Inscribed Radius Bounds for Lower Ricci Bounded Metric Measure Spaces

Inscribed Radius Bounds for Lower Ricci Bounded
Metric Measure Spaces with Mean Convex Boundary††This paper is a contribution
to the Special Issue on Scalar and Ricci Curvature in honor of Misha Gromov on
his 75th Birthday. The full collection is available at
https://www.emis.de/journals/SIGMA/Gromov.html

Annegret BURTSCHER †, Christian KETTERER ‡, Robert J. MCCANN ‡
and Eric WOOLGAR §

A. Burtscher, C. Ketterer, R.J. McCann and E. Woolgar

† Department of Mathematics, IMAPP, Radboud University,
† PO Box 9010, Postvak 59, 6500 GL Nijmegen, The Netherlands
<EMAIL_ADDRESS>
‡ Department of Mathematics, University of Toronto,
‡ 40 St George St, Toronto Ontario, Canada M5S 2E4<EMAIL_ADDRESS><EMAIL_ADDRESS>
§ Department of Mathematical and Statistical Sciences and Theoretical Physics
Institute,
§ University of Alberta, Edmonton AB, Canada T6G 2G1<EMAIL_ADDRESS>
Received June 03, 2020, in final form November 21, 2020; Published online
December 10, 2020

Consider an essentially nonbranching metric measure space with the measure
contraction property of Ohta and Sturm, or with a Ricci curvature lower bound
in the sense of Lott, Sturm and Villani. We prove a sharp upper bound on the
inscribed radius of any subset whose boundary has a suitably signed lower
bound on its generalized mean curvature. This provides a nonsmooth analog to a
result of Kasue (1983) and Li (2014). We prove a stability statement
concerning such bounds and – in the Riemannian curvature-dimension
$(\mathrm{RCD})$ setting – characterize the cases of equality.

curvature-dimension condition; synthetic mean curvature; optimal transport;
comparison geometry; diameter bounds; singularity theorems; inscribed radius;
inradius bounds; rigidity; measure contraction property

51K10; 53C21; 30L99; 83C75

##  1 Introduction

Kasue proved a sharp estimate for the inscribed radius (or inradius, denoted
$\operatorname{InRad}$) of a smooth, $n$-dimensional Riemannian manifold $M$
with nonnegative Ricci curvature and smooth boundary $\partial M$ whose mean
curvature is bounded from below by $n-1$. More precisely, he concluded
$\operatorname{InRad}_{M}\leq 1$ [30]. This result was also rediscovered by Li
[40] and extended to weighted Riemannian manifolds with Bakry–Émery curvature
bounds by Li–Wei [38, 39] and Sakurai [48]. Their result can be seen either as
a manifold-with-boundary analog of Bonnet and Myers’ diameter bound, or as a
Riemannian analog of the Hawking singularity theorem from general relativity
[27] (for the precise statement see [44, Theorem 6.49]). There has been
considerable interest in generalizing Hawking’s result to a nonsmooth setting
[26, 37, 42]. Motivated in part by this goal, we give a generalization of
Kasue’s result which is interesting in itself and can serve as a model for the
Lorentzian case. Independently and simultaneously, Cavalletti and Mondino have
proposed a synthetic new framework for Lorentzian geometry (also under
investigation by one of us independently [43]) in which they establish an
analog of the Hawking result [13].

In this note we generalize Kasue and Li’s estimate to subsets $\Omega$ of a
(potentially nonsmooth) space $X$ satisfying a curvature dimension condition
$\mathrm{CD}(K,N)$ with $K\in\mathbb{R}$ and $N>1$, provided the topological
boundary $\partial\Omega$ has a lower bound on its inner mean curvature in the
sense of [34]. The notion of inner mean curvature in [34] is defined by means
of the $1D$-localisation (needle decomposition) technique of Cavalletti and
Mondino [11] and coincides with the classical mean curvature of a hypersurface
in the smooth context. We also assume that the boundary $\partial\Omega$
satisfies a measure theoretic regularity condition that is implied by an
exterior ball condition. Hence, our result not only covers Kasue’s theorem but
also holds for a large class of domains in Alexandrov spaces or in Finsler
manifolds. Kasue (and Li) were also able to prove a rigidity result analogous
to Cheng’s theorem [15] from the Bonnet–Myers context: namely that, among
smooth manifolds, their inscribed radius bound is obtained precisely by the
Euclidean unit ball. In the nonsmooth case there are also truncated cones that
attain the maximal inradius; under an additional hypothesis known as
$\mathrm{RCD}$, we prove that these are the only nonsmooth optimizers provided
$\Omega$ is compact and its interior is connected.

To state our results first we recall the following definition. For
$\kappa\in\mathbb{R}$ we define $\cos_{\kappa}\colon\R\rightarrow\mathbb{R}$
as the solution of

$\displaystyle v^{\prime\prime}+\kappa v=0,\qquad\text{with}\quad
v(0)=1\qquad\text{and}\qquad v^{\prime}(0)=0.$ (1.1)

The function $\sin_{\kappa}\colon\R\rightarrow\mathbb{R}$ is defined as
solution of the same ordinary differential equation (ODE) with initial values
$v(0)=0$ and $v^{\prime}(0)=1$. We define

$\displaystyle\pi_{\kappa}:=\begin{cases}\dfrac{\pi}{\sqrt{\kappa}}&\mbox{if
}\kappa>0,\\\ \infty&\mbox{otherwise},\end{cases}$ (1.2)

and $\tilde{I}_{\kappa}:=\overline{[0,\pi_{\kappa})}$. Let $K,H\in\mathbb{R}$
and $N>1$. The Jacobian function is

$\displaystyle r\in\R\mapsto
J_{K,H,N}(r):=\left(\cos_{K/{(N-1)}}(r)-\frac{H}{N-1}\sin_{K/(N-1)}(r)\right)_{+}^{N-1},$
(1.3)

where $(a)_{+}:=\max\\{a,0\\}$ for $a\in\R$. Since
$J_{K,H,N}(r)=J_{K,-H,N}(-r)$, its interval of positivity around the origin is
given by $r\in(-r_{K,-H,N},r_{K,H,N})$, where

$\displaystyle r_{K,H,N}:=\inf\\{r\in(0,\infty)\colon J_{K,H,N}(r)=0\\}.$
(1.4)

In [30] and [48] the authors define

$\displaystyle s_{\kappa,\lambda}(r)=\cos_{\kappa}(r)-\lambda\sin_{\kappa}(r)$
(1.5)

for $\kappa,\lambda\in\R$. They say the pair $(\kappa,\lambda)$ satisfies the
ball condition if the equation $s_{\kappa,\lambda}(r)=0$ has a positive
solution. The latter happens if and only if one of the following three cases
holds: (1) $\kappa>0$ and $\lambda\in\R$, (2) $\kappa=0$ and $\lambda>0$ or
(3) $\kappa\leq 0$ and $\lambda>\sqrt{|\kappa|}$. If
$(\kappa,\lambda)=\big{(}\frac{K}{N-1},\frac{H}{N-1}\big{)}$, then $r_{K,H,N}$
coincides with the smallest positive zero of $s_{\kappa,\lambda}$ if any
exists; moreover $s_{\kappa,\lambda}(r)<0$ for all $r>r_{K,H,N}$ if
$\kappa\leq 0$, while $s_{\kappa,\lambda}$ oscillates sinusoidally with mean
zero and period greater than $2r_{K,H,N}$ if $\kappa>0$. In particular,
$r_{K,H,N}<\infty$ if and only if $\big{(}\frac{K}{N-1},\frac{H}{N-1}\big{)}$
satisfies the ball-condition.

For $\Omega\subset X$, letting $\Omega^{c}:=X\setminus\Omega$, our main
theorem reads as follows:

######  Theorem 1.1 (inscribed radius bounds for metric measure spaces).

Let $(X,d,\operatorname{m})$ be an essentially nonbranching
$\mathrm{CD}(K^{\prime},N)$ space with $K^{\prime}\in\R$, $N\in(1,\infty)$ and
$\operatorname{{spt}}\operatorname{m}=X$. Let $K,\chi\in\R$ such that
$\big{(}\frac{K}{N-1},\chi\big{)}$ satisfies the ball condition. Let
$\Omega\subset X$ be closed with $\Omega\neq X$, $\operatorname{m}(\Omega)>0$
and $\operatorname{m}(\partial\Omega)=0$ such that $\Omega$ satisfies the
restricted curvature-dimension condition $\mathrm{CD}_{r}(K,N)$ for $K\in\R$
$($Definition 2.3) and $\partial\Omega=S$ has finite inner curvature
$($Definition 2.17). Assume the inner mean curvature $H_{S}^{-}$ satisfies
$H^{-}_{S}\geq\chi(N-1)$ $\operatorname{m}_{S}$-a.e. where
$\operatorname{m}_{S}$ denotes the surface measure $($Definition 2.15). Then

$\displaystyle\operatorname{InRad}\Omega\leq r_{K,\chi(N-1),N},$ (1.6)

where $\operatorname{InRad}\Omega=\sup\limits_{x\in\Omega}d_{\Omega^{c}}(x)$
is the inscribed radius of $\Omega$.

The theorem generalizes previous results for Riemannian manifolds [30, 40] and
weighted Riemannian manifolds [38, 39, 48]. Moreover Theorem 1.1 also holds in
the context of weighted Finsler manifolds and Alexandrov spaces and seems to
be new in this context.

We also show:

######  Theorem 1.2 (stability).

Consider $(X,d,\operatorname{m})$ and $\Omega\subset X$ as in the previous
theorem. Then, for every $\epsilon>0$ there exists $\delta>0$ such that

$\displaystyle\operatorname{InRad}\Omega\leq
r_{\bar{K},\bar{H},\bar{N}}+\epsilon$

provided $K\geq\bar{K}-\delta$, $H^{-}_{S}\geq\bar{H}-\delta$
$\operatorname{m}_{S}$-a.e. and $N\leq\bar{N}+\delta$ for
$\bar{K},\bar{H}\in\R$ and $\bar{N}\in(1,\infty)$.

######  Remark 1.3 (definitions and improvements).

Let us comment on the definitions in Theorem 1.1 and generalizations.

  1. 1.

The curvature-dimension conditions $\mathrm{CD}(K,N)$ and the restricted
curvature-dimension condition $\mathrm{CD}_{r}(K,N)$ for an essentially
nonbranching metric measure space $(X,d,\operatorname{m})$ are defined in
Definition 2.3. If $(X,d,\operatorname{m})$ satisfies the condition
$\mathrm{CD}(K,N)$ then $\Omega\neq\varnothing$ trivially satisfies
$\mathrm{CD}_{r}(K,N)$ for the same $K$. For this we note that for essentially
nonbranching $\mathrm{CD}(K,N)$ spaces, $L^{2}$-Wasserstein geodesics between
$\operatorname{m}$-absolutely continuous probability measures are unique [10].

  2. 2.

Appendix A extends the conclusions of Theorems 1.1 and 1.2 to the case where
the $\mathrm{CD}(K,N)$ hypothesis is replaced by the measure contraction
property $\mathrm{MCP}(K,N)$ proposed in [46, 49], still under the essentially
nonbranching hypothesis.

  3. 3.

The backward mean curvature bound introduced in Appendix B also suffices for
the conclusion of the above theorems, provided the finiteness assumed of the
inner curvature of $\partial\Omega=S$ is replaced by the requirement that the
surface measure $\operatorname{m}_{S_{0}}$ be Radon. This alternate framework
also suffices for the rigidity result of Theorem 1.4 below. It is related to
but distinct from a notion presented in [13].

  4. 4.

The property “having finite inner curvature” (Definition 2.17) rules out
inward pointing cusps and cones, and is implied by an exterior ball condition
for $\Omega$ (Lemma 2.21). The surface measure $\operatorname{m}_{S}$ is
defined in Definition 2.15.

  5. 5.

For $S$ with finite inner curvature, the definition of generalized inner mean
curvature $H^{-}_{S}$ is given in Definition 2.17. Let us briefly sketch the
idea. Using a needle decomposition associated to the signed distance function
$d_{S}:=d_{\Omega}-d_{{\Omega}^{c}}$, one can disintegrate the reference
measure $\operatorname{m}$ into needles, meaning into conditional measures
$\\{\operatorname{m}_{\alpha}\\}_{\alpha\in Q}$ (for a quotient space $Q$)
that are supported on curves $\gamma_{\alpha}$ of maximal slope of $d_{S}$.
For $\mathfrak{q}$-almost every curve $\gamma_{\alpha}$ with respect to the
quotient measure $\mathfrak{q}$ of $\operatorname{m}$ on $Q$, there exists a
conditional density $h_{\alpha}$ of $\operatorname{m}_{\alpha}$ with respect
to the $1$-dimensional Hausdorff measure $\mathcal{H}^{1}$. Then the inner
mean curvature for $\operatorname{m}_{S}$-a.e. $p=\gamma_{\alpha}(t_{0})\in S$
is defined as $\frac{{\rm d}^{-}}{{\rm d}t}\log
h_{\alpha}(t_{0})=H_{S}^{-}(p)$. This left derivative quantifies the extent to
which a given collection of needles are spreading (i.e., capturing more
measure) as they exit $\Omega$. We postpone details to the Sections 2.3 and
2.4. In the case $(X,d,\operatorname{m})=(M,d_{g},\operatorname{vol}_{g})$ for
a Riemannian manifold $(M,g)$ and $\partial\Omega$ is a hypersurface the inner
mean curvature coincides with the classical mean curvature.

  6. 6.

Our assumptions cover the case of a Riemannian manifold with boundary: If
$(X,d,\operatorname{m})=(M,d_{g},\operatorname{vol}_{g})$ for a
$n$-dimensional Riemannian manifold $(M,g)$ with boundary and Ricci lower
bound $\operatorname{ric}_{M}\geq K$, then one can always construct a
geodesically convex, $n$-dimensional Riemannian manifold $\tilde{M}$ with
boundary such that $M$ isometrically embeds into $\tilde{M}$, and such that
$\operatorname{ric}_{\tilde{M}}\geq K^{\prime}$ [51]. In particular, one can
consider $M$ as a $\mathrm{CD}_{r}(K,n)$ space that is a subset of the
$\mathrm{CD}(K^{\prime},n)$ space
$\big{(}\tilde{M},d_{\tilde{M}},\operatorname{vol}_{\tilde{M}}\big{)}$ (Remark
5.8 in [34]).

###  1.1 Cones and spherical suspension

For smooth, $n$-dimensional Riemannian manifolds with non-negative Ricci
curvature and boundary that has mean curvature bounded from below by $n-1$,
equality in the inradius estimate is obtained precisely by the Euclidean unit
ball (see [30, 39, 40]). In the nonsmooth case, truncated cones also attain
the maximal inradius.

Let $(X,d,\operatorname{m})$ be a metric measure space.

  1. 1.

The Euclidean $N$-cone over $(X,d,\operatorname{m})$ is defined as the metric
measure space

$\displaystyle\big{(}[0,\infty)\times X/\sim,d_{\rm
Eucl},\operatorname{m}^{N}_{\rm
Eucl}\big{)}=:[0,\infty)\times^{N}_{\operatorname{id}}X,$

where the equivalence relation $\sim$ is defined by $(s,x)\sim(t,y)$ if and
only if either $s=t=0$ or $(s,x)=(t,y)$. The tip of the cone is denoted by
$o$. The distance $d_{\rm Eucl}$ is defined by

$\displaystyle d_{\rm
Eucl}^{2}((t,x),(s,y)):=t^{2}+s^{2}-2ts\cos[d(x,y)\wedge\pi],$

where $a\wedge b:=\min\\{a,b\\}$, and the measure $\operatorname{m}^{N}_{\rm
Eucl}$ is given by $r^{N}{\rm d}r\otimes\operatorname{dm}$.

If an Euclidean $N$-cone $[0,\infty)\times^{N}_{\operatorname{id}}X$ is a
manifold then $X=\mathbb{S}^{n}$, $N=n\in\N$ and
$[0,\infty)\times^{n}_{\operatorname{id}}\mathbb{S}^{n}$ is isometric to n+1.

  2. 2.

The hyperbolic $N$-cone is defined similarly:

$\displaystyle\big{(}[0,\infty)\times X/\sim,d_{\rm
Hyp},\operatorname{m}^{N}_{\rm Hyp}\big{)}=:[0,\infty)\times_{\sinh}^{N}X.$

The distance $d_{\rm Hyp}$ is defined by

$\displaystyle\cosh d_{\rm Hyp}((t,x),(s,y)):=\cosh t\cosh s-\sinh t\sinh
s\cos[d(x,y)\wedge\pi],$

and the measure $\operatorname{m}^{N}_{\rm Hyp}$ is given by $\sinh^{N}r\,{\rm
d}r\otimes\operatorname{dm}$.

If a hyperbolic $N$-cone $[0,\infty)\times_{\sinh}^{N}X$ is a manifold then
$X=\mathbb{S}^{n}$, $N=n$ and $[0,\infty)\times_{\sinh}^{n}\mathbb{S}^{n}$ is
isometric to the $n$-dimensional hyperbolic plane.

  3. 3.

Similar, the spherical $N$-suspension over $(X,d,\operatorname{m})$ is defined
as the metric space

$\displaystyle\big{(}[0,\pi]\times X/\sim,d_{\rm
Susp},\operatorname{m}^{N}_{\rm Susp}\big{)}=:[0,\pi]\times_{\sin}^{N}X,$

where the equivalence relation $\sim$ is now defined by $(s,x)\sim(t,y)$ if
and only if either $s=t\in\\{0,\pi\\}$ or $(s,x)=(t,y)$. The distance $d_{\rm
Susp}$ is defined by

$\displaystyle\cos d_{\rm Susp}((t,x),(s,y)):=\cos t\cos s+\sin t\sin
s\cos[d(x,y)\wedge\pi],$

and the measure $\operatorname{m}_{\rm Susp}^{N}$ is given by $\sin^{N}t\,{\rm
d}t\otimes\operatorname{dm}$.

If a sphercial $N$-cone $[0,\pi]\times_{\sin}^{N}X$ is a manifold then
$X=\mathbb{S}^{n}$, $N=n\in\N$ and $[0,\pi]\times_{\sin}^{n}\mathbb{S}^{n}$ is
isometric to the $(n+1)$-dimensional standard sphere $\mathbb{S}^{n+1}$.

In each case $0$ may also be used to denote the equivalence class of the
points $(0,x)$.

The next result shows that in an appropriate setting, cones and suspensions
are the only maximizers of our inscribed radius bound. This requires the
Riemannian curvature-dimension condition $\mathrm{RCD}(K,N)$ (Definition 4.1),
a strengthening of the curvature-dimension condition that rules out Finsler
manifolds and yields isometric rigidity theorems for metric measure spaces.
This condition is crucial in the proof of the next theorem, since it permits
us to exploit the volume cone rigidity theorem by DePhilippis and Gigli [17].
For $K>0$ one can view Theorem 1.4 as a version of the maximal diameter
theorem [32] adapted to mean convex subsets of $\mathrm{RCD}$ spaces. For
rigidity results pertaining to other inequalities in nonsmooth or even
discrete settings, see recent work of Ketterer [33], Nakajima, Shioya [45] and
Cushing et al. [16].

######  Theorem 1.4 (rigidity).

Let $(X,d,\operatorname{m})$ be $\mathrm{RCD}(K,N)$ for $K\in\R$ and
$N\in(1,\infty)$ and let $\Omega\subset X$ be closed with $\Omega\neq X$,
$\operatorname{m}(\Omega)>0$, connected and non-empty interior
$\Omega^{\circ}$, and $\operatorname{m}(\partial\Omega)=0$. We assume that
$K\in\\{N-1,0,-(N-1)\\}$, $\partial\Omega=S$ has finite inner curvature,
$S\neq\\{pt\\}$, and the inner mean curvature $\operatorname{m}_{S}$-a.e.
satisfies $H_{S}^{-}\geq\chi(N-1)\in\R$. Then, there exists $x\in X$ such that

$d_{\Omega^{c}}(x)=\operatorname{InRad}\Omega=r_{K,\chi(N-1),N}$

if and only if $r_{K,\chi(N-1),N}<\infty$ and there exists an
$\mathrm{RCD}(N-2,N-1)$ space $Y$ such that $\Omega^{\circ}$ becomes isometric
to the ball $B_{r_{K,\chi(N-1),N}}(0)$ of radius $r_{K,\chi(N-1),N}$ around
the cone tip in $\tilde{I}_{\frac{K}{N-1}}\times_{\sin_{K/(N-1)}}^{N-1}Y$,
when each is equipped with the induced intrinsic distance which it inherits
from its ambient space. Here
$\tilde{I}_{\frac{K}{N-1}}:=\overline{\big{[}0,\pi_{\frac{K}{N-1}}\big{)}}$.

