# Absolute continuity of Wasserstein barycenters on manifolds with a lower
Ricci curvature bound

Jianyu MA  Institut de Mathématiques de Toulouse

(October 20, 2023)

###### Abstract

Given a complete Riemannian manifold $M$ with a lower Ricci curvature bound,
we consider barycenters in the Wasserstein space $\mathcal{W}_{2}(M)$ of
probability measures on $M$. We refer to them as Wasserstein barycenters,
which by definition are probability measures on $M$. The goal of this article
is to present a novel approach to proving their absolute continuity. We
introduce a new class of displacement functionals exploiting the Hessian
equality for Wasserstein barycenters. To provide suitable instances of such
functionals, we revisit Souslin space theory, Dunford-Pettis theorem and the
de la Vallée Poussin criterion for uniform integrability. Our method shows
that if a probability measure $\mathbb{P}$ on $\mathcal{W}_{2}(M)$ gives mass
to absolutely continuous measures on $M$, then its unique barycenter is also
absolutely continuous. This generalizes the previous results on compact
manifolds by Kim and Pass [31].

00footnotetext:
Keywords: Optimal transports, Riemannian manifolds, Wasserstein barycenters,
Ricci curvature bounds, Weak compactness in $L^{1}$

######

  1. 1 Introduction
  2. 2 Wasserstein barycenters
    1. 2.1 Notation and definitions
    2. 2.2 Construction, existence and uniqueness of Wasserstein barycenters
    3. 2.3 The absolute continuity of Wasserstein barycenters of finitely many measures
      1. 2.3.1 $c$-concave functions
      2. 2.3.2 Lipschitz continuous optimal transport maps of Wasserstein barycenters
      3. 2.3.3 Proof of absolute continuity
  3. 3 Hessian equality for Wasserstein barycenters
    1. 3.1 Local semi-concavity
    2. 3.2 Approximate differentiability
    3. 3.3 Hessian of semi-concave function
    4. 3.4 Differentiating optimal transport maps
    5. 3.5 Proof of Hessian equality
  4. 4 Lower Ricci curvature bounds and displacement functionals
  5. 5 Proof of our main result
    1. 5.1 Wasserstein barycenters’ absolute continuity by approximation
    2. 5.2 Compactness using Souslin space theory
    3. 5.3 Final step of the proof

##  1 Introduction

Barycenter is the notion of mean for probability measures on metric spaces.
Given a probability measure $\mu$ on the Euclidean space $\mathbb{R}^{m}$, if
its first and second moments are finite, then its mean
$\int_{\mathbb{R}^{m}}x\operatorname{d}\mu(x)$ can be equivalently defined as
the unique point where the infimum
$\inf_{y\in\mathbb{R}^{m}}\int_{\mathbb{R}^{m}}\|y-x\|^{2}\operatorname{d}\mu(x)$
is reached. This formulation in terms of minimization and metric is still
valid for general metric spaces, and it leads to our definition of barycenter
(see 2.1). It is worth noting that barycenters’ existence is not guaranteed _a
priori_ for general metric spaces. We restrict our discussions to proper
metric spaces to ensure the existence of barycenters.

Wasserstein spaces are metric spaces extensively studied in the field of
optimal transport theory. Their geometric properties have gained constant
attention. By Wasserstein barycenter we mean a barycenter of some probability
measure on a given Wasserstein space. In the simplest case, given two measures
$\mu,\nu$ in the Wasserstein space $\mathcal{W}_{2}(M)$ over a Riemannian
manifold $M$, all minimal geodesics from $\mu$ to $\nu$ are made of
barycenters of $(1-\lambda)\delta_{\mu}+\lambda\,\delta_{\nu}$ with $\lambda$
varying in $[0,1]$. Their absolute continuity (in possibly generalized
settings) was previously studied as the regularity of displacement in [35, 6,
19, 21, 49]. A more general case was first studied by Agueh and Carlier [1],
where barycenters of $\mathbb{P}:=\sum_{i=1}^{n}\lambda_{i}\,\delta_{\mu_{i}}$
on the Wasserstein space $(\mathcal{W}_{2}(\mathbb{R}^{m}),W_{2})$ were
considered. In this setting, Wasserstein barycenters are solutions to the
following minimization problem:

