$
such that, due to (4.1) and the fact that $E_{1},\dots,E_{j}$ are
isoperimetric in their own spaces,

$I(v)=\sum_{k=1}^{j}I_{X_{k}}(\mathscr{H}^{N}(E_{k})),$

and $\sum_{k=1}^{j}\mathscr{H}^{N}(E_{k})=v$. Using that, for every $v>0$ and
every $k\in\\{1,\dots,j\\}$, we have $I_{X_{k}}(v)\geq I(v)$, see [18], we
thus conclude that

$I(v)\geq\sum_{k=1}^{j}I(\mathscr{H}^{N}(E_{k})),$

which is in contradiction with the fact that $I$ is strictly subadditive on
$(0,v)\subset(0,\varepsilon)$, and $j\geq 2$.

The second item analogously follows from the strict subadditivity of the
profile $I$ of $X$ and the identity in section 4. ∎

As we already remarked, on a smooth Riemannian manifold $(M^{N},g)$, the
barrier $c$ obtained applying Theorem 3.2 to an isoperimetric region $E$ is
unique and it coincides with the value of the (constant) mean curvature of the
regular part of $\partial E$. Moreover, $t\mapsto\operatorname{Per}(E_{t})$ is
always differentiable at $t=0$ and

$\frac{\mathop{}\\!\mathrm{d}}{\mathop{}\\!\mathrm{d}t}|_{t=0}\operatorname{Per}(E_{t})=c\operatorname{Per}(E)\,.$
(4.41)

A well known consequence of this observation is the fact that the
isoperimetric profile is differentiable with derivative $I^{\prime}(v)=c$ at
any volume $v\in(0,\infty)$ such that there exists a unique isoperimetric
region $E$ of volume $v$ (with constant mean curvature of the boundary equal
to $c$). Below we partially generalize this statement to the present context.

######  Corollary .

Let $(X,\mathsf{d},\mathscr{H}^{N})$ be an $\operatorname{RCD}(K,N)$ space
such that $\mathscr{H}^{N}(B_{1}(x))\geq v_{0}>0$ for any $x\in X$.
Let $v\in(0,\mathscr{H}^{N}(X))$ and let
$\Omega,Z_{1},\ldots,Z_{\overline{N}}$ be isoperimetric sets given by Theorem
4.1 applied to the minimization problem at volume $v$111We understand there is
just one isoperimetric region $\Omega\subset X$ if $X$ is compact.. Let $c$ be
any barrier given by Theorem 3.2 applied on either $\Omega,Z_{1},\ldots,$ or
$Z_{\overline{N}}$. Then

$I_{+}^{\prime}(v)\leq c\leq I_{-}^{\prime}(v).$ (4.42)

Hence, if $I$ is differentiable at $v$, if
$E\in\\{\Omega,Z_{1},\ldots,Z_{\overline{N}}\\}$, then the barrier given by
Theorem 3.2 applied to $E$ is unique and equal to $I^{\prime}(v)$. In
particular, if $I$ is differentiable at $v$ and $E$ is an isoperimetric region
on $X$ for the volume $v>0$, we have that the barrier given by Theorem 3.2
applied to $E$ is unique and equal to $I^{\prime}(v)$.

###### Proof.

Let $v\in(0,\mathscr{H}^{N}(X))$ be fixed. Take $\Omega_{i}$ a minimizing
sequence of bounded sets of volume $v$ and let $\Omega,Z_{j}$ be the
isoperimetric regions in $X,X_{j}$ respectively, according to the notation of
Theorem 4.1. Let $c$ be any barrier as in the statement.

Let $E$ be an arbitrary isoperimetric region among $\Omega,Z_{j}$ in the
spaces $X,X_{j}$. Let us set
$\beta(t):=\mathscr{H}^{N}(E_{t})+\sum_{T\in\\{\Omega,Z_{1},\dots,Z_{\overline{N}}\\},T\neq
E}\mathscr{H}^{N}(T)$, where $E_{t}$ is the $t$-tubular neighbourhood of $E$
for $t\in(-\varepsilon,\varepsilon)$, with $\varepsilon>0$ small enough, see
the discussion after (4.10). Arguing as in (4.11) we obtain that, for every
$t\in(-\varepsilon,\varepsilon)$,

$\begin{split}I(\beta(t))&\leq\operatorname{Per}(E_{t})+\sum_{T\in\\{\Omega,Z_{1},\dots,Z_{\overline{N}}\\},T\neq
E}\operatorname{Per}(T)\\\ &\leq
J_{c,K,N}(t)\operatorname{Per}(E)+\sum_{T\in\\{\Omega,Z_{1},\dots,Z_{\overline{N}}\\},T\neq
E}\operatorname{Per}(T).\end{split}$ (4.43)