The theorem generalizes corresponding rigidity results for Riemannian
manifolds [30, 40] and weighted Riemannian manifolds [38, 39, 48].

######  Remark 1.5 (easy direction).

In the above rigidity theorem one direction is obvious. Let us explore this
just for the case $K=0$ and $\chi=1$.

Let $Y$ be an $\mathrm{RCD}(N-2,N-1)$ space. Both the Euclidean $N$-cone
$X=[0,\infty)\times^{N}_{\operatorname{id}}Y$ and its truncation
$\Omega=[0,1]\times^{N}_{\operatorname{id}}Y$ are geodesically convex and
satisfy $\mathrm{RCD}(0,N)$ [31]. The distance function $d_{\Omega^{c}}$ from
$(X,d,\operatorname{m})$ restricted to $\Omega$ is given by
$d_{\Omega^{c}}(t,x)=1-t$. In particular,
$r_{0,N-1,N}=d_{\Omega^{c}}((0,x))=d_{\Omega^{c}}(o)=1$.

Moreover, $S$ has (inner) mean curvature equal to $N-1$ in the sense of
Definition 2.17 in $X=[0,\infty)\times^{N-1}_{\operatorname{id}}Y$. Indeed, we
can see that points $(s,x)$ and $(t,y)$ in $\Omega$ lie on the same needle if
and only if either $x=y$ or $st=0$. Hence, the needles in $\Omega$ for the
corresponding $1D$-localization are $t\in(0,1)\mapsto\gamma(t)=(1-t,x)$, $x\in
Y$. One can also easily check that $h_{\gamma}^{{1}/{N-1}}(t)=t$ for all
needles $\gamma$ in the corresponding disintegration of
$\operatorname{m}|_{\Omega}$. Hence $H_{\partial\Omega}^{-}\equiv N-1$.

######  Remark 1.6.

We may allow $S=\\{pt\\}$ in Theorem 1.4 if we adopt the convention that
$Y=\\{pt\\}$ satisfies $\mathrm{RCD}(N-2,N-1)$ for all $N>1$. This would
correspond to case (1) of De Philippis and Gigli’s classification, Theorem 4.2
below.

##  2 Preliminaries

###  2.1 Curvature-dimension condition

Let $(X,d)$ be a complete and separable metric space and let
$\operatorname{m}$ be a locally finite Borel measure. We call
$(X,d,\operatorname{m})$ a metric measure space. We always assume
$\operatorname{{spt}}\operatorname{m}=X$ and $X\neq\\{pt\\}$.

The length of a continuous curve $\gamma\colon[a,b]\rightarrow X$ is
$L(\gamma)=\sup\big{\\{}\sum_{i=1}^{k-1}d(\gamma(t_{i}),\gamma(t_{i+1}))\big{\\}}\in[0,\infty]$
where the supremum is w.r.t. any subdivision of $[a,b]$ given by $a=t_{1}\leq
t_{2}\leq\dots\leq t_{k-1}\leq t_{k}=b$ and $k\in\mathbb{N}$. Obviously
$L(\gamma)\geq d(\gamma(a),\gamma(b))$; a geodesic refers to any continuous
curve $\gamma\colon[a,b]\rightarrow X$ saturating this bound. We denote the
set of constant speed geodesics $\gamma\colon[0,1]\rightarrow X$ with
$\mathcal{G}(X)$; these are characterized by the identity

$d(\gamma_{s},\gamma_{t})=(t-s)d(\gamma_{0},\gamma_{1})$

for all $0\leq s\leq t\leq 1$. For $t\in[0,1]$ let
$e_{t}\colon\gamma\in\mathcal{G}(X)\mapsto\gamma(t)$ be the evaluation map. A
subset of geodesics $F\subset\mathcal{G}(X)$ is said to be nonbranching if for
any two geodesics $\gamma,\hat{\gamma}\in F$ such that there exists
$\epsilon\in(0,1)$ with $\gamma|_{(0,\epsilon)}=\hat{\gamma}|_{(0,\epsilon)}$,
it follows that $\gamma=\hat{\gamma}$.

######  Example 2.1 (Euclidean geodesics).

When $X\subset{}^{n}$ is convex and $d(x,y)=|x-y|$ then $\mathcal{G}(X)$
consists of the affine maps $\gamma\colon[0,1]\to X$.

The set of (Borel) probability measures on $(X,d,\operatorname{m})$ is denoted
with $\mathcal{P}(X)$, the subset of probability measures with finite second
moment is $\mathcal{P}^{2}(X)$, the set of probability measures in
$\mathcal{P}^{2}(X)$ that are $\operatorname{m}$-absolutely continuous is
denoted with $\mathcal{P}^{2}(X,\operatorname{m})$ and the subset of measures
in $\mathcal{P}^{2}(X,\operatorname{m})$ with bounded support is denoted with
$\mathcal{P}_{b}^{2}(X,\operatorname{m})$.

The space $\mathcal{P}^{2}(X)$ is equipped with the $L^{2}$-Wasserstein
distance $W_{2}$, e.g., [50]. A dynamical optimal coupling is a probability
measure $\Pi\in\mathcal{P}(\mathcal{G}(X))$ such that
$t\in[0,1]\mapsto(e_{t})_{\\#}\Pi$ is a $W_{2}$-geodesic in
$\mathcal{P}^{2}(X)$ where $(e_{t})_{\\#}\Pi$ denotes the push-forward under
the map $\gamma\mapsto e_{t}(\gamma):=\gamma(t)$. The set of dynamical optimal
couplings $\Pi\in\mathcal{P}(\mathcal{G}(X))$ between
$\mu_{0},\mu_{1}\in\mathcal{P}^{2}(X)$ is denoted with
$\operatorname{OptGeo}(\mu_{0},\mu_{1})$.

A metric measure space $(X,d,\operatorname{m})$ is called essentially
nonbranching if for any pair
$\mu_{0},\mu_{1}\in\mathcal{P}^{2}(X,\operatorname{m})$ any
$\Pi\in\operatorname{OptGeo}(\mu_{0},\mu_{1})$ is concentrated on a set of
nonbranching geodesics.

######  Definition 2.2 (distortion coefficients).

For $K\in\mathbb{R}$, $N\in(0,\infty)$ and $\theta\geq 0$ we define the
distortion coefficient as

$\displaystyle
t\in[0,1]\mapsto\sigma_{K,N}^{(t)}(\theta):=\begin{cases}\dfrac{\sin_{K/N}(t\theta)}{\sin_{K/N}(\theta)}\
&\mbox{if }\theta\in[0,\pi_{K/N}),\\\ \infty\ &\mbox{otherwise},\end{cases}$

where $\pi_{\chi}:=\infty\ \textrm{if }\chi\leq 0$ and
$\pi_{\chi}:=\frac{\pi}{\sqrt{\chi}}\ \textrm{if }\chi>0$. Here $\sin_{K/N}$
was defined after (1.1), and $\sigma_{K,N}^{(t)}(0)=t$. Moreover, for
$K\in\mathbb{R}$, $N\in[1,\infty)$ and $\theta\geq 0$ the modified distortion
coefficient is defined as

$\displaystyle
t\in[0,1]\mapsto\tau_{K,N}^{(t)}(\theta):=t^{\frac{1}{N}}\big{[}\sigma_{K,N-1}^{(t)}(\theta)\big{]}^{1-\frac{1}{N}},$

where our conventions are $0\cdot\infty:=0$ and $\infty^{0}:=1$.

######  Definition 2.3 (curvature-dimension conditions [41, 49]).

An essentially nonbranching metric measure space $(X,d,\operatorname{m})$
satisfies the curvature-dimension condition $\mathrm{CD}(K,N)$ for
$K\in\mathbb{R}$ and $N\in[1,\infty)$ if for every
$\mu_{0},\mu_{1}\in\mathcal{P}_{b}^{2}(X,\operatorname{m})$ there exists a
dynamical optimal coupling $\Pi$ between $\mu_{0}$ and $\mu_{1}$ such that for
all $t\in(0,1)$

$\displaystyle\rho_{t}(\gamma_{t})^{-\frac{1}{N}}\geq\tau_{K,N}^{(1-t)}(d(\gamma_{0},\gamma_{1}))\rho_{0}(\gamma_{0})^{-\frac{1}{N}}$
$\displaystyle\hphantom{\rho_{t}(\gamma_{t})^{-\frac{1}{N}}\geq}{}+\tau_{K,N}^{(t)}(d(\gamma_{0},\gamma_{1}))\rho_{1}(\gamma_{1})^{-\frac{1}{N}}\qquad\mbox{for}\quad\Pi\mbox{-a.e.}\
\gamma\in\mathcal{G}(X),$ (2.1)

where $(e_{t})_{\\#}\Pi=\rho_{t}\operatorname{m}$.

Now instead suppose $(X,d,\operatorname{m})$ satisfies
$\mathrm{CD}(K^{\prime},N^{\prime})$ for some $K^{\prime}\in\R$ and
$N^{\prime}\geq 1$. We say a subset $\Omega\subset X$ with
$\operatorname{m}(\Omega)>0$ satisfies the restricted curvature-dimension
condition $\mathrm{CD}_{r}(K,N)$ if for every dynamical optimal coupling $\Pi$
with $(e_{t})_{\\#}\Pi(\Omega)=1$ for all $t\in[0,1]$ and
$\mu_{0},\mu_{1}\in\mathcal{P}_{b}^{2}(X,\operatorname{m})$, (2.1) holds for
all $t\in[0,1]$.

######  Remark 2.4 (locally compact geodesic spaces).

A $\mathrm{CD}(K,N)$ space $(X,d,\operatorname{m})$ for $N\in[1,\infty)$ is
geodesic and locally compact. Hence, by the metric Hopf–Rinow theorem the
space is proper (i.e., Heine–Borel) [7, Theorem 2.5.28].

###  2.2 Disintegration of measures

For further details about the content of this section we refer to [22, Section
452].

Let $(R,\mathcal{R})$ be a measurable space, and let $\mathfrak{Q}\colon
R\rightarrow Q$ be a map for a set $Q$. One can equip $Q$ with the
$\sigma$-algebra $\mathcal{Q}$ that is induced by $\mathfrak{Q}$ where
$B\in\mathcal{Q}$ if $\mathfrak{Q}^{-1}(B)\in\mathcal{R}$. Given a measure
$\operatorname{m}$ on $(R,\mathcal{R})$, one can define its quotient measure
$\mathfrak{q}$ on $Q$ via the push-forward
$\mathfrak{Q}_{\\#}\operatorname{m}=:\mathfrak{q}$.

######  Definition 2.5 (disintegration of measures).

A disintegration of a probability measure $\operatorname{m}$ that is
consistent with $\mathfrak{Q}$ is a map $(B,\alpha)\in\mathcal{R}\times
Q\mapsto\operatorname{m}_{\alpha}(B)\in[0,1]$ such that it follows

  * •

$\operatorname{m}_{\alpha}$ is a probability measure on $(R,\mathcal{R})$ for
every $\alpha\in Q$,

  * •

$\alpha\mapsto\operatorname{m}_{\alpha}(B)$ is $\mathfrak{q}$-measurable for
every $B\in\mathcal{R}$,

and for all $B\in\mathcal{R}$ and $C\in\mathcal{Q}$ the consistency condition

$\operatorname{m}\big{(}B\cap\mathfrak{Q}^{-1}(C)\big{)}=\int_{C}\operatorname{m}_{\alpha}(B)\mathfrak{q}({\rm
d}\alpha)$

holds. We use the notation $\\{\operatorname{m}_{\alpha}\\}_{\alpha\in Q}$ for
such a disintegration. We call the measures $\operatorname{m}_{\alpha}$
conditional probability measures or conditional measures. A disintegration
$\\{\operatorname{m}_{\alpha}\\}_{\alpha\in Q}$ consistent with $\mathfrak{Q}$
is called strongly consistent if for $\mathfrak{q}$-a.e. $\alpha$ we have
$\operatorname{m}_{\alpha}\big{(}\mathfrak{Q}^{-1}(\alpha)\big{)}=1$.

The following theorem is standard:

######  Theorem 2.6 (existence of unique disintegrations).

Assume that $(R,\mathcal{R},\operatorname{m})$ is a countably generated
probability space and $R=\bigcup_{\alpha\in Q}R_{\alpha}$ is a partition of
$R$. Let $\mathfrak{Q}\colon R\rightarrow Q$ be the quotient map associated to
this partition, that is $\alpha=\mathfrak{Q}(x)$ if and only if $x\in
R_{\alpha}$ and assume the corresponding quotient space $(Q,\mathcal{Q})$ is a
Polish space.

Then, there exists a strongly consistent disintegration
$\\{\operatorname{m}_{\alpha}\\}_{\alpha\in Q}$ of $\operatorname{m}$ with
respect to $\mathfrak{Q}\colon R\\!\rightarrow\\!Q$ that is unique in the
following sense: if $\\{\operatorname{m}^{\prime}_{\alpha}\\}_{\alpha\in Q}$
is another consistent disintegration of $\operatorname{m}$ with respect to
$\mathfrak{Q}$ then
$\operatorname{m}_{\alpha}=\operatorname{m}^{\prime}_{\alpha}$ for
$\mathfrak{q}$-a.e. $\alpha\in Q$.

###  2.3 $\boldsymbol{1D}$-localization

In this section we will recall the basics of the localization technique
introduced by Cavalletti and Mondino for 1-Lipschitz functions as a nonsmooth
analog of Klartag’s needle decomposition: needle refers to any geodesic along
which the Lipschitz function attains its maximum slope, also called transport
rays here and by Klartag and others [20, 21, 36]. The presentation follows
Sections 3 and 4 in [11]. We assume familiarity with basic concepts in optimal
transport (for instance [50]).

Let $(X,d,\operatorname{m})$ be a proper metric measure space (with
$\operatorname{{spt}}\operatorname{m}=X$ as we always assume).

Let $u\colon X\rightarrow\mathbb{R}$ be a $1$-Lipschitz function. Then the
transport ordering

$\displaystyle\Gamma_{u}:=\\{(x,y)\in X\times X\colon u(y)-u(x)=d(x,y)\\}$

is a $d$-cyclically monotone set, and one defines $\Gamma_{u}^{-1}=\\{(x,y)\in
X\times X\colon(y,x)\in\Gamma_{u}\\}$.

Note that we switch orientation in comparison to [11] where Cavalletti and
Mondino define $\Gamma_{u}$ as $\Gamma_{u}^{-1}$.

The union $\Gamma_{u}\cup\Gamma_{u}^{-1}$ defines a relation $R_{u}$ on
$X\times X$, and $R_{u}$ induces the transport set with endpoint and branching
points

$\mathcal{T}_{u,e}:=P_{1}(R_{u}\backslash\\{(x,y)\colon x=y\in X\\})\subset
X,$

where $P_{1}(x,y)=x$. For $x\in\mathcal{T}_{u,e}$ one defines
$\Gamma_{u}(x):=\\{y\in X\colon(x,y)\in\Gamma_{u}\\}$, and similarly
$\Gamma_{u}^{-1}(x)$ and $R_{u}(x)$. Since $u$ is $1$-Lipschitz, $\Gamma_{u}$,
$\Gamma_{u}^{-1}$ and $R_{u}$ are closed, as are $\Gamma_{u}(x)$,
$\Gamma_{u}^{-1}(x)$ and $R_{u}(x)$.

The forward and backward branching points are defined respectively as

$\displaystyle
A_{+}:=\\{x\in\mathcal{T}_{u,e}\colon\exists\,z,w\in\Gamma_{u}(x)\mbox{ \&
}(z,w)\notin R_{u}\\},$ $\displaystyle
A_{-}:=\\{x\in\mathcal{T}_{u,e}\colon\exists\,z,w\in\Gamma_{u}^{-1}(x)\mbox{
\& }(z,w)\notin R_{u}\\}.$

Then one considers the $($nonbranched$)$ transport set as
$\mathcal{T}_{u}:=\mathcal{T}_{u,e}\backslash(A_{+}\cup A_{-})$ and the
$($nonbranched$)$ transport relation as the restriction of $R_{u}$ to
$\mathcal{T}_{u}\times\mathcal{T}_{u}$.

The sets $\mathcal{T}_{u,e}$, $A_{+}$ and $A_{-}$ are $\sigma$-compact ([11,
Remark 3.3] and [8, Lemma 4.3] respectively), and $\mathcal{T}_{u}$ is a Borel
set. In [8, Theorem 4.6] Cavalletti shows that the restriction of $R_{u}$ to
$\mathcal{T}_{u}\times\mathcal{T}_{u}$ is an equivalence relation. Hence, from
$R_{u}$ one obtains a partition of $\mathcal{T}_{u}$ into a disjoint family of
equivalence classes $\\{X_{\alpha}\\}_{\alpha\in Q}$. There exists a
measurable section $s\colon\mathcal{T}_{u}\rightarrow\mathcal{T}_{u}$ [8,
Proposition 5.2], such that if $(x,s(x))\in R_{u}$ and $(y,x)\in R_{u}$ then
$s(x)=s(y)$, and $Q$ can be identified with the image of $\mathcal{T}_{u}$
under $s$. Every $X_{\alpha}$ is isometric to an interval
$I_{\alpha}\subset\mathbb{R}$ (cf. [11, Lemma 3.1] and the comment after
Proposition 3.7 in [11]) via a distance preserving map $\gamma_{\alpha}\colon
I_{\alpha}\rightarrow X_{\alpha}$ where $\gamma_{\alpha}$ is parametrized such
that
$d(\gamma_{\alpha}(t),s(\gamma_{\alpha}(t)))=\operatorname{sign}(\gamma_{\alpha}(t))t$,
$t\in I_{\alpha}$, and where $\operatorname{sign}x$ is the sign of
$u(x)-u(s(x))$. The map $\gamma_{\alpha}\colon I_{\alpha}\rightarrow X$
extends to a geodesic also denoted $\gamma_{\alpha}$ and defined on the
closure $\overline{I}_{\alpha}$ of $I_{\alpha}$. We set
$\overline{I}_{\alpha}=[a(X_{\alpha}),b(X_{\alpha})]$.

Then, the quotient map $\mathfrak{Q}\colon\mathcal{T}_{u}\rightarrow Q$ given
by the section $s$ above is measurable, and we set
$\mathfrak{q}:=\mathfrak{Q}_{\\#}\operatorname{m}|_{\mathcal{T}_{u}}$. Hence,
we can and will consider $Q$ as a subset of $X$, namely the image of $s$,
equipped with the induced measurable structure, and $\mathfrak{q}$ as a Borel
measure on $X$. By inner regularity we replace $Q$ with a Borel set
$Q^{\prime}$ such that $\mathfrak{q}(Q\backslash Q^{\prime})=0$ and in the
following we denote $Q^{\prime}$ by $Q$ (compare with [11, Proposition 3.5]
and the following remarks).

In [12, Theorem 3.3], Cavalletti and Mondino extend Definition 2.5 to
disintegrate measures $\operatorname{m}$ which are merely $\sigma$-finite by
using a positive function $f$ on $X$ to relate $\operatorname{m}$ to a
probability measure $f(x)\operatorname{dm}(x)$. Using the framework of this
extension, which we also adopt, they prove:

######  Theorem 2.7 (disintegration into needles/transport rays).

Let $(X,d,\operatorname{m})$ be a geodesic metric measure space with
$\operatorname{{spt}}\operatorname{m}=X$ and $\sigma$-finite
$\operatorname{m}$. Let $u\colon X\rightarrow\mathbb{R}$ be a $1$-Lipschitz
function, let $\\{X_{\alpha}\\}_{\alpha\in Q}$ be the induced partition of
$\mathcal{T}_{u}$ via $R_{u}$, and let
$\mathfrak{Q}\colon\mathcal{T}_{u}\rightarrow Q$ be the induced quotient map
as above. Then, there exists a unique strongly consistent disintegration
$\\{\operatorname{m}_{\alpha}\\}_{\alpha\in Q}$ of
$\operatorname{m}|_{\mathcal{T}_{u}}$ with respect to $\mathfrak{Q}$.