$\min_{\nu}\sum_{i=1}^{n}\lambda_{i}\,W_{2}(\nu,\mu_{i})^{2},\quad\text{for
}\nu\in\mathcal{W}_{2}(\mathbb{R}^{m}).$

They proved barycenters’ existence constructively using a dual formulation and
showed that if at least one of $\mu_{i}$’s is absolutely continuous with
bounded density function, then the unique barycenter is also absolutely
continuous. Kim and Pass [31] conducted the same study for Wasserstein
barycenters on compact Riemannian manifolds $M$ with similar conclusions.
Their generalization is applicable to general probability measures
$\mathbb{P}$ on $\mathcal{W}_{2}(M)$ that give mass to the set of absolutely
continuous measures with a uniform upper density bound. The absolute
continuity of Wasserstein barycenters plays an indispensable role in their
study of Jensen’s type inequalities for Wasserstein barycenters. There is also
a generalization [28] of Agueh and Carlier’s results to compact Alexandrov
spaces with curvature bounded below.

When $\mathbb{P}$ has the form $\sum_{i=1}^{n}\lambda_{i}\,\delta_{\mu_{i}}$
with $\mu_{1}$ absolutely continuous, Kim and Pass’ proof of the absolute
continuity of the (unique) barycenter $\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\mu}$ of $\mathbb{P}$ remains valid for non-compact manifolds $M$. For a
general measure $\mathbb{P}$ giving mass to absolutely continuous measures,
the strategy is to approximate $\mathbb{P}$ with finitely supported measures
$\mathbb{P}_{j}$ whose barycenters $\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\mu}_{j}$ are already shown to be absolutely continuous. Thanks to the law
of large numbers for Wasserstein barycenters (2.2), $\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\mu}_{j}$ converges to $\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\mu}$ weakly. However, this is not sufficient to ensure that
$\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\mu}$ is also absolutely continuous. To overcome this difficulty, Kim and
Pass [31] imposed a uniform upper density bound on $\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\mu}_{j}$’s, which forced them to include the assumption on $\mathbb{P}$.

In our work, instead of following their quantitative approach, we seek for
proper integral functionals $F$ on $\mathcal{W}_{2}(M)$ that admit finite
values only for absolutely continuous measures. The continuity of these
functionals has been studied in various sources, including [13], [49, Theorem
29.20], [41, Chapter 7], and [2, Chapter 15]. We summarize their assumptions
and conclusions in 5.2. Additionally, we aim to control the value of $F$ at
$\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\mu}_{j}$ by those at the support of $\mathbb{P}_{j}$, which enables us to
use the convergence $\mathbb{P}_{j}\rightarrow\mathbb{P}$ effectively. Classic
references, such as Villani’s monograph [49], focus on the $\lambda$-convexity
of $F$, a widely studied property that would satisfy our requirements if we
tolerate some independent constants in its inequality expression of convexity
(4.2). Functionals defined in this way generalize the entropy functional
$f\cdot\operatorname{Vol}\mapsto\int_{M}f\log
f\operatorname{d}\operatorname{Vol}$, which is an important example in the
study of synthetic treatment of Ricci curvature lower bounds developed in [34,
43, 44]. 4.2 reveals how Ricci curvature affects the properties of Wasserstein
barycenters and suggests possible extensions of our current work to general
metric measures spaces.

The methodology previously described leads us to 5.4 on the absolute
continuity of Wasserstein barycenters, where an extra assumption on
$\mathbb{P}$ is needed. With the help of a generalized de la Vallée Poussin
criterion (5.9), this assumption can be further simplified: we ask that
$\mathbb{P}$ gives mass to a compact subset in some weak topology of
absolutely continuous measures. Although this topology is barely mentioned in
the literature of optimal transport, it generates the same Borel sets as the
topology induced by the Wasserstein metric according to the theory of Souslin
space. This helps us to state our main result with a natural assumption on
$\mathbb{P}$:

######  Theorem.

Let $M$ be a complete Riemannian manifold with a lower Ricci curvature bound.
If a probability measure $\mathbb{P}\in\mathcal{W}_{2}(\mathcal{W}_{2}(M))$
gives mass to the set of absolutely continuous probability measures on $M$,
then its unique barycenter is absolutely continuous.