Since $I$ is continuous by section 4 and $\beta(t)$ is continuous at $t=0$ by
the coarea formula, from (4.43) we get that $\operatorname{Per}(E_{t})$ is
continuous at $t=0$. Thus $\beta(t)$ is differentiable at $t=0$. Hence the
function

$\psi(t):=I(\beta(t))-J_{c,K,N}(t)\operatorname{Per}(E)-\sum_{T\in\\{\Omega,Z_{1},\dots,Z_{\overline{N}}\\},T\neq
E}\operatorname{Per}(T)$

is nonpositive on $[0,\varepsilon)$. Moreover, $\psi(0)=0$ and $\psi$ is
right-differentiable at $t=0$, since $I$ is right differentiable everywhere,
by subsection 4.2 and $\beta(t)$ is differentiable at $t=0$. Therefore
$\psi_{+}^{\prime}(0)\leq 0$, that gives $I_{+}^{\prime}(v)\leq c$.

Arguing in the same way on $(-\varepsilon,0]$ we get $c\leq
I_{-}^{\prime}(v)$. Hence (4.42) is shown and the remaining part of the
statement follows. ∎

######  Remark .

Letting $X$, $v\in(0,\mathscr{H}^{N}(X))$ and
$\Omega,Z_{1},\ldots,Z_{\overline{N}}$ as in subsection 4.3, if $I$ is
differentiable at $v$, then the function $t\mapsto\operatorname{Per}(E_{t})$
is differentiable at $t=0$ and its derivative is $c\operatorname{Per}(E)$, for
any $E\in\\{\Omega,Z_{1},\ldots,Z_{\overline{N}}\\}$ where $c$ is a barrier
for $E$. This follows by the argument in the proof of subsection 4.3, see
(4.43).

Moreover, if $I$ is differentiable at $v$, every isoperimetric set
$\Omega,Z_{1},\ldots,Z_{\overline{N}}$ has only one possible barrier
$c=I^{\prime}(v)$.

Another consequence of the sharp concavity properties of the isoperimetric
profile are uniform diameter bounds for isoperimetric regions of small volume,
in great generality, and any volume if the underlying space is
$\operatorname{RCD}(0,N)$ with Euclidean volume growth.

######  Proposition .

For every $N\geq 2$ natural number, $K\leq 0$, and $v_{0}>0$, there exist
constants $\bar{v}=\bar{v}(K,N,v_{0})>0$ and $C=C(K,N,v_{0})>0$ such that the
following holds. Let $(X,\mathsf{d},\mathscr{H}^{N})$ be an
$\operatorname{RCD}(K,N)$ space. Suppose that $\mathscr{H}^{N}(B_{1}(x))\geq
v_{0}$ holds for every $x\in X$. Let $E\subseteq X$ be an isoperimetric
region. Then

$\mathrm{diam}E\leq C\mathscr{H}^{N}(E)^{\frac{1}{N}}\quad\text{ whenever
}\mathscr{H}^{N}(E)\leq\bar{v}\,.$ (4.44)

Moreover, if $K=0$ and

$A\coloneqq{\rm
AVR}(X,\mathsf{d},\mathscr{H}^{N})\coloneqq\lim_{r\to+\infty}\frac{\mathscr{H}^{N}(B_{r}(\bar{x}))}{\omega_{N}r^{N}}>0,$

where $\bar{x}\in X$ and $\omega_{N}$ is the Euclidean volume of the unit ball
in $\mathbb{R}^{N}$, then it holds that

$\mathrm{diam}E\leq\tilde{C}\mathscr{H}^{N}(E)^{\frac{1}{N}}\quad\text{ for
every isoperimetric region }E\subseteq X\,,$ (4.45)

for some constant $\tilde{C}=\tilde{C}(K,N,A)>0$.

###### Proof.

By virtue of subsection 4.2, subsection 4.2, subsection 4.2, and item (2) of
subsection 4.2, we can find constants $\vartheta=\vartheta(K,N,v_{0})>0$,
$v_{1}=v_{1}(K,N,v_{0})>0$, and $C_{1}=C_{1}(K,N,v_{0})>0$ such that:

  * a)

$\mathscr{H}^{N}(F)^{\frac{N-1}{N}}\leq\vartheta^{-1}\operatorname{Per}(F)$
for every set of finite perimeter $F\subseteq X$ with $\mathscr{H}^{N}(F)\leq
v_{1}$.