Now, we assume that $(X,d,\operatorname{m})$ is an essentially nonbranching
$\mathrm{CD}(K,N)$ space for $K\in\R$ and $N>1$. Recall the Bishop–Gromov
inequality holds and $\operatorname{m}$ is therefore $\sigma$-finite. The
following is [12, Lemma 3.4].

######  Lemma 2.8 (negligibility of branching points).

Let $(X,d,\operatorname{m})$ be an essentially nonbranching $\mathrm{CD}(K,N)$
space for $K\in\R$ and $N\in(1,\infty)$ with
$\operatorname{{spt}}\operatorname{m}=X$. Then, for any $1$-Lipschitz function
$u\colon X\rightarrow\R$, it follows
$\operatorname{m}(\mathcal{T}_{u,e}\backslash\mathcal{T}_{u})=0$.

The initial and final points are defined by

$\displaystyle\mathfrak{a}:=\big{\\{}x\in\mathcal{T}_{u,e}\colon\Gamma^{-1}_{u}(x)=\\{x\\}\big{\\}},\qquad\mathfrak{b}:=\\{x\in\mathcal{T}_{u,e}\colon\Gamma_{u}(x)=\\{x\\}\\}.$

In [9, Theorem 7.10] it was proved that under the assumption of the previous
lemma there exists $\hat{Q}\subset Q$ with
$\mathfrak{q}(Q\backslash\hat{Q})=0$ such that for $\alpha\in\hat{Q}$ one has
$\overline{X_{\alpha}}\backslash\mathcal{T}_{u}\subset\mathfrak{a}\cup\mathfrak{b}$.
In particular, for $\alpha\in\hat{Q}$ we have

$\displaystyle R_{u}(x)=\overline{X_{\alpha}}\supset
X_{\alpha}\supset(R_{u}(x))^{\circ}\qquad\forall\,x\in\mathfrak{Q}^{-1}(\alpha)\subset\mathcal{T}_{u},$
(2.2)

where $(R_{u}(x))^{\circ}$ denotes the relative interior of the closed set
$R_{u}(x)$.

The following is [12, Theorem 3.5].

######  Theorem 2.9 (conditional measures inherit curvature-dimension bounds).

Let $(X,d,\operatorname{m})$ be an essentially nonbranching $\mathrm{CD}(K,N)$
space with $\operatorname{{spt}}\operatorname{m}=X$, $K\in\R$ and
$N\in(1,\infty)$. For any $1$-Lipschitz function $u\colon X\rightarrow\R$, let
$\\{\operatorname{m}_{\alpha}\\}_{\alpha\in Q}$ denoted the disintegration of
$\operatorname{m}|_{\mathcal{T}_{u}}$ from Theorem 2.7 which is strongly
consistent with the quotient map $\mathfrak{Q}\colon\mathcal{T}_{u}\rightarrow
Q$. Then there exists $\tilde{Q}$ such that
$\mathfrak{q}(Q\backslash\tilde{Q})=0$ and $\forall\,\alpha\in\tilde{Q}$,
$\operatorname{m}_{\alpha}$ is a Radon measure with
$\operatorname{dm}_{\alpha}=h_{\alpha}\,{\rm d}\mathcal{H}^{1}|_{X_{\alpha}}$
and $(X_{\alpha},d,\operatorname{m}_{\alpha})$ verifies the condition
$\mathrm{CD}(K,N)$. More precisely, for all $\alpha\in\tilde{Q}$ it follows
that

$\displaystyle
h_{\alpha}(\gamma_{t})^{\frac{1}{N-1}}\geq\sigma_{K/N-1}^{(1-t)}(|\dot{\gamma}|)h_{\alpha}(\gamma_{0})^{\frac{1}{N-1}}+\sigma_{K/N-1}^{(t)}(|\dot{\gamma}|)h_{\alpha}(\gamma_{1})^{\frac{1}{N-1}}$
(2.3)

for every affine map
$\gamma\colon[0,1]\rightarrow(a(X_{\alpha}),b(X_{\alpha}))$.

######  Remark 2.10 (semiconcave densities on needles).

The property (2.3) yields that $h_{\alpha}$ is locally Lipschitz continuous on
$(a(X_{\alpha}),b(X_{\alpha}))$ [11, Section 4], and that
$h_{\alpha}\colon(a(X_{\alpha}),b(X_{\alpha}))\rightarrow(0,\infty)$ satisfies

$\displaystyle\frac{{\rm d}^{2}}{{\rm
d}r^{2}}h_{\alpha}^{\frac{1}{N-1}}+\frac{K}{N-1}h_{\alpha}^{\frac{1}{N-1}}\leq
0\qquad\mbox{on $(a(X_{\alpha}),b(X_{\alpha}))$}\mbox{ in distributional
sense;}$ (2.4)

in particular, $h_{\alpha}^{\frac{1}{N-1}}$ is semiconcave on
$(a(X_{\alpha}),b(X_{\alpha}))$, hence admits left and right derivatives at
each point.

Conversely, it is well-known that any function $h_{\alpha}\geq 0$ satisfying
(2.4) can be chosen to be continuous up to its endpoints and that this
extension then satisfies (2.3); see also Remark 2.12.

######  Remark 2.11.

We observe the following from the proof of [11, Theorem 4.2]: when
$\Omega\subset X$ satisfies the restricted condition
$\mathrm{CD}_{r}(K^{\prime},N)$ and the $1$-Lipschitz function
$u=d_{\Omega^{c}}$ is chosen to be the distance function to $\Omega^{c}$, then
$h_{\gamma}$ satisfies (2.3) with $K^{\prime}$ replacing $K$.

Let us be a little bit more precise here. For the proof of Theorem 4.2 in [11]
the authors construct $L^{2}$-Wasserstein geodesics between
$\operatorname{m}$-absolutely continuous probability measures such that the
corresponding optimal dynamical plans are supported on transport geodesics of
the $1$-Lipschitz function $\phi$ that appears in the statement of [11,
Theorem 4.2].

In our situation, when $\phi$ is actually $d_{\Omega^{c}}$, all transport
geodesics of positive length are inside of $\Omega$. Hence, the
$L^{2}$-Wasserstein geodesics constructed by Cavalletti and Mondino are
concentrated in $\Omega$ and the restricted condition
$\mathrm{CD}_{r}(K^{\prime},N)$ applies. Then we can follow verbatim the proof
of Theorem 4.2 in [11].

######  Remark 2.12 (extended densities).

The Bishop–Gromov volume monotonicity implies that $h_{\alpha}$ can always be
extended to continuous function on $[a(X_{\alpha}),b(X_{\alpha})]$ [12, Remark
2.14]. Then (2.3) holds for every affine map
$\gamma\colon[0,1]\rightarrow[a(X_{\alpha}),b(X_{\alpha})]$. We set
$\left(h_{\alpha}\circ\gamma_{\alpha}(r)\right)\cdot
1_{[a(X_{\alpha}),b(X_{\alpha})]}=h_{\alpha}(r)$ and consider $h_{\alpha}$ as
function that is defined everywhere on . We also consider $\frac{{\rm d}}{{\rm
d}r}h_{\alpha}\colon X_{\alpha}\rightarrow\R$ defined a.e. via $\frac{{\rm
d}}{{\rm d}r}(h_{\alpha}\circ\gamma_{\alpha})(r)=:\frac{{\rm d}}{{\rm
d}r}h_{\alpha}(r)$.

It is standard knowledge that the derivatives from the right and from the left

$\displaystyle\frac{{\rm d}^{+}}{{\rm d}r}h_{\alpha}(r)=\lim_{t\downarrow
0}\frac{h_{\alpha}(r+t)-h_{\alpha}(r)}{t},\qquad\frac{{\rm d}^{-}}{{\rm
d}r}h_{\alpha}(r)=\lim_{t\uparrow 0}\frac{h_{\alpha}(r+t)-h_{\alpha}(r)}{t}$

exist for $r\in[a(X_{\alpha}),b(X_{\alpha}))$ and
$r\in(a(X_{\alpha}),b(X_{\alpha})]$ respectively. Moreover, we set $\frac{{\rm
d}^{+}}{{\rm d}r}h_{\alpha}=-\infty$ in $b(X_{\alpha})$ and $\frac{{\rm
d}^{-}}{{\rm d}r}h_{\alpha}=\infty$ in $a(X_{\alpha})$.

######  Remark 2.13 (generic geodesics).

In the following we set $Q^{\dagger}:=\tilde{Q}\cap\hat{Q}$, where $\tilde{Q}$
and $\hat{Q}$ index the transport rays identified between Lemma 2.8 and
Theorem 2.9. Then, $\mathfrak{q}\big{(}Q\backslash Q^{\dagger}\big{)}=0$ and
for every $\alpha\in Q^{\dagger}$ the inequality (2.3) and (2.2) hold. We also
set
$\mathfrak{Q}^{-1}(Q^{\dagger})=:\mathcal{T}_{u}^{\dagger}\subset\mathcal{T}_{u}$
and
$\bigcup_{x\in\mathcal{T}_{u}^{\dagger}}R_{u}(x)=:\mathcal{T}_{u,e}^{\dagger}\subset\mathcal{T}_{u,e}$.

###  2.4 Generalized mean curvature

Let $(X,d,\operatorname{m})$ be a metric measure space as in Theorem 2.9. Let
$\Omega\subset X$ be a closed subset, and let $S=\partial\Omega$ such that
$\operatorname{m}(S)=0$. The function $d_{\Omega}\colon
X\rightarrow\mathbb{R}$ is given by

$\displaystyle d_{\Omega}(x):=\inf_{y\in{\Omega}}d(x,y).$

The signed distance function $d_{S}$ for $S$ is given by

$\displaystyle d_{S}:=d_{{\Omega}}-d_{{\Omega^{c}}}\colon\
X\rightarrow\mathbb{R}.$

It follows that $d_{S}(x)=0$ if and only if $x\in S$, and $d_{S}\leq 0$ if
$x\in\Omega$ and $d_{S}\geq 0$ if $x\in\Omega^{c}$. It is clear that
$d_{S}|_{\Omega}=-d_{\Omega^{c}}$ and $d_{S}|_{\Omega^{c}}=d_{\Omega}$.
Setting $v=d_{S}$ we can also write

$d_{S}(x)=\operatorname{sign}(v(x))d(\\{v=0\\},x),\qquad\forall\,x\in X.$

Since $X$ is proper, $d_{S}$ is $1$-Lipschitz [12, Remarks 8.4 and 8.5]. Let
$\Omega^{\circ}$ denote the topological interior of $\Omega$.

Let $\mathcal{T}_{d_{S},e}$ be the transport set of $d_{S}$ with end- and
branching points. We have $\mathcal{T}_{d_{S},e}\supset X\backslash S$. In
particular, we have $\operatorname{m}(X\backslash\mathcal{T}_{d_{S}})=0$ by
Lemma 2.8 and $\operatorname{m}(S)=0$. Therefore, by Theorem 2.9 the
$1$-Lipschitz function $d_{S}$ induces a partition
$\left\\{X_{\alpha}\right\\}_{\alpha\in Q}$ of $(X,d,\operatorname{m})$ up to
a set of measure zero for a measurable quotient space $Q$, and a
disintegration $\\{m_{\alpha}\\}_{\alpha\in Q}$ that is strongly consistent
with the partition. The subset $X_{\alpha}$, $\alpha\in Q$, is the image of a
distance preserving map $\gamma_{\alpha}\colon I_{\alpha}\rightarrow X$ for an
interval $I_{\alpha}\subset\mathbb{R}$ with
$\overline{I}_{\alpha}=[a(X_{\alpha}),b(X_{\alpha})]\ni 0$.

We consider $Q^{\dagger}\subset Q$ as in Remark 2.13. One has the
representation

$\displaystyle\operatorname{m}(B)=\int_{Q}\operatorname{m}_{\alpha}(B)\,{\rm
d}\mathfrak{q}(\alpha)=\int_{Q^{\dagger}}\int_{\gamma_{\alpha}^{-1}(B)}h_{\alpha}(r)\,{\rm
d}r\,{\rm d}\mathfrak{q}(\alpha)$

for all Borel $B\subset X$.

For any transport ray $X_{\alpha}$ with $\alpha\in Q^{\dagger}$, it follows
that $d_{S}(\gamma_{\alpha}(b(X_{\alpha})))\geq 0$ and
$d_{S}(\gamma_{\alpha}(a(X_{\alpha})))\allowbreak\leq 0$ (for instance compare
with [12, Remark 4.12]).

######  Remark 2.14 (measurability and zero-level selection).

It is easy to see that
$A:=\mathfrak{Q}^{-1}(\mathfrak{Q}(S\cap\mathcal{T}_{d_{S}}))\subset\mathcal{T}_{d_{S}}$
is a measurable subset. The reach $A\subset\mathcal{T}_{d_{S}}$ is defined
such that $\forall\,\alpha\in\mathfrak{Q}(A)$ we have $X_{\alpha}\cap
S=\\{\gamma(t_{\alpha})\\}\neq\varnothing$ for a unique $t_{\alpha}\in
I_{\alpha}$. Then, the map ${\hat{s}}\colon\gamma(t)\in
A\mapsto\gamma(t_{\alpha})\in S\cap\mathcal{T}_{d_{S}}$ is a measurable
section (i.e., selection) on $A\subset\mathcal{T}_{d_{S}}$, and one can
identify the measurable set $\mathfrak{Q}(A)\subset Q$ with $A\cap S$ and can
parameterize $\gamma_{\alpha}$ such that $t_{\alpha}=0$.

This measurable section $\hat{s}$ on $A$ is fixed for the rest of the paper.
The reach $A$ is the union of all disjoint needles that intersect with
$\partial\Omega$ – possibly in $a(X_{\alpha})$ (or in $b(X_{\alpha})$)
provided $a(X_{\alpha})$ (respectively $b(X_{\alpha})$) belongs to
$I_{\alpha}$. We shall also define the inner reach $B_{\rm in}$ as the union
of all needles disjoint from $\overline{\Omega^{c}}$ and the outer reach
$B_{\rm out}$ as the union of all needles disjoint from $\Omega$. The
superscript $\dagger$ will be used indicate intersection with
$\mathcal{T}^{\dagger}_{d_{S}}$. Thus

$\displaystyle{A}\cap\mathfrak{\mathcal{T}}_{d_{S}}^{\dagger}=:A^{\dagger}\qquad\mbox{and}\qquad\bigcup_{x\in
A^{\dagger}}R_{d_{S}}(x)=:A_{e}^{\dagger}.$

The sets $A^{\dagger}$ and $A_{e}^{\dagger}$ are measurable, and also

$\displaystyle B_{\rm
in}^{\dagger}:=\Omega^{\circ}\cap\mathcal{T}^{\dagger}_{d_{S}}\backslash
A^{\dagger}\subset\mathcal{T}^{\dagger}_{d_{S}}\qquad\text{and}\qquad
B^{\dagger}_{\rm out}:=\Omega^{c}\cap\mathcal{T}^{\dagger}_{d_{S}}\backslash
A^{\dagger}\subset\mathcal{T}_{d_{S}}$ (2.5)

as well as $\bigcup_{x\in B^{\dagger}_{\rm
out}}R_{d_{S}}(x)=:B^{\dagger}_{{\rm out},e}$ and $\bigcup_{x\in
B^{\dagger}_{\rm in}}R_{d_{S}}(x)=:B^{\dagger}_{{\rm in},e}$ are measurable.

The map $\alpha\in\mathfrak{Q}(A^{\dagger})\mapsto h_{\alpha}(0)\in\R$ is
measurable (see [9, Proposition 10.4]).

######  Definition 2.15 (surface measures).

Taking $S=\partial\Omega$ as above, we use the disintegration of Remark 2.14
to define the surface measure $\operatorname{m}_{S}$ via

$\int\phi(x)\operatorname{dm}_{S}(x):=\int_{\mathfrak{Q}(A^{\dagger})}\phi(\gamma_{\alpha}(0))h_{\alpha}(0)\,{\rm
d}\mathfrak{q}(\alpha)$

for any bounded and continuous function $\phi\colon X\rightarrow\R$. That is,
$\operatorname{m}_{S}$ is the push-forward of the measure $h_{\alpha}(0)\,{\rm
d}\mathfrak{q}(\alpha)|_{\mathfrak{Q}(A^{\dagger})}$ under the map
$\hat{s}\colon\gamma\in\mathfrak{Q}(A^{\dagger})\mapsto\gamma(0)\in S$.

######  Remark 2.16 (surface measure via ray maps).

Let us briefly explain the previous definition from the viewpoint of the ray
map [11, Definition 3.6] or its precursor from the smooth setting [21]. For
the definition we fix a measurable extension
$s_{0}\colon\mathcal{T}_{d_{S}}\rightarrow\mathcal{T}_{d_{S}}$ such that
$s_{0}|_{A^{\dagger}}=\hat{s}$ as in Remark 2.14. As was explained in Section
2.3 such a section allows us to identify the quotient space $Q$ with a Borel
subset in $X$ up to a set of $\mathfrak{q}$-measure $0$. Following [11,
Definition 3.6] we define the ray map

$\displaystyle g\colon\ \mathcal{V}\subset\mathfrak{Q}(A\cup B_{\rm
in})\times(-\infty,0]\rightarrow X$

into $\Omega$ and its domain $\mathcal{V}$ via its graph

$\displaystyle\mbox{graph}(g)=\\{(\alpha,t,x)\in\mathfrak{Q}(A)\times\R\times\Omega\colon
x\in X_{\alpha},-d(x,\alpha)=t\\}$
$\displaystyle\hphantom{\mbox{graph}(g)=}{}\cup\\{(\alpha,t,x)\in\mathfrak{Q}(B_{\rm
in})\times\R\times\Omega\colon x\in
X_{\alpha},-d(x,\gamma_{\alpha}(b(X_{\alpha})))=t\\}.$

This is exactly the ray map as in [11] up to a reparametrisation for
$\alpha\in\mathfrak{Q}(B_{\rm in})$. Note that
$g(\alpha,0)=\gamma_{\alpha}(0)=\alpha$ and $g(\alpha,t)=\gamma_{\alpha}(t)$
if $\alpha\in\mathfrak{Q}(A)$ but
$\gamma_{\alpha}(t+d(b(X_{\alpha}),\alpha))=g(\alpha,t)$ for
$\alpha\in\mathfrak{Q}(B_{\rm in})$. Then the disintegration for a non-
negative $\phi\in C_{b}(\Omega)$ takes the form

$\displaystyle\int_{\Omega}\phi\thinspace\operatorname{dm}=\int_{Q}\int_{\mathcal{V}_{\alpha}}\phi\circ
g(\alpha,t)h_{\alpha}\circ g(\alpha,t)\,{\rm d}\mathcal{L}^{1}{(t)}\,{\rm
d}\mathfrak{q}(\alpha),$

where
$\mathcal{V}_{\alpha}=P_{2}(\mathcal{V}\cap\\{\alpha\\}\times\R)\subset\R$ and
$P_{2}(\alpha,t)=t$. With Fubini’s theorem the right hand side is

$\displaystyle\int_{\mathcal{V}}\phi\circ g(\alpha,t)h_{\alpha}\circ
g(\alpha,t)\,{\rm
d}\big{(}\mathfrak{q}\otimes\mathcal{L}^{1}\big{)}(\alpha,t)=\iint_{\mathcal{V}_{t}}\phi\circ
g(\alpha,t)h_{\alpha}\circ g(\alpha,t)\,{\rm d}\mathfrak{q}(\alpha)\,{\rm
d}t,$

where $\mathcal{V}_{t}=P_{1}(\mathcal{V}\cap Q\times\\{t\\})\subset Q$ and
$P_{1}(\alpha,t)=\alpha$. In particular, for $\mathcal{L}^{1}$-a.e. $t\in\R$
the map $\alpha\mapsto h_{\alpha}\circ g(\alpha,t)$ is measurable. Hence, for
$\mathcal{L}^{1}$-a.e. $t\in\R$ we define ${\rm
d}\mathfrak{p}_{t}(\alpha)=h_{\alpha}\circ g(\alpha,t)\,{\rm
d}\mathfrak{q}|_{\mathcal{V}_{t}}(\alpha)$ on $Q$. Then disintegration takes
the form

$\displaystyle\operatorname{m}|_{\Omega}=\operatorname{m}|_{\Omega\cap\mathcal{T}_{d_{S}}}=\int(g(\cdot,t)_{\\#}\mathfrak{p}_{t})\,{\rm
d}t.$

Now, we can consider the push-forward
$\operatorname{m}_{S_{0}}=g(\cdot,0)_{\\#}\mathfrak{p}_{0}$. When $B_{\rm
in}\neq\varnothing$ then $\operatorname{m}_{S_{0}}$ may be concentrated on a
larger set than $\operatorname{m}_{S}$ but by construction one recognizes that
$\operatorname{m}_{S}=\operatorname{m}_{S_{0}}|_{A^{\dagger}}$.