### Structure of the paper

In Section 2, we introduce notation and definitions for Wasserstein
barycenters, and then extend Kim and Pass’ proof of their absolute continuity
to non-compact manifolds. In Section 3, we present the Hessian equality for
Wasserstein barycenters (3.13), which is used in Section 4 to justify our
displacement functionals (4.2). Section 5 primarily concerns Polish spaces,
and we use the Souslin space theory to provide appropriate instances of the
previously defined displacement functionals. Our main result, 5.1, is a
consequence of the intermediate result 5.4 after proving several auxiliary
results.

##  2 Wasserstein barycenters

###  2.1 Notation and definitions

######  Definition 2.1 (Barycenter).

Let $(E,d)$ be a metric space and let $\mu$ be a probability measure on $E$
such that $\int_{E}d(x_{0},y)^{2}\operatorname{d}\mu(y)<\infty$ for some point
$x_{0}\in E$. We call $z\in E$ a barycenter of $\mu$ if

$\int_{E}d(z,y)^{2}\operatorname{d}\mu(y)=\min_{x\in
E}\int_{E}d(x,y)^{2}\operatorname{d}\mu(y).$

A metric space is proper if its bounded closed subsets are also compact.
Barycenters always exist in proper spaces since a minimizing sequence is
bounded and thus pre-compact. We refer to Ohta [37] for more details and some
other properties of barycenters in a proper space.

A proper space $(E,d)$ is complete and separable, so are the Wasserstein
spaces built over it. In this article, we consider the ($2$-)Wasserstein space
$(\mathcal{W}_{2}(E),W_{2})$ of probability measures on $E$ with

$\displaystyle\mathcal{W}_{2}(E):$ $\displaystyle=\left\\{\mu\text{ is a
probability measure on }E\,\big{|}\,\exists\,x_{0}\in
E,\int_{E}d(x_{0},y)^{2}\operatorname{d}\mu(y)<\infty\right\\},$
$\displaystyle W_{2}(\mu,\nu)^{2}:$
$\displaystyle=\inf_{\pi\in\Pi(\mu,\nu)}\int_{E\times
E}d(x,y)^{2}\operatorname{d}\pi(x,y),$ (1)

where $\Pi(\mu,\nu)$ is the set of probability measures on $E\times E$ with
marginals $\mu$ and $\nu$. The infimum in (1) is always reached by some
measure $\pi\in\Pi(\mu,\nu)$, and we call it an optimal transport plan between
$\mu$ and $\nu$.

Since Wasserstein spaces are complete and separable, we can construct the
Wasserstein space $(\mathcal{W}_{2}(\mathcal{W}_{2}(E)),\mathbb{W}_{2})$ over
the Wasserstein space $(\mathcal{W}_{2}(E),W_{2})$. Symbols $W_{2}$ and
$\mathbb{W}_{2}$ will always denote Wasserstein metrics in the rest of the
paper. A Wasserstein space $\mathcal{W}_{2}(E)$ is not proper unless the base
space $E$ is compact [4, Remark 7.19]. Classic references on this topic are
[49], [41], and [48].

As mentioned before, Wasserstein barycenters are barycenters of measures on
Wasserstein spaces. We refer to the following result by Le Gouic and Loubes as
the law of large numbers for Wasserstein barycenters since we can set
$\mathbb{P}_{j}$ to be empirical measures for the law $\mathbb{P}$.