  * b)

$I(v)\geq\vartheta v^{\frac{N-1}{N}}$ for every $v\leq v_{1}$.

  * c)

$I(v)/v^{\frac{N-1}{N}}\leq C_{1}$ for every $v\leq v_{1}$.

  * d)

$\tilde{\eta}(v)\coloneqq I(v)-C_{1}v^{\frac{1+N}{N}}$ is concave on
$[0,v_{1}]$.

Given any $x\in E$ and $r>0$, we define
$m_{x}(r)\coloneqq\mathscr{H}^{N}(E\cap B_{r}(x))$ and $E^{x}_{r}\coloneqq
E\setminus B_{r}(x)$. Note that

$a^{\frac{1}{N}}-(a-t)^{\frac{1}{N}}\leq a^{\frac{1-N}{N}}t,\quad\text{ for
every }0<t\leq a.$ (4.46)

Set $v_{E}\coloneqq\mathscr{H}^{N}(E)$ for brevity and suppose $v_{E}\leq
v_{1}$. Since $\lim_{v\searrow 0}\tilde{\eta}(v)=0$, we deduce from d) that
the function $v\mapsto\tilde{\eta}(v)/v$ is non-increasing on $[0,v_{1}]$, so
that

$\frac{I(v_{E})}{v_{E}}-C_{1}v_{E}^{\frac{1}{N}}\leq\frac{I\big{(}v_{E}-m_{x}(r)\big{)}}{v_{E}-m_{x}(r)}-C_{1}\big{(}v_{E}-m_{x}(r)\big{)}^{\frac{1}{N}},\quad\text{
for every }r>0.$

Multiplying both sides by $v_{E}-m_{x}(r)$, rearranging the terms, and using
(4.46), we deduce that

$\begin{split}\operatorname{Per}(E)&=I(v_{E})\\\ &\leq
I\big{(}v_{E}-m_{x}(r)\big{)}+\frac{m_{x}(r)}{v_{E}}I(v_{E})+C_{1}\big{(}v_{E}-m_{x}(r)\big{)}\Big{(}v_{E}^{\frac{1}{N}}-\big{(}v_{E}-m_{x}(r)\big{)}^{\frac{1}{N}}\Big{)}\\\
&\leq\operatorname{Per}(E_{r}^{x})+\frac{m_{x}(r)}{v_{E}}I(v_{E})+C_{1}v_{E}^{\frac{1}{N}}m_{x}(r).\end{split}$
(4.47)

By using [3, Lemma 4.5] we obtain
$\operatorname{Per}(E^{x}_{r})+\operatorname{Per}(E\cap
B_{r}(x))\leq\operatorname{Per}(E)+2m^{\prime}_{x}(r)$ for a.e. $r>0$. Define

$r_{0}\coloneqq\frac{1}{\omega_{N}^{1/N}}\min\bigg{\\{}v_{1}^{1/N},\frac{\vartheta
v_{E}}{4I(v_{E})},\frac{\vartheta}{4C_{1}v_{E}^{1/N}}\bigg{\\}}>0.$

For every $r\in(0,r_{0})$ we have $\mathscr{H}^{N}(E\cap
B_{r}(x))\leq\mathscr{H}^{N}(B_{r}(x))\leq\omega_{N}r^{N}\leq v_{1}$, so that
a) yields

$\operatorname{Per}(E^{x}_{r})\leq\operatorname{Per}(E)-\operatorname{Per}(E\cap
B_{r}(x))+2m^{\prime}_{x}(r)\leq\operatorname{Per}(E)-\vartheta
m_{x}(r)^{\frac{N-1}{N}}+2m^{\prime}_{x}(r)$ (4.48)

holds for a.e. $r\in(0,r_{0})$. Combining (4.47) with (4.48), we thus obtain
for a.e. $r\in(0,r_{0})$ that

$\vartheta
m_{x}(r)^{\frac{N-1}{N}}-\frac{I(v_{E})}{v_{E}}m_{x}(r)-C_{1}v_{E}^{\frac{1}{N}}m_{x}(r)\leq
2m^{\prime}_{x}(r).$ (4.49)

Our choice of $r_{0}$ yields
$m_{x}(r)^{1/N}\leq\omega_{N}^{1/N}r_{0}\leq(\vartheta v_{E})/(4I(v_{E}))$ and
$m_{x}(r)^{1/N}\leq\vartheta/(4C_{1}v_{E}^{1/N})$ for every $r\in(0,r_{0})$.
Hence we deduce that