######  Definition 2.17 (inner mean curvature).

Set $S=\partial\Omega$ and let $\\{X_{\alpha}\\}_{\alpha\in Q}$ be the
disintegration induced by $u:=d_{S}$. Recalling (2.5), we say that $S$ has
finite inner (respectively outer) curvature if
$\operatorname{m}\big{(}B^{\dagger}_{\rm in}\big{)}=0$ (respectively
$\operatorname{m}\big{(}B^{\dagger}_{\rm out}\big{)}=0$), and $S$ has finite
curvature if $\operatorname{m}\big{(}B^{\dagger}_{\rm out}\cup
B^{\dagger}_{\rm in}\big{)}=0$. If $S$ has finite inner curvature we define
the inner mean curvature of $S$ $\operatorname{m}_{S}$-almost everywhere as

$\displaystyle p\in S\mapsto H_{S}^{-}(p):=\begin{cases}\dfrac{{\rm
d}^{-}}{{\rm d}r}|_{r=0}\log h_{\alpha}\circ\gamma_{\alpha}(r)&\mbox{if }\
p=\gamma_{\alpha}(0)\in S\cap A^{\dagger},\\\ \infty&\mbox{if }\ p\in
B^{\dagger}_{{\rm out},e}\cap S,\end{cases}$

where we set $\frac{{\rm d}^{-}}{{\rm d}r}\log
h_{\alpha}(\gamma_{\alpha}(0))=-\infty$ if $h_{\alpha}(\gamma_{\alpha}(0))=0$.

######  Remark 2.18 ((sign) conventions).

We point out two differences in comparison to [34]: For the definition of
$A^{\dagger}$ we do not remove points that lie in $\mathfrak{a}$ and
$\mathfrak{b}$, and we switched signs in the definition of inner mean
curvature. The latter allows us to work with mean curvature bounded below
instead of bounded above.

######  Remark 2.19 (smooth case).

Let us briefly address the case of a Riemannian manifold $(M,g)$ equipped with
a measure of the form $\operatorname{dm}=\Psi{\rm d}\\!\operatorname{vol}_{g}$
for $0<\Psi\in C^{\infty}(M)$ and $\Omega$ with a boundary $S$ which is a
smooth compact submanifold. For every $x\in S$ there exist $a_{x}<0$ and
$b_{x}>0$ such that $\gamma_{x}(r)=\exp_{x}(r\nabla d_{S}(x))$ is a minimal
geodesic on $(a_{x},b_{x})\subset\R$, and we define

$\displaystyle\mathcal{U}=\\{(x,r)\in S\times\R\colon
r\in(a_{x},b_{x})\\}\subset S\times\R$

and the map $T\colon\mathcal{U}\rightarrow M$ via $T(x,r)=\gamma_{x}(r)$. The
map $T$ is a diffeomorphism on $\mathcal{U}$, with
$\operatorname{vol}_{g}(M\backslash T(\mathcal{U}))=0$ and the integral of
$\phi\in C(M)$ can be computed effectively by the following formula:

$\displaystyle\int\phi\operatorname{dm}=\int_{S}\int_{a_{x}}^{b_{x}}\phi\circ
T(x,r)\det DT_{(x,r)}|_{T_{x}S}\Psi\circ T(x,r)\,{\rm d}r\,{\rm
d}\\!\operatorname{vol}_{S}(x),$

where $\operatorname{vol}_{S}$ is the induced Riemannian surface measure on
$S$. By comparison with the needle technique disintegration it is not
difficult to see that $\operatorname{dm}_{S}=\Psi\,{\rm
d}\\!\operatorname{vol}_{S}$. Moreover, the open needles for $d_{S}$ are the
geodesics $\gamma_{x}\colon(a_{x},b_{x})\rightarrow M$ and the densities
$h_{x}(r)$ are given by $c(x)\det DT_{(x,r)}\Psi\circ T(x,r)$ for some
normalization constant $c(x)$, $x\in S$.

A direct computation then yields

$\displaystyle\frac{{\rm d}}{{\rm d}r}\log h_{x}(0)=H_{S}(x)+\langle\nabla
d_{S}(x),\nabla\log\Psi\rangle(x),\qquad\forall\,x\in S,$

where $H_{S}$ is the standard mean curvature, i.e., the trace of the second
fundamental form of $S$.

######  Definition 2.20 (exterior ball condition).

Let $\Omega\subset X$ and $\partial\Omega=S$. Then $S$ satisfies the exterior
ball condition if for all $x\in S$ there exists $r_{x}>0$ and
$p_{x}\in\Omega^{c}$ such that $d(x,p_{x})=r_{x}$ and
$B_{r_{x}}(p_{x})\subset\Omega^{c}$. We say $S$ satisfies a uniform exterior
ball condition if there exists $\delta>0$ such that $r_{x}\geq\delta$ for all
$x\in S$.

######  Lemma 2.21 (exterior ball criterion for finite inner curvature).

Let $\Omega\subset X$. If $S=\partial\Omega$ satisfies the exterior ball
condition, then $S$ has finite inner curvature.

###### Proof.

Let $S$ satisfy the exterior ball condition. Then for every $x\in S$ there
exists a point $p_{x}\in\Omega^{c}$ and a geodesic
$\gamma_{x}\colon[0,r_{x}]\rightarrow\Omega^{c}$ from $x$ to $p_{x}$ such that
$L(\gamma_{x})=d(x,p_{x})=r_{x}$ and $d(p_{x},y)>r_{x}$ for any $y\in
S\backslash\\{x\\}$. Hence, $d_{S}(p_{x})=r_{x}$ and the image of $\gamma_{x}$
is contained in $R_{d_{S}}(x)$.

Recall the definition of $Q^{\dagger}\subset Q$ (Remark 2.13). Since
$Q^{\dagger}$ has full $\mathfrak{q}$-measure, it is enough to show that for
all $\alpha\in Q^{\dagger}$ the endpoint $b(X_{\alpha})>0$. Then also $B_{\rm
in}^{\dagger}=\varnothing$. Assume the contrary. Let $\alpha^{\prime}\in
Q^{\dagger}$ and let $\gamma^{\prime}:=\gamma_{\alpha^{\prime}}$ be the
corresponding geodesic such that $b(X_{\alpha^{\prime}})=0$, that is
$\mbox{Im}(\gamma^{\prime}|_{(a(X_{\alpha^{\prime}}),0)})\subset\Omega$. The
concatenation
$\gamma^{\prime\prime}\colon(a(X_{\alpha^{\prime}}),r_{x})\rightarrow X$ of
$\gamma^{\prime}$ with $\gamma_{x}$ for $x=\gamma^{\prime}(0)$ satisfies
$\gamma^{\prime\prime}(0)=x$ and

$\displaystyle d(\gamma^{\prime\prime}(s),\gamma^{\prime\prime}(t))\leq
d(\gamma^{\prime\prime}(s),x)+d(x,\gamma^{\prime\prime}(t))=d_{S}(\gamma^{\prime\prime}(t))-d_{S}(\gamma^{\prime\prime}(s))\leq
d(\gamma^{\prime\prime}(s),\gamma^{\prime\prime}(t))$ (2.6)

for $s\in(a(X_{\alpha^{\prime}}),0]$ and $t\in[0,r_{x})$.

Thus the inequalities in (2.6) are actually equalities. Hence,
$\mbox{Im}(\gamma^{\prime\prime})\subset R_{d_{S}}(\gamma^{\prime\prime}(s))$,
the points that are $R_{d_{S}}$-related to $\gamma^{\prime\prime}(s)$. These
are exactly the points $y$ that satisfy (2.6) with $\gamma^{\prime\prime}(t)$
replaced with $y$. But this contradicts the requirement
$\overline{X_{\alpha^{\prime}}}=R_{d_{S}}(\gamma^{\prime\prime}(s))$ from the
definition of $Q^{\dagger}$. ∎

######  Remark 2.22 (partial converse).

As was pointed out to us by one of the referees the converse implication in
Lemma 2.21 holds in the following sense. If for $x\in S\cap A$ then there
exists $p\in X$ that either belongs to $\Omega^{c}$ or $\Omega^{\circ}$ such
that $B_{r}(p)$ is either fully contained in $\Omega^{c}$ or $\Omega^{\circ}$
with $r=d(p,x)$.

######  Remark 2.23 (related literature).

We note that the previous notion of mean curvature under the assumption that
$S$ has finite inner curvature, allows to assign to any point $p\in S\cap
A^{\dagger}$ a number that is the mean curvature of $S$ at $p$. This was
useful for proving the Heintze–Karcher inequality in [34].

If one is just interested in lower bounds for the mean curvature, one can
adapt a definition of Cavalletti–Mondino [13]. They define achronal future
timelike complete Borel subsets in a Lorentz length space having forward mean
curvature bounded below. We will not recall their definition for Lorentz
length spaces but we give a corresponding definition for $\mathrm{CD}(K,N)$
metric measure spaces in the appendix of this article and outline how analogs
of our results also hold for this notion of lower mean curvature bounds.

##  3 Proof of inradius bounds and stability (Theorems 1.1 and 1.2)

Recall the Jacobian $J_{K,H,N}(r)$ and its maximal interval
$r\in(-r_{K,-H,N},r_{K,H,N})$ of positivity around the origin defined in
(1.1)–(1.4). To prove our main theorems requires a sort of below-tangent
implication (3.2) of distorted power concavity (3.1) from [34]:

######  Lemma 3.1 (comparison inequality).

Let $h\colon[a,b]\rightarrow[0,\infty)$ be continuous such that $a\leq 0<b$
and every affine map $\gamma\colon[0,1]\rightarrow[a,b]$ satisfies

$\displaystyle
h(\gamma_{t})^{\frac{1}{N-1}}\geq\sigma_{K/(N-1)}^{(1-t)}(|\dot{\gamma}|)h(\gamma_{0})^{\frac{1}{N-1}}+\sigma_{K/(N-1)}^{(t)}(|\dot{\gamma}|)h(\gamma_{1})^{\frac{1}{N-1}}\qquad\forall\,t\in[0,1].$
(3.1)

Then

$\displaystyle(h(r))^{\frac{1}{N-1}}\leq(h(0))^{\frac{1}{N-1}}\cos_{\frac{K}{N-1}}(r)+\frac{{\rm
d}^{+}}{{\rm
d}s}\Big{|}_{s=0}(h(s))^{\frac{1}{N-1}}\sin_{\frac{K}{N-1}}(r)\qquad\forall\,r\in[a,b].$
(3.2)

If $h(0)>0$, it follows $h(r)h(0)^{-1}\leq J_{K,H,N}(r)$ where $H=-\frac{{\rm
d}^{+}}{{\rm d}r}\big{|}_{r=0}\log h(r)$ and in particular $b\leq r_{K,H,N}$.

###### Proof.

If $a<0$, the lemma is exactly the statement of Corollary 4.3 in [34].

For $a=0$ we pick $r_{n}\downarrow 0$. Then, the statement follows since
$\frac{{\rm d}^{+}}{{\rm d}r}h(r)$ is continuous from the left for a
semiconcave function $h$. ∎

######  Remark 3.2 (reverse parameterization).

If instead $h\colon[a,b]\rightarrow[0,\infty)$ is continuous and every affine
map $\gamma\colon[0,1]\rightarrow[a,b]$ satisfies (3.1) but $a<0\leq b$, then
applying Lemma 3.1 to $\tilde{h}(r):=h(-r)$ yields $-a\leq r_{K,\tilde{H},N}$
with $\tilde{H}=\frac{{\rm d}^{-}}{{\rm d}r}\big{|}_{r=0}\log h(r)$.

###### Proof of Theorem 1.1.

Let $(X,d,\operatorname{m})$ be a $\mathrm{CD}(K^{\prime},N)$ space and
consider $\Omega\subset X$ satisfying $\mathrm{CD}_{r}(K,N)$ as assumed in
Theorem 1.1. Let $u=d_{S}$ be corresponding signed distance function. Let
$\\{X_{\alpha}\\}_{\alpha\in Q}$ be the decomposition of $\mathcal{T}_{d_{S}}$
and $\int\operatorname{m}_{\alpha}\,{\rm d}\mathfrak{q}(\alpha)$ be the
disintegration of $\operatorname{m}$ given by Theorem 2.9 and Remark 2.14. In
Remark 2.13 we define $Q^{\dagger}\subset Q$. Recall that $Q^{\dagger}$ is a
subset of $Q$ with full $\mathfrak{q}$-measure and for all $\alpha\in
Q^{\dagger}$ one has ${\operatorname{dm}}_{\alpha}=h_{\alpha}\,{\rm
d}\mathcal{H}^{1}$, $X_{\alpha,e}=\overline{X}_{\alpha}$ and $h_{\alpha}$
satisfies

$\displaystyle\big{(}h_{\alpha}^{\frac{1}{N-1}}\big{)}^{\prime\prime}+\frac{K}{N-1}h_{\alpha}^{\frac{1}{N-1}}\leq
0\qquad\mbox{on}\quad(a(X_{\alpha}),0)$ (3.3)

in the distributional sense.

Assume $K\in\R$ and $H^{-}_{S}\geq\chi(N-1)$ $\operatorname{m}_{S}$-a.e.
Recall that

$\displaystyle H^{-}_{S}(\gamma_{\alpha}(0))=\frac{{\rm d}^{-}}{{\rm
d}r}\Big{|}_{r=0}\log h_{\alpha}(r).$

In particular, $h_{\alpha}(0)>0$ for $\mathfrak{q}$-a.e. $\alpha\in
Q^{\dagger}$, since $h_{\alpha}(0)=0$ yields $\frac{{\rm d}^{-}}{{\rm d}r}\log
h_{\alpha}(r)=-\infty<\chi(N-1)$ by Definition 2.17. Now (3.3) implies
$h_{\alpha}$ can be represented by a continuous function on
$[a(X_{\alpha}),0]$ which satisfies the hypotheses of Remark 3.2. That remark
then asserts $-a(X_{\alpha})\leq r_{K,\chi(N-1),N}$ for any $\alpha\in
Q^{\dagger}$.

Since $(\operatorname{m}_{\alpha}|_{\Omega})_{\alpha\in Q^{\dagger}}$ is a
disintegration of $\operatorname{m}|_{\Omega}$, and since
$\operatorname{m}_{\alpha}|_{\Omega}$ is supported on
$\mbox{Im}(\gamma_{\alpha}|_{[a(X_{\alpha}),0]})$ where
$\gamma_{\alpha}\colon[a(X_{\alpha},0]\rightarrow\overline{\Omega}$ is a
geodesic, it follows that for $\operatorname{m}$-almost every $x\in\Omega$
there exists $\alpha\in Q^{\dagger}$ and $t\in[a(X_{\alpha}),0]$ such that
$x=\gamma_{\alpha}(t)$. For such $x$ it follows that
$d_{\Omega^{c}}(x)=-t\leq-a(X_{\alpha})\leq r_{K,\chi(N-1),N}$. Hence
$d_{\Omega^{c}}\leq r_{K,\chi(N-1),N}$ for $\operatorname{m}$-almost
everywhere in $\Omega$.

By continuity of $d_{\Omega^{c}}$ and $X=\operatorname{{spt}}\operatorname{m}$
it follows that $d_{\Omega^{c}}(x)\leq r_{K,\chi(N-1),N}$ for all
$x\in\Omega$. In particular, the inscribed radius satisfies
$\operatorname{InRad}\Omega\leq r_{K,\chi(N-1),N}$. ∎

###### Proof of Theorem 1.2.

Assume $K\geq\bar{K}-\delta$, $H^{-}_{S}\geq\bar{H}-\delta$
$\operatorname{m}_{S}$-a.e. and $N\leq\bar{N}+\delta$. Since $\delta>0$ and
$X\in\mathrm{CD}(K,N)$ imply $X\in\mathrm{CD}(\bar{K}-\delta,\bar{N}+\delta)$,
Theorem 1.1 yields

$\operatorname{InRad}\Omega\leq
r_{\bar{K}-\delta,\bar{H}-\delta,\bar{N}+\delta}.$

Now for any $\epsilon>0$ there exists $\delta>0$ such that
$r_{\bar{K}-\delta,\bar{H}-\delta,\bar{N}+\delta}\leq
r_{\bar{K},\bar{H},\bar{N}}+\epsilon$ since the function
$s_{\frac{K}{N-1},\frac{H}{N-1}}(r)=\cos_{\frac{K}{N-1}}(r)-\frac{H}{N-1}\sin_{\frac{K}{N-1}}(r)$
whose first positive zero defines $r_{K,H,N}$ is continuously differentiable
with respect to $K$, $H$, $N$ and $r$, and its derivative is non-zero at
$r=r_{\bar{K},\bar{H},\bar{N}}$; the implicit function theorem then gives
continuous differentiability of $r_{\bar{K},\bar{H},\bar{N}}$ with respect to
its parameters near any
$\big{(}\frac{\bar{K}}{N-1},\frac{\bar{H}}{N-1}\big{)}$ satisfying the ball
condition. If the ball condition is not satisfied, then
$r_{\bar{H},\bar{K},\bar{N}}=\infty$ and the theorem holds trivially. ∎

##  4 Rigidity

###  4.1 The Riemannian curvature-dimension condition

We recall briefly the Riemannian curvature-dimension condition that is a
strengthening of the $\mathrm{CD}(K,N)$ condition and the result of the
combined efforts by several authors [1, 4, 5, 9, 19, 23].

The Cheeger energy $\operatorname{Ch}\colon
L^{2}(\operatorname{m})\rightarrow[0,\infty]$ of a metric measure space
$(X,d,\operatorname{m})$ is defined as

$\displaystyle 2\operatorname{Ch}(f):=\liminf_{\operatorname{Lip}(X)\ni
u_{n}\overset{L^{2}}{\rightarrow}f}\int(\operatorname{Lip}u_{n})^{2}\operatorname{dm},$
(4.1)

where $\operatorname{Lip}(X)$ is the space of Lipschitz functions on
$(X,d,\operatorname{m})$ and
$\operatorname{Lip}u(x):=\limsup\limits_{y\rightarrow
x}\frac{|u(x)-u(y)|}{d(x,y)}$ is the local slope of
$u\in\operatorname{Lip}(X)$. The $L^{2}$-Sobolev space is defined as
$W^{{1,2}}(X)=\\{f\in L^{2}(\operatorname{m})\colon$
$\operatorname{Ch}(f)<\infty\\}$ and equipped with the norm
$\|f\|^{2}:={\|f\|_{\scriptscriptstyle{L^{2}(\operatorname{m})}}^{2}+2\operatorname{Ch}(f)}$
[2, 3].

For $u\in W^{1,2}(X)$ the Cheeger energy can be written as

$\displaystyle 2\operatorname{Ch}(u)=\int_{X}|\nabla u|^{2}\operatorname{dm}$

for a measurable density $|\nabla u|\colon X\rightarrow[0,\infty)$ that is
identified as the minimal weak upper gradient of $u$. For more details about
the minimal weak upper gradients and its characterizations we refer to [2,
14].