######  Theorem 2.2 (Law of large numbers for Wasserstein barycenters, [33]).

Let $(E,d)$ be a proper metric space. Fix a probability measure
$\mathbb{P}\in\mathcal{W}_{2}(\mathcal{W}_{2}(E))$ on $\mathcal{W}_{2}(E)$.
Given a sequence of measures
$\mathbb{P}_{j}\in\mathcal{W}_{2}(\mathcal{W}_{2}(E))$ with their
corresponding barycenters $\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\mu}_{j}\in\mathcal{W}_{2}(E)$, if
$\mathbb{W}_{2}(\mathbb{P}_{j},\mathbb{P})\rightarrow 0$, then
$W_{2}(\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\mu}_{j},\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\mu})\rightarrow 0$ for some barycenter $\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\mu}$ of $\mathbb{P}$ up to extracting a subsequence of $\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\mu}_{j}$.

For two topological spaces $E_{1}$ and $E_{2}$, we denote by $p_{1}$ and
$p_{2}$ the canonical projection maps defined on $E_{1}\times E_{2}$, where
$p_{1}$ maps $(x,y)\in E_{1}\times E_{2}$ to $x\in E_{1}$ and $p_{2}$ maps
$(x,y)$ to $y\in E_{2}$. Recall that these projection maps are continuous and
open (mapping open sets to open sets). The map $p_{1}$ (respectively $p_{2}$)
is closed if $E_{2}$ (respectively $E_{1}$) is compact [11, Proposition 8.2].
By convention, $\operatorname{Id}$ denotes the identity map $x\mapsto x$.

Consider the Wasserstein space $(\mathcal{W}_{2}(E),W_{2})$ over some metric
space $(E,d)$. For $\mu\in\mathcal{W}_{2}(E)$, we define
$W_{2}(\mu,E):=\inf_{x\in E}W_{2}(\mu,\delta_{x})$. For a given point $z\in
E$, since
$W_{2}(\mu,\delta_{z})^{2}=\int_{M}d(z,y)^{2}\operatorname{d}\mu(y)$, $z$ is a
barycenter of $\mu$ if and only if $W_{2}(\mu,\delta_{z})=W_{2}(\mu,E)$. The
following lemma is useful when compactness arguments are needed, and it uses
the notation introduced above.

######  Lemma 2.3.

Let $(E,d)$ be a proper space. Given an integer $n\geq 1$, let
$\lambda_{i}>0,1\leq i\leq n$, be $n$ positive real numbers such that
$\sum_{i=1}^{n}\lambda_{i}=1$. The set

$\Gamma:=\left\\{(x_{1},\ldots,x_{n},z)\in
E^{n+1}\,\bigg{|}\,W_{2}(\mu,\delta_{z})=W_{2}(\mu,E),\,\mu:=\sum_{i=1}^{n}\lambda_{i}\,\delta_{x_{i}}\right\\}$

is closed. Denote by $\operatorname{bary}(\boldsymbol{A})$ the set of all
barycenters of the measures $\sum_{i=1}^{n}\lambda_{i}\,\delta_{x_{i}}$ when
$(x_{1},\ldots,x_{n})$ runs through a subset $\boldsymbol{A}\subset E^{n}$. If
$\boldsymbol{A}$ is compact, then $\operatorname{bary}(\boldsymbol{A})$ is
compact.

###### Proof.

For $\boldsymbol{x}:=(x_{1},x_{2},\ldots,x_{n})\in E^{n}$, we define
$\eta(\boldsymbol{x}):=\sum_{i=1}^{n}\lambda_{i}\,\delta_{x_{i}}\in\mathcal{W}_{2}(E)$.
The map $\eta:(E,d)\rightarrow(\mathcal{W}_{2}(E),W_{2})$ is continuous by
definition of Wasserstein metric: for $\boldsymbol{x},\boldsymbol{y}\in
E^{n}$,

$W_{2}(\eta(\boldsymbol{x}),\eta(\boldsymbol{y}))^{2}\leq\sum_{i=1}^{n}\lambda_{i}\,d(x_{i},y_{i})^{2}.$

It follows from the triangle inequality that the map $\boldsymbol{x}\in
E^{n}\mapsto W_{2}(\eta(\boldsymbol{x}),E)$ is also continuous, which implies
that the set $\Gamma$ is closed.