$\begin{split}\frac{\vartheta}{2}m_{x}(r)^{\frac{N-1}{N}}&\overset{\phantom{\eqref{eq:BoundDiam3}}}{=}\vartheta
m_{x}(r)^{\frac{N-1}{N}}-\frac{\vartheta}{4}m_{x}(r)^{\frac{N-1}{N}}-\frac{\vartheta}{4}m_{x}(r)^{\frac{N-1}{N}}\\\
&\overset{\phantom{\eqref{eq:BoundDiam3}}}{\leq}\vartheta
m_{x}(r)^{\frac{N-1}{N}}-\bigg{(}\frac{I(v_{E})}{v_{E}}m_{x}(r)^{\frac{1}{N}}-C_{1}v_{E}^{\frac{1}{N}}m_{x}(r)^{\frac{1}{N}}\bigg{)}m_{x}(r)^{\frac{N-1}{N}}\\\
&\overset{\eqref{eq:BoundDiam3}}{\leq}2m^{\prime}_{x}(r)\end{split}$ (4.50)

holds for a.e. $r\in(0,r_{0})$. Therefore we can estimate

$\begin{split}m_{x}(r_{0})^{1/N}&\geq
m_{x}(r_{0})^{1/N}-m_{x}(r_{0}/2)^{1/N}=\int_{r_{0}/2}^{r_{0}}\big{(}m_{x}(r)^{1/N}\big{)}^{\prime}\,\mathop{}\\!\mathrm{d}r\\\
&=\frac{1}{N}\int_{r_{0}/2}^{r_{0}}\frac{m^{\prime}_{x}(r)}{m_{x}(r)^{(N-1)/N}}\,\mathop{}\\!\mathrm{d}r\overset{\eqref{eq:BoundDiam4}}{\geq}\frac{\vartheta
r_{0}}{8N}.\end{split}$ (4.51)

Pick a maximal family $\mathcal{F}$ of pairwise disjoint open balls $B$ having
radius $r_{0}$ and center $c(B)$ in $E$. The maximality ensures that
$\\{B_{2r_{0}}(c(B))\,:\,B\in\mathcal{F}\\}$ is a covering of $E$. The
estimate

$v_{E}=\mathscr{H}^{N}(E)\geq\sum_{B\in\mathcal{F}}\mathscr{H}^{N}(E\cap
B)=\sum_{B\in\mathcal{F}}m_{c(B)}(r_{0})\overset{\eqref{eq:BoundDiam5}}{\geq}\bigg{(}\frac{\vartheta
r_{0}}{8N}\bigg{)}^{N}\\#\mathcal{F}$

shows that the cardinality of $\mathcal{F}$ cannot exceed the quantity
$(8N)^{N}v_{E}/(\vartheta r_{0})^{N}$. Given that the set
$U\coloneqq\bigcup_{B\in\mathcal{F}}B_{2r_{0}}(c(B))$ contains $E$, which is
connected by subsection 4.3, we deduce that also $U$ must be connected. Then

$\mathrm{diam}E\leq\mathrm{diam}U\leq\sum_{B\in\mathcal{F}}\mathrm{diam}B_{2r_{0}}(c(B))\leq
4r_{0}\\#\mathcal{F}\leq\frac{4(8N)^{N}v_{E}}{\vartheta^{N}r_{0}^{N-1}}.$
(4.52)

Now let us define the quantity $\bar{v}=\bar{v}(K,N,v_{0})>0$ as

$\bar{v}\coloneqq\min\bigg{\\{}v_{1},4^{N}v_{1},\frac{\vartheta^{N/2}}{C_{1}^{N/2}}\bigg{\\}}.$

Observe that if $v_{E}\leq\bar{v}$, then it holds that

$\begin{split}\frac{\vartheta v_{E}}{4I(v_{E})}&\leq\frac{\vartheta
v_{E}^{\frac{N-1}{N}}}{I(v_{E})}\frac{\bar{v}^{\frac{1}{N}}}{4}\overset{\rm
b)}{\leq}\frac{\bar{v}^{\frac{1}{N}}}{4}\leq v_{1}^{1/N},\qquad\frac{\vartheta
v_{E}}{4I(v_{E})}\leq\frac{\vartheta
v_{E}^{\frac{N-1}{N}}}{I(v_{E})}\frac{\bar{v}^{\frac{2}{N}}}{4v_{E}^{\frac{1}{N}}}\overset{\rm
b)}{\leq}\frac{\bar{v}^{\frac{2}{N}}}{4v_{E}^{\frac{1}{N}}}\leq\frac{\vartheta}{4C_{1}v_{E}^{\frac{1}{N}}}.\end{split}$

This implies that $r_{0}=\vartheta v_{E}/(4\omega_{N}^{1/N}I(v_{E}))$ whenever
$v_{E}\leq\bar{v}$. Then (4.52) and c) yield