######  Definition 4.1 (Riemannian curvature-dimension condition).

A metric measure space $(X,d,\operatorname{m})$ satisfies the Riemannian
curvature-dimension condition $\mathrm{RCD}(K,N)$ if $(X,d,\operatorname{m})$
satisfies the condition $\mathrm{CD}(K,N)$ and $W^{1,2}(X)$ is a Hilbert
space, meaning the $2$-homogeneous Cheeger energy (4.1) satisfies the
parallelogram law:

$\operatorname{Ch}(f+g)+\operatorname{Ch}(f-g)=2\operatorname{Ch}(f)+2\operatorname{Ch}(g).$

When $(X,d,\operatorname{m})$ is an $\mathrm{RCD}$ space, one can introduce a
symmetric bilinear form $\langle\cdot,\cdot\rangle$ on the Sobolev space
$W^{1,2}(X)$ with values in $L^{1}(\operatorname{m})$ via

$\displaystyle(f,g)\in W^{1,2}(X)\times W^{1,2}(X)\mapsto\langle\nabla
f,\nabla
g\rangle:=\frac{1}{4}|\nabla(f+g)|^{2}-\frac{1}{4}|\nabla(f-g)|^{2}\in
L^{1}(\operatorname{m}).$

###  4.2 Volume cone implies metric cone

The following theorem by Gigli and De Philippis will be crucial in the proof
of the rigidity result.

######  Theorem 4.2 (volume cone implies metric cone [17, Theorem 4.1]).

Let $K\\!\in\\!\\{-(N-1),0,N-1\\}$, $N\in[1,\infty)$ and
$(X,d,\operatorname{m})$ an $\mathrm{RCD}(K,N)$ space with
$\operatorname{{spt}}\operatorname{m}=X$. Assume there exists $o\in X$ and
$R>r>0$ such that

$\displaystyle\operatorname{m}(B_{R}(o))=\frac{{\displaystyle\int_{0}^{R}\big{(}\sin_{\frac{K}{N-1}}u\big{)}^{N-1}{\rm
d}u}}{{\displaystyle\int_{0}^{r}\big{(}\sin_{\frac{K}{N-1}}v\big{)}^{N-1}{\rm
d}v}}\operatorname{m}(B_{r}(o)).$ (4.2)

Then exactly one of the following three cases holds:

  * $(1)$

If $\partial B_{R/2}(o)$ contains only one point, then $X$ is isometric to
$[0,\operatorname{diam}_{X}]$ $($or $[0,\infty)$ if $X$ is unbounded$)$ with
an isometry that sends $o$ to $0$ either way. The measure
$\operatorname{m}|_{B_{R}(o)}$ is proportional to
$\big{(}\sin_{\frac{K}{N-1}}^{N-1}x\big{)}{\rm d}x$.

  * $(2)$

If $\partial B_{R/2}(o)$ contains exactly two points then $X$ is a
$1$-dimensional Riemannian manifold, possibly with boundary, and there exists
a bijective, locally distance preserving map from $B_{R}(o)$ to $(-R,R)$ that
sends $o$ to $0$ under which the measure $\operatorname{m}|_{B_{R}(o)}$
becomes proportional to $\big{(}\sin_{\frac{K}{N-1}}^{N-1}|x|\big{)}{\rm d}x$.

  * $(3)$

If $\partial B_{R/2}(o)$ contains more than two points then $N\\!\geq\\!2$ and
there exists an $\mathrm{RCD}(N{-}2,N{-}1)$ space $Z$ with
$\operatorname{diam}_{Z}\leq\pi$ and a local isometry ${U}\colon
B_{R}(o)\rightarrow[0,R)\times_{\sin_{\frac{K}{N-1}}}^{N}Z$ sending $o$ to $0$
that is also a measure preserving bijection.

######  Remark 4.3 (excluding the middle case).

In the second case the conclusion also implies that $N=1$: otherwise $X$ is
locally isomorphic to $(-R,R)$ equipped with a measure proportional to
$\big{(}\sin_{{K}/{N-1}}^{N-1}|x|\big{)}{\rm d}x$. But for $N>1$ this space
does not satisfy the $\mathrm{CD}$ condition because the density of the
reference measure vanishes at $0$. This means the density is not semi-concave
which is a necessary condition for the measure on a $1D$ space to satisfy the
$\mathrm{CD}$ condition.

######  Remark 4.4.

In the proof of Theorem 4.2 Gigli and De Philippis show that the map $U$ has
an inverse ${V}\colon B_{R}(0)\rightarrow B_{R}(o)$ that is also a local
isometry.

###  4.3 Distributional Laplacian and strong maximum principle

We recall the notion of the distributional Laplacian for $\mathrm{RCD}$ spaces
(cf. [12, 23]).

Let $(X,d,\operatorname{m})$ be an $\mathrm{RCD}$ space, and
$\operatorname{Lip}_{c}(\Omega)$ denote the set of Lipschitz functions
compactly supported in an open subset $\Omega\subset X$. A Radon functional
over $\Omega$ is a linear functional
$T:\operatorname{Lip}_{c}(\Omega)\rightarrow\R$ such that for every compact
subset $W$ in $\Omega$ there exists a constant $C_{W}\geq 0$ such that

$\displaystyle|T(f)|\leq{C_{W}}\max_{W}|f|\qquad\forall\,f\in\operatorname{Lip}_{c}(\Omega)\quad\mbox{with}\
\operatorname{{spt}}f\subset W.$ (4.3)

One says $T$ is non-negative if $T(f)\geq 0$ for all
$f\in\operatorname{Lip}_{c}(\Omega)$ satisfying $f\geq 0$.

The classical Riesz–Markov–Kakutani representation theorem says that for every
non-negative Radon functional $T$ from (4.3) there exists a non-negative Radon
measure $\mu_{T}$ such that $T(f)=\int f\,{\rm d}\mu_{T}$ for all
$f\in\operatorname{Lip}_{c}(\Omega)$.

######  Definition 4.5 (nonsmooth Laplacian).

Let $\Omega\subset X$ be an open subset and let $u\in\operatorname{Lip}(X)$.
One says $u$ is in domain of the distributional Laplacian on $\Omega$ provided
there exists a Radon functional $T$ over $\Omega$ such that

$\displaystyle T(f)=\int\langle\nabla u,\nabla
f\rangle\operatorname{dm}\qquad\forall\,f\in\operatorname{Lip}_{c}(\Omega).$

In this case we write $u\in D({\bf\Delta},\Omega)$. If $T$ is represented as a
measure $\mu_{T}$, one writes $\mu_{T}\in{\bf\Delta}u|_{\Omega}$, and if there
is only one such measure $\mu_{T}$ by abuse of notation we will identify
$\mu_{T}$ with $T$ and write $\mu_{T}={\bf\Delta}u|_{\Omega}$.

We also recall that $u\in W_{\rm loc}^{1,2}(\Omega)$ for an open set
$\Omega\subset X$ if and only if for any Lipschitz function $\phi$ with
compact support in $\Omega$ we have $\phi\cdot u\in W^{1,2}(X)$. In
particular, if $u\in\operatorname{Lip}(X)$ then $u\in W_{\rm
loc}^{1,2}(\Omega)$.

######  Remark 4.6 (locality and linearity).

  * (i)

If $u\in D({\bf\Delta},\Omega)$ and $\Omega^{\prime}$ is open in $X$ with
$\Omega^{\prime}\subset\Omega$, then $u\in D({\bf\Delta},\Omega^{\prime})$ and
for $\mu\in{\bf\Delta}u|_{\Omega}$ it follows that
$\mu|_{\Omega^{\prime}}\in{\bf\Delta}u|_{\Omega^{\prime}}$.

  * (ii)

If $u,v\in D({\bf\Delta},\Omega)$, then $u+v\in D({\bf\Delta},\Omega)$ and for
$\mu_{u}\in{\bf\Delta}u|_{\Omega}$ and $\mu_{v}\in{\bf\Delta}v_{\Omega}$ it
follows that $\mu_{u}+\mu_{v}\in{\bf\Delta}(u+v)|_{\Omega}$.

Recall that $u\in W^{1,2}(\Omega)$ is sub-harmonic if

$\displaystyle\int_{\Omega}\\!|\nabla
u|^{2}\operatorname{dm}\leq\\!\int_{\Omega}\\!|\nabla(u+g)|^{2}\operatorname{dm}\\!\\!\\!\qquad\forall\,g\in
W^{1,2}(\Omega)\ \mbox{with}\ g\leq 0\ \mbox{compactly supported in
$\Omega^{\circ}$.}$

One says $u$ is super-harmonic if $-u$ is sub-harmonic, and $u$ is harmonic if
it is both sub- and super-harmonic. The following can be found in [24, Theorem
4.3].

######  Theorem 4.7 (characterizing super-harmonicity).

Let $X$ be an $\mathrm{RCD}(K,N)$ space with $K\in\R$ and $N\in[1,\infty)$,
let $\Omega\subset X$ be open and $u\in W_{\rm loc}^{1,2}(\Omega)$. Then $u$
is super-harmonic if and only if ${u}\in D({\bf\Delta},\Omega)$ and there
exists $\mu\in{\bf\Delta}u|_{\Omega}$ such that $\mu\leq 0$.

The following is [6, Theorem 9.13] (see also [25]):

######  Theorem 4.8 (strong maximum principle).

Let $X$ be an $\mathrm{RCD}(K,N)$ space with $K\in\R$ and $N\in[1,\infty)$,
let $\Omega\subset X$ be a connected open set with compact closure and let
$u\in W_{\rm loc}^{1,2}(\Omega)\cap C(\Omega)$ be sub-harmonic. If there
exists $x_{0}\in\Omega$ such that $u(x_{0})=\max_{\bar{\Omega}}u$ then $u$ is
constant.

Let us recall another result of Cavalletti–Mondino:

######  Theorem 4.9 (Laplacian of a signed distance, [12, Corollary 4.16]).

Let $(X,d,\operatorname{m})$ be a $\mathrm{CD}(K,N)$ space, and $\Omega$ and
$S=\partial\Omega$ as above. Then $d_{S}\in D({\bf\Delta},X\backslash S)$, and
one element of ${\bf\Delta}d_{S}|_{X\backslash S}$ that we also denote with
${\bf\Delta}d_{S}|_{X\backslash S}$ is the Radon functional on $X\backslash S$
given by the representation formula

$\displaystyle({\bf\Delta}d_{S})|_{X\backslash S}=(\log
h_{\alpha})^{\prime}\operatorname{m}|_{X\backslash
S}+\int_{Q}(h_{\alpha}\delta_{a(X_{\alpha})\cap\\{d_{S}<0\\}}-h_{\alpha}\delta_{b(X_{\alpha})\cap\\{d_{S}>0\\}})\,{\rm
d}\mathfrak{q}(\alpha).$

We note that the Radon functional ${\bf\Delta}d_{S}|_{X\backslash S}$ can be
represented as the difference of two measures $[{\bf\Delta}d_{S}]^{+}$ and
$[{\bf\Delta}d_{S}|_{X\backslash S}]^{-}$ such that

$\displaystyle[{\bf\Delta}d_{S}|_{X\backslash S}]^{+}_{\rm
reg}-[{\bf\Delta}d_{S}|_{X\backslash S}]^{-}_{\rm reg}=(\log
h_{\alpha})^{\prime}\quad\operatorname{m}\mbox{-a.e.},$

where $[{\bf\Delta}d_{S}|_{X\backslash S}]^{\pm}_{\rm reg}$ denotes the
$\operatorname{m}$-absolutely continuous part in the Lebesgue decomposition of
$[{\bf\Delta}d_{S}|_{X\backslash S}]^{\pm}$. In particular, $-(\log
h_{\alpha})^{\prime}$ coincides with a measurable function
$\operatorname{m}$-a.e.

To prove the rigidity asserted in Theorem 1.4, we need one more lemma:

######  Lemma 4.10 (Riccati comparison).

Let $u\colon[0,b]\rightarrow\R$ be non-negative and continuous such that
$u^{\prime\prime}+\kappa u\leq 0$ in the distributional sense, $u(0)=1$ and
$u^{\prime}(0)\leq d$. Let $v\colon[0,\bar{b}]\rightarrow\R$ be the maximal
non-negative solution of $v^{\prime\prime}+\kappa v=0$ with $v(0)=1$ and
$v^{\prime}(0)=d$. That is, $v=s_{\kappa,-d}$ from (1.5). Then $\bar{b}\geq b$
and $\frac{{\rm d}^{+}}{{\rm d}t}\log u\leq(\log v)^{\prime}$ on $[0,b)$.

###### Proof.

Note that $v(r)=\cos_{\kappa}(r)+d\sin_{\kappa}(r)$ and
$\bar{b}=r_{\kappa(N-1),-d(N-1),N}$. Then Lemma 3.1 already yields that
$b\leq\bar{b}$ and $u\leq v$ on $[0,b]$. Therefore, without loss of generality
we restrict $v$ to $[0,b]$.

We pick $\varphi\in C^{2}_{c}(\R)$ compactly supported in $(-1,1)$ with
$\int\varphi\,{\rm d}\mathcal{L}^{1}=1$ and define
$\varphi_{\epsilon}(x)=\epsilon\varphi\big{(}\frac{x}{\epsilon}\big{)}$. Let
$\bar{\epsilon}>{0}$ and $\epsilon\in(0,\bar{\epsilon})$, and let
$u_{\epsilon}=\int\varphi_{\epsilon}(t)u(s-t)\,{\rm d}t$ be the mollification
of $u$ by $\varphi_{\epsilon}$. One can check that $u_{\epsilon}$ is well-
defined on $[\bar{\epsilon},b-\bar{\epsilon}]$ and $u_{\epsilon}\in
C^{2}([\bar{\epsilon},b-\bar{\epsilon}])$ satisfies

$\displaystyle u_{\epsilon}^{\prime\prime}+(\kappa+\delta)u_{\epsilon}\leq 0$

in the classical sense with $\delta=\delta(\epsilon)\rightarrow 0$ for
$\epsilon\rightarrow 0$. Since $u$ is continuous, $u_{\epsilon}(t)\rightarrow
u(t)$ for all $t\in[\bar{\epsilon},b-\bar{\epsilon}]$. Moreover,
$u_{\epsilon}^{\prime}(t)\rightarrow u^{\prime}(t)$ for every
$t\in[\bar{\epsilon},b-\bar{\epsilon}]$ where $u$ is differentiable.

Let $v_{\epsilon}\colon[0,\bar{b}_{\epsilon}]\rightarrow[0,\infty)$ be the
maximal positive solution of
$v_{\epsilon}^{\prime\prime}+(\kappa+\delta(\epsilon))v_{\epsilon}=0$ with
$v_{\epsilon}(0)=1$ and $v_{\epsilon}^{\prime}(0)=d$. Since
$\delta(\epsilon)\rightarrow 0$ for $\epsilon\rightarrow 0$ we have
$\bar{b}_{\epsilon}\rightarrow\bar{b}$, $v_{\epsilon}\rightarrow v$ and
$v^{\prime}_{\epsilon}\rightarrow v$ pointwise on $[0,\bar{b}]$ if
$\epsilon\rightarrow 0$.

We pick $\bar{\epsilon}\in(0,b)$ and $t\in[\bar{\epsilon},b-\bar{\epsilon}]$
where $u$ is differentiable. Then

$\displaystyle 0$
$\displaystyle\geq\int_{\bar{\epsilon}}^{t}[v_{\epsilon}(u_{\epsilon}^{\prime\prime}+(\kappa+\delta)u_{\epsilon})-u_{\epsilon}(v_{\epsilon}^{\prime\prime}+(\kappa+\delta)v_{\epsilon})]\,{\rm
d}\mathcal{L}^{1}=\int_{\bar{\epsilon}}^{t}\\{[v_{\epsilon}u_{\epsilon}^{\prime}]^{\prime}-[u_{\epsilon}v_{\epsilon}^{\prime}]^{\prime}\\}\,{\rm
d}\mathcal{L}^{1}$
$\displaystyle=v_{\epsilon}(t)u_{\epsilon}^{\prime}(t)-u_{\epsilon}(t)v_{\epsilon}^{\prime}(t)+u_{\epsilon}(\bar{\epsilon})v_{\epsilon}^{\prime}(\bar{\epsilon})-v_{\epsilon}(\bar{\epsilon})u_{\epsilon}^{\prime}(\bar{\epsilon})$
$\displaystyle\rightarrow
v(t)u^{\prime}(t)-u(t)v^{\prime}(t)+u(\bar{\epsilon})v^{\prime}(\bar{\epsilon})-v(\bar{\epsilon})u^{\prime}(\bar{\epsilon}).$

Since $u$ is semiconcave and continuous on $[0,b]$, the right derivative
$\frac{{\rm d}^{+}}{{\rm d}t}u\colon[0,b]\rightarrow\R\cup\\{\infty\\}$ is
continuous from the right. Hence, for $\bar{\epsilon}\downarrow 0$ and any
$t\in(0,b)$ it follows

$\displaystyle 0\geq v(t)\frac{{\rm d}^{+}}{{\rm
d}t}u(t)-u(t)v^{\prime}(t)+u(0)v^{\prime}(0)-v(0)\frac{{\rm d}^{+}}{{\rm
d}t}u(0)\geq v(t)\frac{{\rm d}^{+}}{{\rm d}t}u(t)-u(t)v^{\prime}(t).$

Hence $\frac{{\rm d}^{+}}{{\rm d}t}\log u=\frac{\frac{{\rm d}^{+}}{{\rm
d}t}u}{u}\leq\frac{v^{\prime}}{v}=(\log v)^{\prime}$ as desired. ∎

We obtain the following improved Laplace comparison statement for distance
functions in $\mathrm{CD}(K,N)$ spaces that may be of interest in its own
right. In the smooth context the result was obtained by Kasue [29, Corollary
2.44] for Riemannian manifolds and by Sakurai [48, Lemma 3.3] for weighted
Riemannian manifolds. For weighted Finsler manifolds that satisfy a lower
Bakry–Émery Ricci curvature bound in the sense of [47] the result seems to be
new.

######  Corollary 4.11 (improved Laplace comparison).

Let $(X,d,\operatorname{m})$ be an essentially nonbranching
$\mathrm{CD}(K^{\prime},N)$ space with $K^{\prime}\in\R$, $N\in(1,\infty)$ and
$\operatorname{{spt}}\operatorname{m}=X$. For $K,\chi\in\R$, let
$\Omega\subset X$ be closed with $\Omega\neq X$, $\operatorname{m}(\Omega)>0$
and $\operatorname{m}(\partial\Omega)=0$ such that $\Omega$ satisfies
$\mathrm{CD}_{r}(K,N)$ and $\partial\Omega=S$ has finite inner curvature.
Assume the inner mean curvature $H_{S}^{-}$ satisfies $H_{S}^{-}\geq(N-1)\chi$
$\operatorname{m}_{S}$-almost everywhere. Then

$\displaystyle({\bf\Delta}d_{\Omega^{c}})|_{\Omega^{\circ}}\leq(N-1)\frac{s^{\prime}_{\frac{K}{N-1},\chi}(d_{\Omega^{c}})}{s_{\frac{K}{N-1},\chi}(d_{\Omega^{c}})}\operatorname{m}|_{\Omega^{\circ}},$

where $s_{\kappa,\lambda}$ was defined in equation (1.5).

###### Proof.

Let $u=d_{S}$ be the signed distance function of $S$. Let
$\\{X_{\alpha}\\}_{\alpha\in Q}$ be the decomposition of $\mathcal{T}_{u}$ and
$\int\operatorname{m}_{\alpha}\,{\rm d}\mathfrak{q}(\alpha)$ be the
disintegration of $\operatorname{m}$ given by Theorem 2.7 and Remark 2.14.
Recall that $m_{\alpha}=h_{\alpha}\mathcal{H}^{1}$ for $\mathfrak{q}$-a.e.
$\alpha\in Q$. We consider $Q^{\dagger}\subset Q$ that has full
$\mathfrak{q}$-measure as defined in Remark 2.13. For every $\alpha\in
Q^{\dagger}$ we have that
$\operatorname{m}_{\alpha}=h_{\alpha}\mathcal{H}^{1}$,
$X_{\alpha,e}=\overline{X}_{\alpha}$ and $h_{\alpha}$ is continuous on
$[a(X_{\alpha}),0]$ by Remark 2.12 and satisfies

$\displaystyle\big{(}h_{\alpha}^{\frac{1}{N-1}}\big{)}^{\prime\prime}+\frac{K}{N-1}h_{\alpha}^{\frac{1}{N-1}}\leq
0\qquad\mbox{on}\ (a(X_{\alpha}),0)\qquad\forall\,\alpha\in Q^{\dagger},$
(4.4)

in the distributional sense. Note that we have the constant $K$ because of
Remark 2.11. As usual we write $h_{\alpha}=h_{\alpha}\circ\gamma_{\alpha}$. We
also have the properties of $h_{\alpha}$ as discussed in Remark 2.10. The
function $r\in[0,-a(X_{\alpha})]\mapsto\tilde{h}_{\alpha}(r):=h_{\alpha}(-r)$
is also continuous and (4.4) is still holds on $(0,-a(X_{\alpha}))$. Recall
that by the lower mean curvature bound the set of $\alpha$’s in $Q$ with
$h_{\alpha}(0)=0$ has $\mathfrak{q}$-measure $0$. Hence $h_{\alpha}(r)>0$ for
$\mathfrak{q}$-a.e. $\alpha$.

Therefore, for $\mathfrak{q}$-a.e. $\alpha\in Q^{\dagger}$ we have that
$\big{[}\tilde{h}_{\alpha}(r)/\tilde{h}_{\alpha}(0)\big{]}^{\frac{1}{N-1}}=:u(r)$
satisfies $u^{\prime\prime}(r)+\frac{K}{N-1}u(r)\leq 0$ in the distributional
sense with $u(0)=1$. Moreover, we have

$\displaystyle\chi(N-1)\leq H^{-}_{S}(\gamma_{\alpha}(0))=\frac{{\rm
d}^{-}}{{\rm d}r}\Big{|}_{r=0}\log h_{\alpha}(r)=-\frac{{\rm d}^{+}}{{\rm
d}r}\Big{|}_{r=0}\log\tilde{h}_{\alpha}(r)$

and therefore

$\displaystyle\frac{{\rm d}^{+}}{{\rm d}r}\Big{|}_{r=0}u(r)=\frac{{\rm
d}^{+}}{{\rm
d}r}\Big{|}_{r=0}\left[\frac{\tilde{h}_{\alpha}(r)}{\tilde{h}_{\alpha}(0)}\right]^{\frac{1}{N-1}}\leq-\chi.$

By Theorem 4.9, $d_{S}\in D({\bf\Delta},X\backslash S)$ and

$\displaystyle({\bf\Delta}d_{S})|_{X\backslash S}=(\log
h_{\alpha})^{\prime}\operatorname{m}|_{X\backslash
S}+\int_{Q}(h_{\alpha}\delta_{a(X_{\alpha})\cap\\{d_{S}{\color[rgb]{0,0,0}<}0\\}}-h_{\alpha}\delta_{b(X_{\alpha})\cap\\{d_{S}{\color[rgb]{0,0,0}>}0\\}})\,{\rm
d}\mathfrak{q}(\alpha).$

Recall that $-d_{S}|_{\Omega^{\circ}}=d_{\Omega^{c}}|_{\Omega^{\circ}}$ and by
locality of the distributional Laplacian $(({\bf\Delta}d_{S})|_{X\backslash
S})|_{\Omega^{\circ}}={\bf\Delta}(d_{S}|_{\Omega^{\circ}})={\bf\Delta}(-d_{\Omega^{c}}|_{\Omega^{\circ}})$.
Hence

$\displaystyle{\bf\Delta}(-d_{\Omega^{c}}|_{\Omega^{\circ}})=(\log
h_{\alpha})^{\prime}\operatorname{m}|_{\Omega^{\circ}}+\int_{Q}{(h_{\alpha}\delta_{a(X_{\alpha})\cap\\{d_{S}<0\\}}-h_{\alpha}\delta_{b(X_{\alpha})\cap\\{d_{S}>0\\}}}\,{\rm
d}\mathfrak{q}(\alpha).$

Any $\gamma_{\alpha}$ for $\alpha\in Q$ that starts inside of $\Omega^{\circ}$
satisfies $\gamma_{\alpha}(b(X_{\alpha}))\in\overline{\Omega^{c}}$. Hence
$\int_{Q}h_{\alpha}\delta_{b(X_{\alpha})\cap\Omega^{\circ}}\,{\rm
d}\mathfrak{q}(\alpha)\allowbreak=0$. Recall that $-(\log
h_{\alpha})^{\prime}(r)=(\log\tilde{h}_{\alpha})^{\prime}(-r)$. It follows
that

$\displaystyle({\bf\Delta}d_{\Omega^{c}})|_{\Omega^{\circ}}=-{\bf\Delta}(-d_{\Omega^{c}})|_{\Omega^{\circ}}\leq-(\log
h_{\alpha})^{\prime}\operatorname{m}|_{\Omega^{\circ}}=(\log\tilde{h}_{\alpha})^{\prime}\operatorname{m}|_{\Omega^{\circ}}.$

In the first equality it seems as one would use linearity of the Laplacian to
pull the minus sign in front of ${\bf\Delta}$. But if one examines the proof
of Theorem 4.9, one can easily observe that this is possible also in
$\mathrm{CD}$ (or even $\mathrm{MCP}$ context) because replacing $d_{S}$ with
$-d_{S}$ just results in reversing the parametrization of the geodesics
$\gamma_{\alpha}$.

Now the corollary follows immediately from the curvature bounds and the
Riccati comparison lemma (Lemma 4.10). ∎

###  4.4 Proof of rigidity (Theorem 1.4)

1\. We assume that $K\in\\{N-1,0,-(N-1)\\}$.

Now $d_{S}|_{{\Omega^{\circ}}}=-d_{\Omega^{c}}|_{{\Omega^{\circ}}}\in
D({\bf\Delta},{\Omega^{\circ}})$ and by Corollary 4.11 we have

$\displaystyle\frac{1}{N-1}{\bf\Delta}d_{\Omega^{c}}|_{{\Omega^{\circ}}}\leq\frac{s_{\frac{K}{N-1},\chi}^{\prime}(d_{\Omega^{c}})}{s_{\frac{K}{N-1},\chi}(d_{\Omega^{c}})}\operatorname{m}|_{\Omega^{\circ}}=\frac{-\frac{K}{N-1}\sin_{\frac{K}{N-1}}d_{\Omega^{c}}-\chi\cos_{\frac{K}{N-1}}d_{\Omega^{c}}}{\cos_{\frac{K}{N-1}}d_{\Omega^{c}}-\chi\sin_{\frac{K}{N-1}}d_{\Omega^{c}}}\operatorname{m}|_{{\Omega^{\circ}}}.$
(4.5)

Assume equality holds in the inradius bound (1.6), meaning
$d_{\Omega^{c}}(p)=r_{K,\chi(N-1),N}<\infty$ for some $p\in{\Omega^{\circ}}$
by Remark 2.4. In particular, there exists a geodesic
$\gamma^{*}\colon[a,b]\rightarrow\Omega$ of length
${L}(\gamma^{*})=d(\gamma^{*}(a),\gamma^{*}(b))={r_{K,\chi(N-1),N}}$ such that
$\gamma^{*}(a)=p$ and $\gamma^{*}(b)\in S:=\partial\Omega$. Moreover
$B_{r_{K,\chi(N-1),N}}(p)\subset{\Omega^{\circ}}$ and

$\displaystyle\frac{1}{N-1}{\bf\Delta}d_{p}|_{{\Omega^{\circ}\backslash\\{p\\}}}\leq\frac{\cos_{\frac{K}{N-1}}d_{p}}{\sin_{\frac{K}{N-1}}d_{p}}\operatorname{m}|_{{\Omega^{\circ}\backslash\\{p\\}}}$
(4.6)

by the Laplace comparison theorem [23, Corollary 5.15] (heuristically the
limit $\chi\to\infty$ in inequality (4.5); see also [12, Theorem 1.1]).

We will add the previous inequalities (4.5) and (4.6). We first note that

$\displaystyle\sin_{\frac{K}{N-1}}(d_{p})\big{[}\cos_{\frac{K}{N-1}}d_{\Omega^{c}}-\chi\sin_{\frac{K}{N-1}}d_{\Omega^{c}}\big{]}=\sin_{\frac{K}{N-1}}(d_{p})\big{[}s_{\frac{K}{N-1},\chi}(d_{\Omega^{c}})\big{]}>0\qquad\mbox{$\operatorname{m}$-a.e.\
on }\Omega^{\circ}.$

This is true because on the one hand
$J_{K,\chi(N-1),N}(d_{\Omega^{c}})=(s_{{K}/{(N-1)},\chi}(d_{\Omega^{c}}))^{N-1}>0$
$\operatorname{m}$-a.e. on $\Omega^{\circ}$. The latter is easy to see using
the disintegration induced by $d_{\Omega^{c}}$ on $\Omega^{\circ}$ and Theorem
1.1. For the other factor recall when $K=N-1>0$ that $d_{p}\leq\pi$ for any
$p\in X$ by the Bonnet–Myers diameter estimate (e.g., [49]) with at most one
point $q\neq p$ where $d_{p}(q)=\pi$ [46].

Adding the inequalities (4.5) and (4.6) and using the linearity of the Laplace
operator yields

$\displaystyle\big{(}\sin_{\frac{K}{N-1}}(d_{p})\big{[}s_{\frac{K}{N-1},\chi}(d_{\Omega^{c}})\big{]}\big{)}\big{|}_{\Omega^{\circ}\backslash\\{p\\}}\frac{{\bf\Delta}(d_{p}+d_{\Omega^{c}})}{N-1}\Big{|}_{{\Omega^{\circ}\backslash\\{p\\}}}$
$\displaystyle\qquad{}\leq\left(\big{[}\cos_{\frac{K}{N-1}}d_{p}\big{]}\big{[}\cos_{\frac{K}{N-1}}d_{\Omega^{c}}\big{]}-\big{[}\sin_{\frac{K}{N-1}}d_{p}\big{]}\frac{K}{N-1}\big{[}\sin_{\frac{K}{N-1}}d_{\Omega^{c}}\big{]}\right)\operatorname{m}|_{{\Omega^{\circ}\backslash\\{p\\}}}$
$\displaystyle\qquad\quad{}-\chi\big{(}\big{[}\sin_{\frac{K}{N-1}}d_{p}\big{]}\big{[}\cos_{\frac{K}{N-1}}d_{\Omega^{c}}\big{]}+\big{[}\cos_{\frac{K}{N-1}}d_{p}\big{]}\big{[}\sin_{\frac{K}{N-1}}d_{\Omega^{c}}\big{]}\big{)}\operatorname{m}|_{{\Omega^{\circ}\backslash\\{p\\}}}$
$\displaystyle\qquad{}=\big{(}{\cos_{\frac{K}{N-1}}(d_{p}+d_{\Omega^{c}})-\chi\sin_{\frac{K}{N-1}}(d_{p}+d_{\Omega^{c}})}\big{)}\operatorname{m}|_{{\Omega^{\circ}\backslash\\{p\\}}}$
$\displaystyle\qquad{}={s_{\frac{K}{N-1},\chi}(d_{p}+d_{\Omega^{c}})}\operatorname{m}|_{\Omega^{\circ}\backslash\\{p\\}}\leq
0.$

The last inequality follows from $r_{K,\chi(N-1),N}\leq
d_{p}+d_{\Omega^{c}}\leq 2r_{K,\chi(N-1),N}$ by the triangle inequality, the
definition of $r_{K,\chi(N-1),N}$, the period $<2r_{K,\chi(N-1),N}$ of the
sinusoid $s_{\frac{K}{N-1},\chi}$, and Theorem 1.1.

Hence $d_{p}+d_{\Omega^{c}}$ on ${\Omega^{\circ}\backslash\\{p\\}}$ is a
super-harmonic function that attains its minimum $r_{K,\chi(N-1),N}$ inside
${\Omega^{\circ}}\backslash\\{p\\}$ along the geodesic $\gamma^{*}$.
Therefore, $d_{p}+d_{\Omega^{c}}=r_{K,\chi(N-1),N}$ on the connected component
$\Omega^{*}$ of $\gamma^{*}$ in ${\Omega^{\circ}}\backslash\\{p\\}$ by the
strong maximum principle (Theorem 4.8).

In particular, it follows that ${\Omega^{*}}\subset B_{r_{K,\chi(N-1),N}}(p)$
and $a(X_{\alpha})=-r_{K,\chi(N-1),N}$ for $\mathfrak{q}$-a.e. $\alpha\in Q$
such that $\mbox{Im}\gamma_{\alpha}$ intersects with $\Omega^{*}$ (and
therefore $\gamma_{\alpha}(I_{\alpha})\subset\Omega^{*}$). Moreover, both
inequalities (4.5) and (4.6) are saturated throughout ${\Omega^{*}}$:

$\displaystyle{\bf\Delta}d_{\Omega^{c}}|_{\Omega^{*}}=(N-1)\frac{-\frac{K}{N-1}\sin_{\frac{K}{N-1}}d_{\Omega^{c}}-\chi\cos_{\frac{K}{N-1}}d_{\Omega^{c}}}{\cos_{\frac{K}{N-1}}d_{\Omega^{c}}-\chi\sin_{\frac{K}{N-1}}d_{\Omega^{c}}}\operatorname{m}|_{{\Omega^{*}}}.$

By Theorem 4.9,

$\displaystyle({\bf\Delta}d_{p})\circ\gamma_{\alpha}(r)=-({\bf\Delta}d_{\Omega^{c}})\circ\gamma_{\alpha}(r)=(\log
h_{\alpha})^{\prime}\circ\gamma_{\alpha}(r)$

for $r\in[-r_{K,\chi(N-1),N},0)$ and $\mathfrak{q}$-a.e. $\alpha\in Q$. Recall
that $(\log h_{\alpha})^{\prime}\circ\gamma_{\alpha}(r)$ is in fact given by
$(\log h_{\alpha})^{\prime}(r)$ for the density $h_{\alpha}$ of
$\operatorname{m}_{\alpha}$ with respect to $\mathcal{L}^{1}$ on
$[a(X_{\alpha}),b(X_{\alpha})]$.

Solving the resulting ODE for $h_{\alpha}$ yields

$h_{\alpha}(r)=h_{\alpha}(0)J_{K,\chi(N-1),N}(-r)\qquad\mbox{for
$r\in[-r_{K,\chi(N-1),N},0)$ and $\mathfrak{q}$-a.e.\ $\alpha\in Q$}.$

Proportionality of $s_{\frac{K}{N-1},\chi}(r_{K,\chi(N-1),N}-r)$ to
$\sin_{\frac{K}{N-1}}(r)$ therefore provides $\lambda>0$ such that

$\displaystyle\operatorname{m}(B_{R}(p)\cap\Omega^{*})=\lambda\int_{0}^{R}(\sin_{\frac{K}{N-1}}r)^{N-1}\,{\rm
d}r\qquad\forall\,R\in[0,r_{K,\chi(N-1),N}].$

Hence $B_{r_{K,\chi(N-1),N}}(p)\cap\Omega^{*}=\Omega^{*}$ is a volume cone in
the sense of (4.2).

2\. We will show that $\Omega^{\circ}\backslash\\{p\\}$ is connected. We argue
by contradiction and outline the idea first. We can construct explicitly a
Wasserstein geodesic between $\delta_{q}$ for a point
$q\in\Omega^{\circ}\backslash(\Omega^{*}\cup\\{p\\})$ and another
$\operatorname{m}$-absolutely continuous measure concentrated in $\Omega^{*}$.
If $\Omega^{\circ}\backslash\\{p\\}$ is not connected but $\Omega^{\circ}$ is,
this Wasserstein geodesic is concentrated on branching geodesics unless the
metric measure space is isomorphic to an interval equipped with a measure that
has a non-negative and semiconcave density w.r.t. $\mathcal{L}^{1}$. Since the
former contradicts the essentially non-branching property of $\mathrm{RCD}$
spaces, the latter must hold. But in this case the volume cone property of
$\operatorname{m}$ on $\Omega^{*}$ contradicts that
$\operatorname{{spt}}\operatorname{m}=X$ and that the $\mathcal{L}^{1}$
density of $\operatorname{m}$ must be positive in the interior of the interval
because of semiconcavity.

Now, we give the precise construction. Assume
$\Omega^{\circ}\backslash\\{p\\}$ is not connected. Then we can pick
${q^{\prime}}\in\Omega^{**}:=(\Omega^{\circ}\backslash\\{p\\})\backslash(\Omega^{*})$.
One can see that $\Omega^{**}$ is open. Since $\Omega^{**}$ is open, it has
positive measure. Thus we can pick $q^{\prime}$ as above such that there is a
unique arclength parameterized geodesic
$r\in[0,d(p,q^{\prime})]\mapsto\hat{\gamma}$ between $p=\gamma(0)$ and
$q^{\prime}$, and such that $\hat{\gamma}$ is inside $\Omega$. Choose
$\delta>0$ small enough such that $B_{2\delta}(p)\subset\Omega$ and set
$q:=\hat{\gamma}(\delta)$.

The set $\Omega^{*}\cap B_{\delta}(p)$ is open and hence has positive
$\operatorname{m}$ measure. Let
$\mu_{0}=\frac{1}{\operatorname{m}(\Omega^{*}\cap
B_{\delta}(p))}\operatorname{m}|_{\Omega^{*}\cap B_{\delta}(p)}$ and
$(\mu_{t})_{t\in[0,1]}$ be the Wasserstein geodesic between $\mu_{0}$ and
$\delta_{q}=\mu_{1}$. Since we assume that
$\Omega^{\circ}=\Omega^{*}\cup\Omega^{**}\cup\\{p\\}$ is connected, but
$\Omega^{*}\cup\Omega^{**}=\Omega^{\circ}\backslash\\{p\\}$ is not connected,
we have that the unique $L^{2}$-Wasserstein geodesic $(\mu_{t})_{t\in[0,1]}$
(for uniqueness see [10] for instance) must be given by
$\mu_{t}=(e_{t})_{\\#}\Pi$ with $\Pi\in\mathcal{P}(\mathcal{G}(X))$ supported
on geodesics that are reparametrized concatenations of the geodesic segments
$r\in[0,\delta]\mapsto\gamma_{\alpha}(-r_{K,\chi(N-1),N}+\delta-r)$ and
$\hat{\gamma}|_{[0,\delta]}$. (Note that any geodesic that connects $q$ with a
point in $\Omega^{*}$ must stay in $B_{2\delta}(p)\subset\Omega$ and since we
assume $\Omega\backslash\\{p\\}$ is not connected, such geodesics must go
through $p$. And for $\operatorname{m}$-a.e. $x$ in $\Omega^{*}\cap
B_{\delta}(p)$ the unique geodesic that connects $x$ with $p$ is given by the
restriction of some geodesic $\gamma_{\alpha}$ for $\alpha\in Q$.)

But $\mathrm{RCD}$ spaces are essentially non-branching (see also Remark 4.12
below). Hence it follows that $\Pi$ must be concentrated on non-branching
geodesics. Hence, there exists a single geodesic $\tilde{\gamma}$ such that
$\Pi$ is concentrated on $\tilde{\gamma}$. In particular $(\Omega^{*}\cap
B_{\delta}(p),\operatorname{m}|_{\Omega^{*}\cap B_{\delta}(p)})$ is isomorphic
to an interval. Then by [35, Theorem 1.1] $(X,d,\operatorname{m})$ is
isomorphic to a $1$-dimensional manifold. Let us assume $p=0\in\R$ and
$q=-d(0,q)=-\delta\in\R$ and $\Omega^{*}\subset(0,\infty)$. By the volume cone
property that we proved in the previous step we have
$\operatorname{dm}|_{\Omega^{*}}(r)$ is proportional to
$\sin_{{K}/{(N-1)}}^{N-1}|r|\,{\rm d}r$. Since the density of
$\operatorname{m}$ w.r.t. $\mathcal{L}^{1}$ must be semi-concave [35, Theorem
1.1] and positive for interior points but must vanish at the origin, we have a
contradiction. Hence $\Omega^{\circ}\backslash\\{p\\}=\Omega^{*}$ is
connected.