$\begin{split}\mathrm{diam}E&\leq\frac{4(8N)^{N}v_{E}}{\vartheta^{N}}\frac{(4\omega_{N}^{1/N})^{N-1}I^{N-1}(v_{E})}{\vartheta^{N-1}v_{E}^{N-1}}=C^{\prime}\frac{I^{N-1}(v_{E})}{v_{E}^{N-2}}=C^{\prime}\bigg{(}\frac{I(v_{E})}{v_{E}^{\frac{N-2}{N-1}}}\bigg{)}^{N-1}\\\
&=C^{\prime}\bigg{(}\frac{I(v_{E})}{v_{E}^{\frac{N-1}{N}}}v_{E}^{\frac{1}{N(N-1)}}\bigg{)}^{N-1}=C^{\prime}\bigg{(}\frac{I(v_{E})}{v_{E}^{\frac{N-1}{N}}}\bigg{)}^{N-1}v_{E}^{\frac{1}{N}}\leq
C^{\prime}C_{1}^{N-1}v_{E}^{\frac{1}{N}}\end{split}$

whenever $v_{E}\leq\bar{v}$, where $C^{\prime}=C^{\prime}(K,N,v_{0})>0$. This
proves (4.44) with $C\coloneqq C^{\prime}C_{1}^{N-1}$.

Finally, suppose that $K=0$ and $A={\rm AVR}(X,\mathsf{d},\mathscr{H}^{N})>0$.
Fix an isoperimetric set $E\subseteq X$ such that
$v_{E}>\tilde{v}\coloneqq\min\big{\\{}\bar{v}(0,N,v_{0}),\bar{v}(0,N,A\omega_{N})\big{\\}}$.
We denote by $\tilde{\mathsf{d}}$ the rescaled distance
$\tilde{v}^{1/N}\mathsf{d}/v_{E}^{1/N}$ on $X$ and all the metric quantities
associated with $\tilde{\mathsf{d}}$ will be marked with a tilde. The metric
measure space $(X,\tilde{\mathsf{d}},\tilde{\mathscr{H}}^{N})$ is an
$\operatorname{RCD}(0,N)$ space and for any $x\in X$ it holds that

$\tilde{\mathscr{H}}^{N}\big{(}\tilde{B}_{1}(x)\big{)}=\tilde{\mathscr{H}}^{N}\big{(}B_{(v_{E}/\tilde{v})^{1/N}}(x)\big{)}=\frac{\mathscr{H}^{N}\big{(}B_{(v_{E}/\tilde{v})^{1/N}}(x)\big{)}}{v_{E}/\tilde{v}}\geq
A\omega_{N}.$

Moreover, we have that
$\tilde{\mathscr{H}}^{N}(E)=\tilde{v}\mathscr{H}^{N}(E)/v_{E}=\tilde{v}$, thus
(4.44) applied to $(X,\tilde{\mathsf{d}},\tilde{\mathscr{H}}^{N})$ yields

$\mathrm{diam}E=\frac{v_{E}^{\frac{1}{N}}}{\tilde{v}^{\frac{1}{N}}}\tilde{\mathrm{diam}}E\leq\frac{v_{E}^{\frac{1}{N}}}{\tilde{v}^{\frac{1}{N}}}C(0,N,A\omega_{N})\tilde{\mathscr{H}}^{N}(E)^{\frac{1}{N}}=C(0,N,A\omega_{N})\mathscr{H}^{N}(E)^{\frac{1}{N}}.$

This gives (4.45) with
$\tilde{C}\coloneqq\max\big{\\{}C(0,N,v_{0}),C(0,N,A\omega_{N})\big{\\}}$. The
statement is achieved. ∎

######  Remark .

subsection 4.3 is a three-fold generalization of [78, Lemma 4.9]. Indeed, we
prove the statement in the non smooth $\operatorname{RCD}(K,N)$ case with
reference measure $\mathscr{H}^{N}$ and $\mathscr{H}^{N}(B_{1}(x))\geq
v_{0}>0$ for every $x\in X$, while the authors in [78] deal with smooth
Riemannian manifolds. Moreover, in the smooth case, our setting corresponds to
the weak bounded geometry hypothesis in [78], while [78, Lemma 4.9] is proved
under the stronger mild bounded geometry hypothesis. Finally, we are able to
drop the hypothesis (H) in [78, Lemma 4.9]. Hence, subsection 4.3 is new even
in the smooth setting and sharpens the previous [78, Lemma 4.9].