3\. We can finish the proof of the main theorem by application of Theorem 4.2.
Recall that Theorem 4.2 provides a measure space isomorphism $U$ between
$(\Omega^{\circ},\operatorname{m}|_{\Omega^{\circ}})$ and a truncated cone
such that $U$ and its inverse are locally distance preserving. This yields
that $U$ must be an isometry with respect to the induced intrinsic distances.
Though this conclusion might be obvious to experts, we provide the proof in
the following.

We observe first that $S=\partial B_{r_{K,\chi(N-1),N}}(p)$ must contain more
than two points: our hypotheses rule out $S\subset\\{pt\\}$, while if $S$
consists of precisely two points, Remark 4.3 asserts $N=1$ and that is also
excluded by assumption.

Hence, only the last case in Theorem 4.2 remains relevant to us. By Remark 4.4
there exist local isometries $U$ and $V$ between
$\Omega^{\circ}=B_{r_{K,\chi(N-1),N}}(p)$ and the ball
$B_{r_{K,\chi(N-1),N}}(o)$ in the corresponding cone that are also measure
preserving bijections.

Now, it is standard knowledge that ${U}$ is an isometry with respect to the
induced intrinsic distances.

Let us be more precise. Set $r_{K,\chi(N-1),N}=R$ and let
$\tilde{d}_{\epsilon}$ be the induced intrinsic distance on
$\bar{B}_{R-\epsilon}(p)$. We denote by $d^{*}$ the cone (or suspension
distance) and by $\tilde{d}^{*}$ and $\tilde{d}^{*}_{\epsilon}$ the induced
intrinsic distances of $B_{R}(o)$ and $\bar{B}_{R-\epsilon}(o)$, respectively.
Then $U$ is an isometry between $\bar{B}_{R-\epsilon}(p)$ and
$\bar{B}_{R-\epsilon}(o)$ with respect to the induced intrinsic distances.

To prove this let $\gamma\colon[0,1]\rightarrow\bar{B}_{R-\epsilon}(p)$ be a
geodesic with respect to $\tilde{d}_{\epsilon}$ between
$x,y\in\bar{B}_{R-\epsilon}(p)$. We can divide $\gamma$ into $k\in\mathbb{N}$
small pieces $\gamma|_{[t_{i-1},t_{i}]}$ with $i=1,\dots,k$ and $t_{0}=0$,
$t_{k}=1$ such that each piece stays inside a small ball that is mapped
isometrically with respect to $d$ via $U$ to a small ball in $B_{R}(o)$. We
obtain

$\displaystyle\sum_{i=1}^{k}d^{*}({U}(\gamma(t_{i-1})),{U}(\gamma(t_{i})))=\sum_{i=1}^{k}d(\gamma(t_{i-1}),\gamma(t_{i}))\leq\sum_{i=1}^{k}\tilde{d}_{\epsilon}(\gamma(t_{i-1}),\gamma(t_{i}))=\tilde{d}_{\epsilon}(x,y).$

The first equality holds because $U$ is an isometry with respect to $d$ and
$d^{*}$ on the small balls that contain $\gamma|_{[t_{i-1},t_{i}]}$. The last
equality holds because $\gamma$ is geodesic with respect to
$\tilde{d}_{\epsilon}$ and the inequality holds because the intrinsic distance
is always equal or larger than $d$ itself.

On the left hand side we can take the supremum with respect to all such
subdivisions $(t_{i})_{i=0,\dots,k-1}$. This yields
$\tilde{d}_{\epsilon}^{*}({U}(x),{U}(y))\leq
L({U}\circ\gamma)\leq\tilde{d}_{\epsilon}(x,y)$ where $L(U\circ\gamma)$ is the
length of the continuous curve $U\circ\gamma$. In particular ${U}\circ\gamma$
is a rectifiable curve(that means has finite length) in
$\bar{B}_{R-\epsilon}(o)$.

We can argue in the same way for the inverse map ${V}$ and obtain that
$U\colon\bar{B}_{R-\epsilon}(p)\rightarrow\bar{B}_{R-\epsilon}(o)$ is an
isometry with respect to the induced intrinsic distances
$\tilde{d}_{\epsilon}$ and $\tilde{d}^{*}_{\epsilon}$.

Finally, we let $\epsilon\rightarrow 0$ and observe that
$\tilde{d}_{\epsilon}\rightarrow\tilde{d}$ on $\bar{B}_{R-\epsilon}(p)$ and
the same for $\tilde{d}^{*}_{\epsilon}$ and $\tilde{d}^{*}$. This finishes the
proof.

######  Remark 4.12.

A deep new result by Qin Deng [18] shows that $\mathrm{RCD}$ spaces are in
fact non-branching. In this case the middle step of the previous proof
simplifies: If $\Omega^{\circ}\backslash\\{p\\}$ is not connected but
$\Omega^{\circ}$ is connected, this yields almost immediately the existence of
a branching geodesic unless the space is isomorphic to an interval equipped
with a measure. The proof of Deng’s result is quite long and involved.
Therefore we provide a proof that only relies on the relatively weak property
that the space is essentially non-branching.

##  Appendix A Substituting measure contraction for lower Ricci bounds

In this appendix we will sketch why the results of Theorem 1.1 also hold when
one replaces the condition $\mathrm{CD}(K,N)$ with the weaker measure
contraction property $\mathrm{MCP}(K,N)$ that was introduced in [46, 49]. We
will not repeat the technical details but focus on necessary modifications for
this setup.

For a proper metric measure space $(X,d,\operatorname{m})$ that is essentially
nonbranching there are several equivalent ways to define the
$\mathrm{MCP}(K,N)$. The following one can be found in [9, Section 9].

######  Definition A.1 (measure contraction property).

Let $(X,d,\operatorname{m})$ be proper and essentially nonbranching. The
measure contraction property $\mathrm{MCP}(K,N)$, $K\in\R$ and
$N\in(1,\infty)$ holds if for every pair
$\mu_{0},\mu_{1}\in\mathcal{P}^{2}(X)$ such that $\mu_{0}$ is
$\operatorname{m}$-absolutely continuous there exists a dynamical optimal plan
$\Pi$ such that $(e_{t})_{\\#}\Pi=\mu_{t}\in\mathcal{P}^{2}(\operatorname{m})$
and

$\displaystyle\rho_{t}(\gamma_{t})^{-\frac{1}{N}}\geq\tau_{K,N}^{(1-t)}({d(\gamma_{0},\gamma_{1})})\rho_{0}(\gamma_{0})^{-\frac{1}{N}}\qquad\mbox{for
}\Pi\mbox{-a.e.\ geodesic $\gamma$},$

where $\mu_{t}=\rho_{t}\operatorname{m}$.

For $\Omega\subset X$ with $\operatorname{m}(\Omega)>0$ the restricted measure
contraction property $\mathrm{MCP}_{r}$ is defined similarly as the condition
$\mathrm{CD}_{r}$ (compare with Definition 2.3).

All the technical results in Section 2.3 still hold when we replace the
condition $\mathrm{CD}(K,N)$ with $\mathrm{MCP}(K,N)$ (cf. [12]). Only in
Theorem 2.9 the density $h_{\alpha}$ of the conditional measure
$\operatorname{m}_{\alpha}$ need not satisfy (2.3) and therefore need not be
semiconcave, although it does remain locally Lipschitz on
$(a(X_{\alpha}),b(X_{\alpha}))$ and extends continuously to the endpoints.
Instead one has only

$\displaystyle
h_{\alpha}(\gamma_{t})^{\frac{1}{N-1}}\geq\sigma_{K,N}^{(1-t)}({|\gamma_{0}-\gamma_{1}|})h_{\alpha}(\gamma_{0})^{\frac{1}{N-1}}$
(A.1)

for every affine function
$\gamma\colon[0,1]\rightarrow[a(X_{\alpha}),b(X_{\alpha})]$ in general where
we consider $h_{\alpha}$ as a continuous function on
$[a(X_{\alpha}),b(X_{\alpha})]$.

Considering $\Omega\subset X$ with $\partial\Omega=S$ and
$\operatorname{m}(S)=0$ then Definition 2.15 for $\operatorname{m}_{S}$
continues to make sense. We also can define the notion of finite inner
curvature of $\Omega$. However, since $h_{\alpha}$ is not semiconcave in
general, the right and the left derivative might not exist for every
$t\in[a(X_{\alpha}),b(X_{\alpha})]$. Therefore, for a continuous function
$f\colon[a,b]\rightarrow\mathbb{R}$ we set

$\displaystyle\frac{{\rm d}^{-}}{{\rm d}t}f(t)=\limsup_{h\uparrow
0}\frac{1}{h}\left[f(t+h)-f(t)\right]\qquad\mbox{for }t\in(a,b].$

We can set up a definition of mean curvature for subsets in $\mathrm{MCP}$
spaces in the following way.

######  Definition A.2 (inner mean curvature revisited in the $\mathrm{MCP}$
setting).

Set $S=\partial\Omega$ and let $\\{X_{\alpha}\\}_{\alpha\in Q}$ be the
disintegration induced by $u:=d_{S}$. Recalling (2.5), we say that $S$ has
finite inner (respectively outer) curvature if
$\operatorname{m}\big{(}B^{\dagger}_{\rm in}\big{)}=0$ (respectively
$\operatorname{m}(B^{\dagger}_{\rm out})=0$). If $S$ has finite inner
curvature we define the inner mean curvature of $S$
$\operatorname{m}_{S}$-almost everywhere as

$\displaystyle p\in S\mapsto H_{S}^{-}(p):=\begin{cases}\dfrac{{\rm
d}^{-}}{{\rm d}r}|_{r=0}\log h_{\alpha}\circ\gamma_{\alpha}&\mbox{if}\
p=\gamma_{\alpha}(0)\in S\cap A^{\dagger},\\\ \infty&\mbox{if}\ p\in
B^{\dagger}_{{\rm out},e}\cap S,\end{cases}$ (A.2)

where we set $\frac{{\rm d}^{-}}{{\rm d}r}\log
h_{\alpha}(\gamma_{\alpha}(0))=-\infty$ if $h_{\alpha}(\gamma_{\alpha}(0))=0$.

######  Theorem A.3 (inscribed radius bounds under $\mathrm{MCP}$).

Let $(X,d,\operatorname{m})$ be an essentially nonbranching
$\mathrm{MCP}(K^{\prime},N)$ space with $K^{\prime}\in\R$, $N\in{(}1,\infty)$
and $\operatorname{{spt}}\operatorname{m}=X$. Consider $K,H\in\R$ such that
$\big{(}\frac{K}{N-1},\frac{H}{N-1}\big{)}$ satisfies the ball condition. Let
$\Omega\subset X$ be closed with $\Omega\neq X$, $\operatorname{m}(\Omega)>0$
and $\operatorname{m}(\partial\Omega)=0$ such that $\Omega$ satisfies the
restricted curvature-dimension condition $\mathrm{MCP}_{r}(K,N)$ for $K\in\R$
and $\partial\Omega=S$ has finite inner curvature. Assume the inner mean
curvature $H_{S}^{-}$ satisfies $H^{-}_{S}\geq H$ $\operatorname{m}_{S}$-a.e.,
where $\operatorname{m}_{S}$ denotes the surface measure. Then

$\displaystyle\operatorname{InRad}\Omega\leq r_{K,H,N},$

where $\operatorname{InRad}\Omega=\sup\limits_{x\in\Omega}d_{\Omega^{c}}(x)$
is the inscribed radius of $\Omega$.

###### Proof.

Let $\eta\colon[0,1]\rightarrow[a(X_{\alpha}),0]$ be an affine function with
$a(X_{\alpha})<\eta_{1}<\eta_{0}:=0$. We note that
$|\eta_{0}-\eta_{1}|=d(\gamma_{\alpha}(\eta_{0}),\gamma_{\alpha}(\eta_{1}))$.
From (A.1) we have

$\displaystyle
h_{\alpha}(\eta(t))^{\frac{1}{N-1}}\geq\sigma_{K/N-1}^{(1-t)}(|\eta_{0}-\eta_{1}|)h_{\alpha}(\eta_{0})^{\frac{1}{N-1}}$

for any $\alpha\in Q^{\dagger}$. It follows

$\displaystyle\frac{{\rm d}^{-}}{{\rm d}r}\Big{|}_{r=\eta_{0}}h_{\alpha}$
$\displaystyle=\limsup_{\eta_{t}\rightarrow\eta_{0}}\frac{1}{\eta_{t}-\eta_{0}}\left[h_{\alpha}(\eta_{t})-h_{\alpha}(\eta_{0})\right]$
$\displaystyle\leq\frac{1}{\eta_{1}-\eta_{0}}\frac{{\rm d}}{{\rm
d}t}\Big{|}_{t=0}\sigma_{K/N-1}^{(1-t)}(|\eta_{0}-\eta_{1}|)^{N-1}h_{\alpha}(\eta_{0}).$

If the inner mean curvature is bounded below by $H$, then

$\displaystyle H\leq\frac{1}{\eta_{1}-\eta_{0}}\frac{{\rm d}}{{\rm
d}t}\Big{|}_{t=0}\sigma_{K/N-1}^{(1-t)}(|\eta_{0}-\eta_{1}|)^{{N-1}}=\frac{-|\eta_{0}-\eta_{1}|}{\eta_{1}-\eta_{0}}(N-1)\frac{\cos_{K/(N-1)}(|\eta_{0}-\eta_{1}|)}{\sin_{K/(N-1)}(|\eta_{0}-\eta_{1}|)}.$

Hence
$\frac{H}{N-1}\leq\frac{\cos_{K/(N-1)}(|\eta_{0}-\eta_{1}|)}{\sin_{K/(N-1)}(|\eta_{0}-\eta_{1}|)}$
for $\mathfrak{q}$-a.e. $\alpha\in Q^{\dagger}$. The sharp Bonnet–Myers
diameter bound [41, 49] for $\mathrm{CD}(K,N)$ spaces yields
$|\eta_{0}-\eta_{1}|<\pi_{\frac{K}{N-1}}$ from (1.2). Thus the denominator
above is non-negative and

$\displaystyle
0\leq\cos_{K/(N-1)}(|\eta_{0}-\eta_{1}|)-\frac{H}{N-1}\sin_{K/(N-1)}(|\eta_{0}-\eta_{1}|).$

Since this expression holds for all $\eta_{1}\in(a(X_{\alpha}),0)$ we conclude
$d(\gamma_{\alpha}(\eta_{0}),\gamma_{\alpha}(\eta_{1}))=|\eta_{0}-\eta_{1}|\leq
r_{K,H,N}$. Otherwise $|\eta_{0}-\eta_{1}|>r_{K,H,N}$ implies that the right
hand side in the last inequality is negative by the definition of $r_{K,H,N}$.

Noting $\eta_{0}=0$, taking $\eta_{1}\rightarrow a(X_{\alpha})$, one can
finish the argument exactly as for $\mathrm{CD}(K,N)$ spaces. ∎

##  Appendix B A different form of mean curvature bound also suffices

Inspired by [13], in this appendix we introduce another new notion of mean
curvature bounded from below which yields Theorems 1.1 and 1.4 without
requiring finite inner curvature of $\Omega$ but assuming that the measure
$\mathfrak{p}_{0}$ in Remark 2.16 is a Radon measure on $Q$. Let $L_{-{\rm
loc}}^{1}(Q,{\rm d}\mathfrak{p}_{0})$ denote the class of
$\mathfrak{p}_{0}$-measurable functions $k\colon
Q\longrightarrow[-\infty,\infty]$ whose negative part $\min\\{0,k\\}$ belongs
to $L^{1}_{\rm loc}(Q,{\rm d}\mathfrak{p}_{0})$, i.e., is locally
$\mathfrak{p}_{0}$-summable.

######  Definition B.1 (backward mean curvature bounded below).

Let $(X,d,\operatorname{m})$ be an essentially nonbranching metric measure
space that satisfies $\mathrm{MCP}$ or $\mathrm{CD}$. Recall the family of
measures $\\{\mathfrak{p}_{t}\\}_{t\in(-\infty,0]}$ on $Q$ given by ${\rm
d}\mathfrak{p}_{t}(\alpha)=h_{\alpha}\circ g(\alpha,t){\rm
d}\mathfrak{q}|_{\mathcal{V}_{t}}(\alpha)$ that we introduced in Remark 2.16,
and its image $\operatorname{m}_{S_{t}}=g(\cdot,t)_{\\#}\mathfrak{p}_{t}$ on
$X$. Recall that $Q$ is constructed as a Borel subset of $X$ and
$(\alpha,t)\mapsto g(\alpha,t)$ is the ray map constructed in that remark.

Then $S=\partial\Omega$ has _backward mean curvature bounded from below by
$k\in L^{1}_{-{\rm loc}}(Q,{\rm d}\mathfrak{p}_{0})$_ if the measure
$\mathfrak{p}_{0}$ is a Radon measure, $h_{\alpha}\circ g(\alpha,0)>0$ for
$\mathfrak{q}$-a.e. $\alpha\in Q$, and

$\displaystyle\frac{{\rm d}^{-}}{{\rm d}t}\Big{|}_{t=0}\int_{Y}{\rm
d}\mathfrak{p}_{t}:=\limsup_{t\uparrow 0}\frac{1}{t}\left(\int_{Y}{\rm
d}\mathfrak{p}_{t}-\int_{Y}{\rm
d}\mathfrak{p}_{0}\right)\geq\int_{Y}k(q)\,{\rm d}\mathfrak{p}_{0}(q)$ (B.1)

for any bounded measurable subset $Y\subset Q$. Moreover, $S$ has _backward-
lower_ mean curvature bounded from below by $k$ if the same inequality holds
when $\limsup$ is replaced by $\liminf$. Heuristically, these limits quantify
the relative rate of change of surface area of the level sets of $d_{S}$, as
when $Y=Q\subset X$ is bounded. We may denote the greatest lower bounds
$k^{\pm}\in L^{1}_{-{\rm loc}}(Q,{\rm d}\mathfrak{p}_{0})$ for the backwards
and backwards-lower mean curvature for $k^{+}$ and $k^{-}$ respectively.

######  Remark B.2.

Since it is not assumed that $\mathfrak{p}_{t}$ for $t<0$ is a Radon measure,
$\int_{Y}{\rm d}\mathfrak{p}_{t}$ can be infinite.

Similarly, when we want to distinguish between upper and lower limits, we
refer to (A.2) as the inner mean curvature, and to the analogous quantity with
$\liminf$ in place of $\limsup$, as the inner-lower mean curvature. In a
$\mathrm{CD}(K,N)$ space, these two notions coincide.

######  Lemma B.3 (backward versus inner mean curvature in $\mathrm{MCP}$).

Let $(X,d,\operatorname{m})$ be an essentially nonbranching
$\mathrm{MCP}(K,N)$ space. Assume that $S=\partial\Omega$ for some Borel set
$\Omega$ that has finite inner curvature in the sense of Definition A.2 and
that $h_{\alpha}\circ g(\alpha,0){\rm d}\mathfrak{q}(\alpha)={\rm
d}\mathfrak{p}_{0}(\alpha)$ is a Radon measure with $h_{\alpha}\circ
g(\alpha,0)>0$ for $\mathfrak{q}$-a.e. $\alpha\in Q$. Then
$\operatorname{m}_{S}$-almost everywhere, the inner mean curvature
$H^{-}_{S}(x)$ of $S$ is bounded from below by $k(x)$ if $S$ has backward mean
curvature bounded from below by $k\circ g(\cdot,0)\in L^{1}_{-{\rm
loc}}(Q,{\rm d}\mathfrak{p}_{0})$.