Notice that subsection 4.3 generalizes also [67, Lemma 5.5] to the setting of
$\operatorname{RCD}(0,N)$ spaces with a uniform bound below on the volume of
unit balls. The generalization is by far non-trivial since the results in [67]
hold for convex bodies with uniform bound below on the volume of unit balls.

###  4.4. Stability of isoperimetric sets

In this final part we derive a stability result for sequences of isoperimetric
sets converging in $L^{1}$ to a limit set, improving the $L^{1}$ convergence
to Hausdorff convergence. First, we need the following technical lemma,
independently observed also in the recent [61], relating convergence in
Hausdorff sense and convergence of mean curvature barriers.

######  Lemma .

Let $(X_{i},\mathsf{d}_{i},\mathscr{H}^{N},x_{i})$ be
$\operatorname{RCD}(K,N)$ spaces converging to
$(Y,\mathsf{d}_{Y},\mathscr{H}^{N},y)$ in pmGH sense, and let
$(Z,\mathsf{d}_{Z})$ be a space realizing the convergence. Assume that
$\mathscr{H}^{N}(B_{1}(p))\geq v_{0}>0$ for any $p\in X_{i}$ and any $i$. Let
$E_{i}\subset X_{i}$, $F\subset Y$, and let $f_{i},f$ be the signed distance
functions from $E_{i},F$, respectively. Then the following holds.

  1. (1)

If $E_{i}$’s and $F$ are open and bounded, $E_{i}\to F$ in $L^{1}$-strong, and
$\partial E_{i}\to\partial F$ in Hausdorff distance in $Z$, then
$\overline{E}_{i}\to\overline{F}$ in Hausdorff distance in $Z$ and $f_{i}\to
f$ locally uniformly in $Z$.

  2. (2)

If $E_{i}$’s and $F$ are open and bounded, $\overline{E}_{i}\to\overline{F}$
in Hausdorff distance in $Z$, and $f_{i}$ satisfies (3.2) for some barrier
$c_{i}$ for any $i$ and $c_{i}\to c$, then $f$ satisfies (3.2) with barrier
$c$.

###### Proof.

For simplicity, we identify in this proof the spaces $X_{i},Y$ and balls
$B_{r}(x_{i}),B_{r}(y)$ with their image in $Z$ via the maps $\Psi_{i},\Psi$
given by subsection 2.1, denoting by $B^{Z}_{r}(z)$ open balls in $Z$. Hence,
by hypotheses, we notice that
$\overline{B}_{r}(y)\cup\bigcup_{i}\overline{B}_{r}(x_{i})$ is compact in $Z$
for any $r>0$. For $A\subset Z$, we denote by $(A)_{\varepsilon}$ the open
tubular neighbourhoods in $Z$. We prove the items separately.

  1. (1)

Let $R>0$ be such that $F\subset B_{R}(y)$. Let $y_{i}\in E_{i}$ be any
sequence and define $r_{i}\coloneqq\mathsf{d}_{i}(y_{i},\partial E_{i})$. If
$\lim_{i}r_{i}=0$, then $y_{i}\in(\partial F)_{\varepsilon}$ for some
$\varepsilon>0$ and any $i$ sufficiently large. If instead
$r_{i_{k}}\geq\overline{r}>0$ for some subsequence, then
$\limsup_{k}\mathsf{d}_{i_{k}}(y_{i_{k}},x_{i_{k}})\leq R+2$. Otherwise, if
$\overline{\rho}\coloneqq\min\\{1,\overline{r}\\}$, then

$\begin{split}0&=\mathscr{H}^{N}(F\setminus
B_{R}(y))\geq\limsup_{k}\mathscr{H}^{N}(E_{i_{k}}\setminus
B_{R}(x_{i_{k}}))\\\ &\geq\limsup_{k}\mathscr{H}^{N}(E_{i_{k}}\cap
B_{\overline{\rho}}(y_{i_{k}}))=\limsup_{k}\mathscr{H}^{N}(B_{\overline{\rho}}(y_{i_{k}}))\geq
C(\overline{\rho},K,N,v_{0})>0.\end{split}$

Hence, up to taking a larger $R$, we may assume that $E_{i}\subset
B_{R}(x_{i})$ for any $i$. Since
$\overline{B}_{R}(y)\cup\bigcup_{i}\overline{B}_{R}(x_{i})$ is compact in $Z$,
if we show that $\overline{E}_{i}\to\overline{F}$ in the Kuratowski sense,
then convergence also holds in Hausdorff distance in $Z$. By convergence in
$L^{1}$-strong, it is clear that if $w\in\overline{F}$, then there is a
sequence of points in $\overline{E}_{i}$ converging to $w$. On the other hand,
if $y_{i_{k}}\in E_{i_{k}}$ converges to some $w$ in $Z$, if by contradiction
$w\not\in\overline{F}$, then $\mathscr{H}^{N}(E_{i_{k}}\cap
B_{r}(y_{i_{k}}))\to 0$ for some $r>0$. But then we can find a sequence
$r_{k}\searrow 0$ and a sequence of points $z_{i_{k}}\in\partial E_{i_{k}}\cap
B_{r_{k}}(y_{i_{k}})$. Hence $z_{i_{k}}\to w$ in $Z$, and then $w\in\partial
F$, which gives a contradiction. Finally Hausdorff convergence immediately
implies that $f_{i}\to f$ locally uniformly.

  2. (2)

Fix a non negative function $\varphi\in\operatorname{LIP}_{c}(B_{D}(w))$,
where $B_{4D}(w)\Subset Y\setminus\overline{F}$. Then, for large $i$, there
exist a sequence of balls $B_{3D}(w_{i})\Subset
X_{i}\setminus\overline{E}_{i}$ and non negative functions
$\varphi_{i}\in\operatorname{LIP}_{c}(B_{2D}(w_{i}))$ such that
$\varphi_{i}\to\varphi$ strongly in $H^{1,2}$. Indeed one can construct
$\varphi_{i}$ by first taking a sequence of non negative functions
$\widetilde{\varphi}_{i}\in H^{1,2}(X_{i})$ converging to $\varphi$ in
$H^{1,2}$-strong, see [12, Theorem 8.1], and a sequence of cut-off functions
$\eta_{i}\in\operatorname{LIP}_{c}(B_{2D}(w_{i}))$ such that $\eta_{i}\equiv
1$ on $B_{D}(w_{i})$. By density of $\operatorname{LIP}_{c}(X_{i})$ in
$H^{1,2}(X_{i})$, by a diagonal argument we may assume that
$\widetilde{\varphi}_{i}\in\operatorname{LIP}_{c}(X_{i})$; hence the desired
sequence is given by the product
$\varphi_{i}\coloneqq\eta_{i}\widetilde{\varphi}_{i}$. By assumption we know
that there hold the non-sharp adimensional bounds

$-\int\nabla
f_{i}\cdot\nabla\varphi_{i}\,\mathrm{d}\mathscr{H}^{N}\leq\int(c_{i}-Kf_{i})\varphi_{i}\,\mathrm{d}\mathscr{H}^{N}.$
(4.53)

Observe that $f_{i}$ belongs to the local Sobolev space
$H^{1,2}(B_{r}(w_{i}))$ for any $r>0$. Hence, recalling that $f_{i}\to f$
locally uniformly, since also $|\nabla f_{i}|=1$ and $|\nabla f|=1$ almost
everywhere, then $f_{i}\to f$ (locally) in $H^{1,2}$-strong. Therefore
$\int\nabla
f_{i}\cdot\nabla\varphi_{i}\,\mathrm{d}\mathscr{H}^{N}\to\int\nabla
f\cdot\nabla\varphi\,\mathrm{d}\mathscr{H}^{N}$ by [12, Theorem 5.7]. On the
other hand, also the right hand side of (4.53) passes to the limit. By
arbitrariness of $\varphi\in\operatorname{LIP}_{c}(B_{D}(w))$ and balls
$B_{D}(w)\Subset Y\setminus\overline{F}$, we conclude that $f$ satisfies the
adimensional bound $\Delta f\leq c-Kf$ on $Y\setminus\overline{F}$. Eventually
the inequality can be improved arguing as in Step 2 of the proof of Theorem
3.2 and then $f$ satisfies (3.2) with barrier $c$. An analogous argument
clearly yields the desired inequality also on $F$.

∎

The next Theorem 4.3 yields the aforementioned convergence in Hausdorff
distance of a sequence of isoperimetric sets $E_{i}$, and it crucially
exploits the previously discussed fine properties on the isoperimetric
profile. Observe that no uniform hypotheses on the mean curvature barriers for
the $E_{i}$’s are assumed; instead a uniform bound on such barriers is derived
exploiting the uniform properties we proved on the isoperimetric profile.

######  Theorem 4.3.

Let $(X_{i},\mathsf{d}_{i},\mathscr{H}^{N},x_{i})$ be a sequence of
$\operatorname{RCD}(K,N)$ spaces converging to
$(Y,\mathsf{d}_{Y},\mathscr{H}^{N},y)$ in pmGH sense, and let
$(Z,\mathsf{d}_{Z})$ be a space realizing the convergence. Assume that
$\mathscr{H}^{N}(B_{1}(p))\geq v_{0}>0$ for any $p\in X_{i}$ and any $i$. Let
$E_{i}\subset X_{i}$, $F\subset Y$.