Conversely, the backward-lower mean curvature is bounded from below by $k\circ
g(\cdot,0)$ if $k\circ g(\cdot,0)\in L^{1}_{-{\rm loc}}(Q,{\rm
d}\mathfrak{p}_{0})$ and there exists $\delta>0$ such that $(i)$ the inner-
lower mean curvature of $S$ is bounded from below by $k(x)$ for
$\operatorname{m}_{S}$-almost every $x\in S$, and $(ii)$
$b(X_{\alpha})>\delta$ holds for $q$-$a.e.\ \alpha\in Q$. Note $(ii)$ is
satisfied if $\Omega$ satisfies a uniform exterior ball condition.

###### Proof.

1. Assume backward mean curvature bounded below by $k\circ g(\cdot,0)\in L^{1}_{-{\rm loc}}(Q,{\rm d}\mathfrak{p}_{0})$. Then, by monotonicity of the right hand side in (B.1) the backward mean curvature is also bounded below by $k^{M}:=\min\\{k,M\\}\in L^{1}_{\rm loc}(Q,{\rm d}\mathfrak{p}_{0})$ for $M>0$ arbitrary.

We can compute for $t<0$ and a bounded measurable set $Y\subset Q$:

$\displaystyle\int_{Y}{\rm d}\mathfrak{p}_{t}-\int_{Y}{\rm
d}\mathfrak{p}_{0}=\int_{Y}\left(1_{\mathcal{V}_{t}}(\alpha)h_{\alpha}\circ
g(\alpha,t)-1_{\mathcal{V}_{0}}(\alpha)h_{\alpha}\circ g(\alpha,0)\right){\rm
d}\mathfrak{q}(\alpha).$

There exists a measurable subset $Q^{*}\subset Q^{\dagger}$ with
$\mathfrak{q}[Q^{\dagger}\setminus Q^{*}]=0$ such that the map
$\mathcal{M}\colon\alpha\in Q^{*}\mapsto-a(X_{\alpha})\in[0,\infty)$ is
measurable (for instance compare with step 1 in the proof of Theorem 7.10 in
[9] or Remark 3.4 in [28]). Then, we consider the family of measurable sets
$Q_{m}=\mathcal{M}^{-1}\big{(}\big{[}\frac{1}{m},m\big{]}\big{)}$ for
$m\in\mathbb{N}$ that satisfy $\bigcup_{m\in\mathbb{N}}Q_{m}=Q^{*}$. As in [9]

$\displaystyle\frac{1}{h_{\alpha}\circ g(\alpha,0)}\frac{1}{r}(h_{\alpha}\circ
g(\alpha,r)-h_{\alpha}\circ
g(\alpha,0))\leq(N-1)\frac{\cos_{-|K|/(N-1)}(-a(X_{\alpha}))}{\sin_{-|K|/(N-1)}(-a(X_{\alpha}))}\leq
C(K,N,m)\\!$

$\forall\,r\in(a(X_{\alpha}),0),\ \forall\,\alpha\in Q_{m}$. Thus Fatou’s
lemma yields

$\displaystyle\int_{{Y}\cap
Q_{m}\cap\mathcal{V}_{0}}\big{(}k^{M}h_{\alpha}\big{)}\circ g(\alpha,0)\,{\rm
d}\mathfrak{q}(\alpha)=\int_{{Y}\cap Q_{m}}k^{M}\circ g(\alpha,0)\,{\rm
d}\mathfrak{p}_{0}(\alpha)$ $\displaystyle\qquad{}\leq\int_{{Y}\cap
Q_{m}}\limsup_{t\uparrow
0}\frac{1}{t}\left(1_{\mathcal{V}_{t}}(\alpha)h_{\alpha}\circ
g(\alpha,t)-1_{\mathcal{V}_{0}}(\alpha)h_{\alpha}\circ g(\alpha,0)\right){\rm
d}\mathfrak{q}(\alpha)$ $\displaystyle\qquad{}\leq\int_{{Y}\cap
Q_{m}}\limsup_{t\uparrow
0}\frac{1}{t}\left(1_{\mathcal{V}_{t}\cap\mathcal{V}_{0}}(\alpha)h_{\alpha}\circ
g(\alpha,t)-1_{\mathcal{V}_{0}}(\alpha)h_{\alpha}\circ g(\alpha,0)\right){\rm
d}\mathfrak{q}(\alpha)$ $\displaystyle\qquad{}=\int_{{Y}\cap
Q_{m}\cap\mathcal{V}_{0}}\frac{{\rm d}^{-}}{{\rm
d}t}\Big{|}_{t=0}h_{\alpha}\circ g(\alpha,t)\,{\rm d}\mathfrak{q}(\alpha)$

for any bounded measurable $Y\subset Q$. It follows that

$\displaystyle\big{(}k^{M}h_{\alpha}\big{)}\circ g(\alpha,0)\leq\frac{{\rm
d}^{-}}{{\rm d}t}\Big{|}_{t=0}h_{\alpha}\circ g(\alpha,t)\qquad\mbox{for
}\mathfrak{q}\mbox{-a.e. }\alpha\in\mathcal{V}_{0}.$ (B.2)

We assumed $h_{\alpha}\circ g(\alpha,0)>0$ for $\mathfrak{q}$-almost every
$\alpha\in\mathcal{V}_{0}$. Therefore, it follows that

$\displaystyle\int_{{Y}\cap\mathcal{V}_{0}}\big{(}k^{M}h_{\alpha}\big{)}\circ
g(\alpha,0)\,{\rm d}\mathfrak{q}(\alpha)$
$\displaystyle\leq\int_{{Y}\cap\mathcal{V}_{0}}\frac{{\rm d}^{-}}{{\rm
d}t}\Big{|}_{t=0}h_{\alpha}\circ g(\alpha,t)\,{\rm d}\mathfrak{q}(\alpha)$
$\displaystyle=\int_{{Y}\cap\mathcal{V}_{0}}\frac{{\rm d}^{-}}{{\rm
d}t}\Big{|}_{t=0}\log(h_{\alpha}\circ g(\alpha,t))h_{\alpha}\circ
g(\alpha,0)\,{\rm d}\mathfrak{q}(\alpha).$

Now, we recall that
$\mathcal{V}_{0}\subset\mathfrak{Q}\big{(}A^{\dagger}\big{)}\cup B_{\rm
in}^{\dagger}$ with $\mathfrak{q}\big{(}\mathfrak{Q}\big{(}A^{\dagger}\cup
B_{\rm in}^{\dagger}\big{)}\backslash\mathcal{V}_{0}\big{)}=0$ and
$\operatorname{m}\big{(}B^{\dagger}_{\rm
in}\big{)}=\mathfrak{q}\big{(}\mathfrak{Q}\big{(}B^{\dagger}_{\rm
in}\big{)}\big{)}=0$ (because we assume finite inner curvature). Moreover
$g(\alpha,t)=\gamma_{\alpha}(t)$, $h_{\alpha}\circ
g(\alpha,t)=h_{\alpha}\circ\gamma_{\alpha}(t)=h_{\alpha}(t)$ and
$\alpha=\gamma_{\alpha}(0)$ if $\alpha\in\mathfrak{Q}(A)$. Hence
$\operatorname{m}_{S}=\operatorname{m}_{S_{0}}=g(\cdot,0)_{\\#}\mathfrak{p}_{0}$
and

$\displaystyle\int_{Y}k^{M}\operatorname{dm}_{S}$
$\displaystyle=\int_{{Y}\cap\mathfrak{Q}(A^{\dagger})}k^{M}\circ\gamma_{\alpha}(0)h_{\alpha}(0)\,{\rm
d}\mathfrak{q}(\alpha)$
$\displaystyle\leq\int_{{Y}\cap\mathfrak{Q}(A^{\dagger})}H^{-}_{S}\circ\gamma_{\alpha}(0)h_{\alpha}(0)\,{\rm
d}\mathfrak{q}(\alpha)=\int_{Y}H^{-}_{S}\operatorname{dm}_{S},$

and consequently $k^{M}\leq H_{S}^{-}$ $\operatorname{m}_{S}$-almost
everywhere. Letting $M\rightarrow\infty$ yields $k\leq H_{S}^{-}$
$\operatorname{m}_{S}$-almost everywhere.

2. Now assume for some $k\circ g(\cdot,0)\in L^{1}_{-{\rm loc}}(Q,{\rm d}\mathfrak{p}_{0})$ that $\operatorname{m}_{S}$-a.e. the inner-lower mean curvature is bounded from below by $k$. We also assume that $b(X_{\alpha})>\delta>0$ for $\mathfrak{q}$-almost every $\alpha\in Q$ and some $\delta>0$. This implies $B_{\rm in}=\varnothing$ and hence $\operatorname{m}_{S_{0}}=\operatorname{m}_{S}$. Recall from the proof of Lemma 2.21 that this is true if $\Omega$ satisfies a uniform exterior ball condition.

Again as in [9]

$\displaystyle-C(K,N,\delta)\leq-(N-1)\frac{\cos_{-|K|/(N-1)}(b(X_{\alpha})-a(X_{\alpha}))}{\sin_{-|K|/(N-1)}(b(X_{\alpha})-a(X_{\alpha}))}\leq\frac{(h_{\alpha}\circ
g(\alpha,r)-h_{\alpha}\circ g(\alpha,0))}{r\ \cdot\ h_{\alpha}\circ
g(\alpha,0)}$

$\forall\,r\in(a(X_{\alpha}),b(X_{\alpha}))$.

Using Fatou’s lemma again, it follows that

$\displaystyle\int_{Y}k\circ\gamma_{\alpha}(0)\,{\rm
d}\mathfrak{p}_{0}(\alpha)$
$\displaystyle=\int_{Y\cap\mathcal{V}_{0}}(kh_{\alpha})\circ g(\alpha,0)\,{\rm
d}\mathfrak{q}(\alpha)=\int_{Y\cap\mathfrak{Q}(A^{\dagger})}k(\gamma_{\alpha}(0))h_{\alpha}(0)\,{\rm
d}\mathfrak{q}(\alpha)$
$\displaystyle\leq\int_{Y\cap\mathcal{V}_{0}}\liminf_{t\uparrow 0}\frac{\log
h_{\alpha}(t)-\log h_{\alpha}(0)}{t}h_{\alpha}(0)\,{\rm
d}\mathfrak{q}(\alpha)$ $\displaystyle\leq\liminf_{t\uparrow
0}\int_{Y}\frac{1}{t}\left(1_{\mathcal{V}_{t}}(\alpha)h_{\alpha}(t)-1_{\mathcal{V}_{0}}(\alpha)h_{\alpha}(0)\right){\rm
d}\mathfrak{q}(\alpha)$ $\displaystyle=\liminf_{t\uparrow
0}\frac{1}{t}\left(\int_{Y}{\rm d}\mathfrak{p}_{t}-\int_{Y}{\rm
d}\mathfrak{p}_{0}\right),$

for any bounded measurable set $Y\subset Q$. Thus the backward-lower mean
curvature is bounded below by $k\circ g(\cdot,0)$ as desired. ∎

######  Corollary B.4 (backward versus inner mean curvature in $\mathrm{CD}$).

Let $(X,d,\operatorname{m})$ be an essentially nonbranching $\mathrm{CD}(K,N)$
space. Assume that $S=\partial\Omega$ for some Borel set $\Omega$ satisfies a
uniform external ball condition and that $h_{\alpha}\circ g(\alpha,0){\rm
d}\mathfrak{q}(\alpha)={\rm d}\mathfrak{p}_{0}(\alpha)$ is a Radon measure
with $h_{\alpha}\circ g(\alpha,0)>0$ for $\mathfrak{q}$-a.e. $\alpha\in Q$.
Then $S$ has a backwards mean curvature bound $k\in L^{1}_{-{\rm loc}}(Q,{\rm
d}\mathfrak{p}_{0})$ from below if and only if $S$ has inner mean curvature
$H^{-}_{S}\in L^{1}_{-{\rm loc}}(S,\operatorname{dm}_{S})$. When either holds,
then $\mathfrak{p}_{0}$-almost everywhere on $Q$, the backward and backward-
lower mean curvatures $k^{\pm}$ of $S$ both coincide with $H^{-}_{S}\circ
g(\cdot,0)$.

###### Proof.

Recall that the uniform external ball condition assumed implies $S$ has finite
inner curvature by Lemma 2.21, and that
$\operatorname{m}_{S}=\operatorname{m}_{S_{0}}$. Let $H^{-}_{S}$ denote the
inner mean curvature of $S$, which agrees with its inner-lower mean curvature
$\operatorname{m}_{S}$-a.e. due to the semiconcavity of
$h_{\alpha}^{1/{(N-1)}}$ in $\mathrm{CD}(K,N)$ spaces.

If $S$ has a backwards mean curvature bound $k\circ g(\cdot,0)\in L^{1}_{-{\rm
loc}}(Q,{\rm d}\mathfrak{p}_{0})$ from below, it admits a greatest such bound
$k^{+}\circ g(\cdot,0)$. Lemma B.3 asserts $H^{-}_{S}\ \circ g(\cdot,0)\geq
k^{+}\circ g(\cdot,0)$ holds $\mathfrak{p}_{0}$-a.e., which implies
$H^{-}_{S}\in L^{1}_{-{\rm loc}}(S,\operatorname{dm}_{S})$.

Conversely, if $S$ has inner mean curvature $H^{-}_{S}\in L^{1}_{-{\rm
loc}}(S,\operatorname{dm}_{S})$, then Lemma B.3 asserts $S$ has backwards-
lower mean curvature $k^{-}\circ g(\cdot,0)\geq H^{-}_{S}\circ g(\cdot,0)\in
L^{1}_{-{\rm loc}}(Q,{\rm d}\mathfrak{p}_{0})$. We conclude $k^{-}\circ
g(\cdot,0)$ is also a backwards (i.e., backwards-upper) mean curvature lower
bound for $S$.

Since $k^{+}\circ g(\cdot,0)\geq k^{-}\circ g(\cdot,0)$ by definition, in
either (and hence both) cases above we conclude equalities hold
$\mathfrak{p}_{0}$-a.e. in all three of the inequalities preceding, to
conclude the proof. ∎

We state a theorem under $\mathrm{MCP}$. The corresponding statement for
$\mathrm{CD}$ then follows since $\mathrm{CD}$ implies $\mathrm{MCP}$ for
essentially nonbranching proper metric measure spaces.

######  Theorem B.5 (inradius bounds under backward mean curvature bounded
below).

Let $(X,d,\operatorname{m})$ be an essentially nonbranching
$\mathrm{MCP}(K,N)$ space with $K\in\R$, $N\in{(}1,\infty)$ and
$\operatorname{{spt}}\operatorname{m}=X$. Let $\Omega\subset X$ be closed with
$\Omega\neq X$, $\operatorname{m}(\Omega)>0$ and
$\operatorname{m}(\partial\Omega)=0$. Assume $\partial\Omega=S$ has backward
mean curvature bounded from below by $H\in\mathbb{R}$. Then

$\displaystyle\operatorname{InRad}\Omega\leq r_{K,H,N}.$

###### Proof.

As in the previous appendix we have

$\displaystyle
h_{\alpha}(t)^{\frac{1}{N-1}}\geq\sigma_{K/N-1}^{(1-\frac{t}{a(X_{\alpha})})}(-a(X_{\alpha}))h_{\alpha}(0)^{\frac{1}{N-1}}$

$\mbox{for any }t\in(a(X_{\alpha}),0)\mbox{and any }\alpha\in Q^{\dagger}$.
Therefore, it follows that

$\displaystyle\frac{{\rm d}^{-}}{{\rm d}t}\Big{|}_{t=0}h_{\alpha}\circ
g(\alpha,t)$ $\displaystyle=\limsup_{t\uparrow
0}\frac{1}{t}\left(h_{\alpha}(g(\alpha,t))-h_{\alpha}(g(\alpha,0))\right)$
$\displaystyle\leq\frac{{\rm d}^{-}}{{\rm
d}t}\Big{|}_{t=0}\sigma_{K/N-1}^{\left(\frac{a(X_{\alpha})-t}{a(X_{\alpha})}\right)}(-a(X_{\alpha}))^{N-1}h_{\alpha}(g(\alpha,0)).$

Since the backward mean curvature is bounded below by $H$, for $Y\subset Q$
bounded and measurable it follows that

$\displaystyle H\int_{{Y}\cap\mathcal{V}_{0}}h_{\alpha}\circ g(\alpha,0)\,{\rm
d}\mathfrak{q}(\alpha)\leq\int_{{Y}\cap\mathcal{V}_{0}}\frac{{\rm d}^{-}}{{\rm
d}t}\Big{|}_{t=0}h_{\alpha}\circ g(\alpha,t)\,{\rm d}\mathfrak{q}(\alpha).$

We obtain the inequality (B.2) exactly as in the beginning of step 1 of the
proof of Lemma B.3. By the definition of backward-lower mean curvature bounds
we have $h_{\alpha}(0)=h_{\alpha}\circ g(\alpha,0)>0$ for
$\mathfrak{q}$-almost every $\alpha$. Hence

$\frac{H}{N-1}\leq\frac{1}{N-1}\frac{{\rm d}^{-}}{{\rm d}t}\Big{|}_{t=0}\log
h_{\alpha}\circ
g(\alpha,t)\leq\frac{\cos_{K/(N-1)}(-a(X_{\alpha}))}{\sin_{K/(N-1)}({-a(X_{\alpha})})}$

for $\mathfrak{q}$-a.e. $\alpha\in\mathcal{V}_{0}=\mathfrak{Q}(A\cup B_{\rm
in})$.

At this point it is clear that we can finish the proof as in Theorem A.3. ∎

######  Theorem B.6 (rigidity under backward mean curvature bounded from
below).

Let $(X,d,\operatorname{m})$ be $\mathrm{RCD}(K,N)$ for $K\in\R$ and
$N\in(1,\infty)$ and let $\Omega\subset X$ be compact with $\Omega\neq X$,
$\operatorname{m}(\Omega)>0$, connected and non-empty interior
$\Omega^{\circ}$ and $\operatorname{m}(\partial\Omega)=0$. We assume that
$K\in\\{N-1,0,-(N-1)\\}$, $\partial\Omega=S\neq\\{pt\\}$ and $S$ has backward
mean curvature bounded below by $\chi(N-1)\in\mathbb{R}$. Then, there exists
$x\in X$ such that

$d_{S}(x)=\operatorname{InRad}\Omega=r_{K,\chi(N-1),N}$

if and only if $r_{K,\chi(N-1),N}<\infty$ and there exists an
$\mathrm{RCD}(N-2,N-1)$ space $Y$ such that
$\big{(}\Omega^{\circ},\tilde{d}_{\Omega^{\circ}}\big{)}$ is isometric to
$\big{(}B_{r_{K,\chi(N-1),N}}(0),\tilde{d}\big{)}$ in
$\tilde{I}_{\frac{K}{N-1}}\times_{\sin_{K/(N-1)}}^{N-1}Y$, where
$\tilde{d}_{\Omega}$ and $\tilde{d}$ are the induced intrinsic distances of
$\Omega^{\circ}$ and ${B}_{r_{K,\chi(N-1),N}}(0)$, respectively.

###### Proof.

In the end of the proof of Theorem B.5 we obtained $H\leq\frac{{\rm
d}^{-}}{{\rm d}t}\big{|}_{t=0}\log h_{\alpha}\circ g(\alpha,t)$ for
$\mathfrak{q}$-a.e. $\alpha\in\mathcal{V}_{0}$ with $H=\chi(N-1)$, so in
particular for $\alpha\in\mathfrak{Q}(B_{\rm in})$. Then using the Riccati
comparison and the maximum principle we can follow verbatim the same proof as
in Section 4. ∎

### Acknowledgements

The authors are grateful to Yohei Sakurai for directing us to the work of
Kasue, and to the anonymous referees for very constructive comments. AB is
supported by the Dutch Research Council (NWO) – Project number
VI.Veni.192.208. CK is funded by the Deutsche Forschungsgemeinschaft (DFG) –
Projektnummer 396662902, “Synthetische Krümmungsschranken durch Methoden des
optimal Transports”. RM’s research is supported in part by NSERC Discovery
Grants RGPIN–2015–04383 and 2020–04162. EW’s research is supported in part by
NSERC Discovery Grant RGPIN-2017-04896.

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