If $E_{i}$ is isoperimetric, $E_{i}\subset B_{R}(x_{i})$ for some $R>0$ for
any $i$, $c_{i}$ is a mean curvature barrier for $E_{i}$ for any $i$,
$E_{i}\to F$ in $L^{1}$-strong, and
$0<\lim_{i}\mathscr{H}^{N}(E_{i})<\lim_{i}\mathscr{H}^{N}(X_{i})$, then

$\begin{split}\text{$F$ is isoperimetric}\,,&\\\ |c_{i}|\leq L&\qquad\text{for
any $i$ large enough}\,,\\\ |D\chi_{E_{i}}|\to|D\chi_{F}|&\qquad\text{in
duality with $C_{\rm bs}(Z)$}\,,\\\ \partial E_{i}\to\partial
F\,,\,\,\overline{E}_{i}\to\overline{F}&\qquad\text{in Hausdorff distance in
$Z$}\,,\end{split}$ (4.54)

where
$L=L(K,N,v_{0},\lim_{i}\mathscr{H}^{N}(E_{i}),\lim_{i}\mathscr{H}^{N}(X_{i}))>0$.
In particular, the mean curvature barriers $c_{i}$ converge up to subsequence
to a mean curvature barrier for $F$.

###### Proof.

As in the proof of subsection 4.4, we identify in this proof the spaces
$X_{i},Y$ and balls $B_{r}(x_{i}),B_{r}(y)$ with their images in $Z$ via the
maps $\Psi_{i},\Psi$ given by subsection 2.1. Hence, by hypotheses, we notice
that $\overline{B}_{r}(y)\cup\bigcup_{i}\overline{B}_{r}(x_{i})$ is compact in
$Z$ for any $r>0$.

By a classical contradiction argument as in [17, 18], it follows that
$\operatorname{Per}(E_{i})\to\operatorname{Per}(F)$ and $F$ is isoperimetric.
In particular, $|D\chi_{E_{i}}|\to|D\chi_{F}|$ in duality with $C_{\rm
bs}(Z)$.

Let $V_{1}\vcentcolon=\lim_{i}\mathscr{H}^{N}(E_{i})/2$. If
$\lim_{i}\mathscr{H}^{N}(X_{i})<+\infty$, let

$V_{2}\vcentcolon=(2\lim_{i}\mathscr{H}^{N}(E_{i})+\lim_{i}\mathscr{H}^{N}(X_{i}))/3,$

and
$V_{3}\vcentcolon=(\lim_{i}\mathscr{H}^{N}(E_{i})+2\lim_{i}\mathscr{H}^{N}(X_{i}))/3$,
otherwise let $V_{2}\vcentcolon=2\lim_{i}\mathscr{H}^{N}(E_{i})$ and
$V_{3}\vcentcolon=3\lim_{i}\mathscr{H}^{N}(E_{i})$. In any case we have that
$0<V_{1}\leq\mathscr{H}^{N}(E_{i})\leq V_{2}<V_{3}<\mathscr{H}^{N}(X_{i})$ for
large $i$.

In order to get the desired Hausdorff convergence, by compactness of
$\overline{B}_{R}(y)\cup\bigcup_{i}\overline{B}_{R}(x_{i})$ and by subsection
4.4, it is sufficient to prove convergence of $\partial E_{i}$ to $\partial F$
in the sense of Kuratowski. For large $i$, by applying subsection 4.2 with
$V_{1},V_{2},V_{3}$ chosen as before, we have that $E_{i}$ satisfies uniform
density estimates at boundary points, independently on $i$. This readily
implies that any converging sequence $q_{i}\in\partial E_{i}$ must converge to
a point in $\partial F$ (compare with [73, Theorem 2.43]), and then Kuratowski
convergence of $\partial E_{i}$ to $\partial F$ is achieved.

Finally, by subsection 4.2, we know that $I_{X_{i}}$ is $L$-Lipschitz on
$[V_{1},V_{2}]$ for any $i$, with $L=L(K,N,v_{0},V_{1},V_{2})$. Combining with
subsection 4.3, we get that any mean curvature barrier $c_{i}$ for $E_{i}$
satisfies $|c_{i}|\leq L$ for any $i$ such that
$V_{1}\leq\mathscr{H}^{N}(E_{i})\leq V_{2}$. The subconvergence of $c_{i}$ to
a mean curvature barrier for $F$ follows from subsection 4.4. ∎

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