Note that
$\operatorname{bary}(\boldsymbol{A})=p_{2}\left(\Gamma\cap(\boldsymbol{A}\times
E)\right)$, where $p_{2}:\boldsymbol{A}\times E\rightarrow E$ is the canonical
projection map. If $\boldsymbol{A}$ is compact, $p_{2}$ is a closed map and
thus $\operatorname{bary}(\boldsymbol{A})$ is closed as $\Gamma$ is closed.
The set $\operatorname{bary}(\boldsymbol{A})$ is bounded since barycenters are
located within the union of $n$ bounded balls with centers $x_{i}$. ∎

For a metric space $E$ we denote by $\mathcal{B}(E)$ the $\sigma$-algebra of
its Borel sets. We shall apply the following widely used measurable selection
theorem to construct Wasserstein barycenters in the next subsection. Its proof
could be found in [9, Theorem 6.9.3], [22], and [32].

######  Theorem 2.4 (Kuratowski and Ryll-Nardzewski measurable selection
theorem).

Let $E$ be a complete separable metric space, and let $\Psi$ be a map on a
measurable space $(\Omega,\mathcal{B})$ with values in the set of nonempty
closed subsets of $E$. Suppose that for every open set $U\subset E$, we have

$\left\\{\omega\in\Omega\,\mid\,\Psi(\omega)\cap
U\neq\emptyset\right\\}\in\mathcal{B}.$

Then $\Psi$ has a selection that is measurable with respect to the pair of
$\sigma$-algebras $\mathcal{B}$ and $\mathcal{B}(E)$.

The notion of conditional measures [9, Definition 10.4.2] will be used to
prove 2.22.

######  Definition 2.5 (Conditional probability measures).

Let $E$ be a complete separable metric space and let $n\geq 2$ be a positive
integer. Denote by $\boldsymbol{x}^{\prime}=(x_{2},\ldots,x_{n})\in E^{n-1}$
the last $n-1$ components of a point
$\boldsymbol{x}=(x_{1},x_{2},\ldots,x_{n})\in E^{n}$. Given a Borel
probability measures $\gamma$ on $E^{n}$, define the measure
$\pi:={p_{2}}_{\\#}\gamma$ on $E^{n-1}$, where $p_{2}$ is the projection
$\boldsymbol{x}\in E\times E^{n-1}\mapsto\boldsymbol{x}^{\prime}\in E^{n-1}$.
We call $\gamma(\cdot,\cdot):\mathcal{B}(E)\times
E^{n-1}\rightarrow\mathbb{R}$ a conditional measure for $\gamma$, written as
$\operatorname{d}\gamma(\boldsymbol{x})=\gamma(\operatorname{d}\boldsymbol{x},\boldsymbol{x}^{\prime})\,\operatorname{d}\pi(\boldsymbol{x}^{\prime})$,
if

  1. 1.

for all $\boldsymbol{x}^{\prime}\in E^{n-1}$,
$\gamma(\cdot,\boldsymbol{x}^{\prime})$ is a Borel probability measure on
$E^{n}$,

  2. 2.

for $\pi$-almost every $\boldsymbol{x}^{\prime}\in E^{n-1}$,
$\gamma(\cdot,\boldsymbol{x}^{\prime})$ is concentrated on
$E\times\\{\boldsymbol{x}^{\prime}\\}$,

  3. 3.

for any Borel set $\boldsymbol{R}\subset E^{n}$, the function
$\boldsymbol{x}^{\prime}\mapsto\gamma(\boldsymbol{R},\boldsymbol{x}^{\prime})$
is measurable, and

  4. 4.

for any Borel set $\boldsymbol{S}\subset E^{n-1}$,
$\gamma[\boldsymbol{R}\cap(E\times\boldsymbol{S})]=\int_{\boldsymbol{S}}\gamma(\boldsymbol{R},\boldsymbol{x}^{\prime})\operatorname{d}\pi(\boldsymbol{x}^{\prime})$.

Under our assumption that $E$ is complete and separable, conditional measures
always exist [9, Corollary 10.4.10]. For $\pi$-almost every
$\boldsymbol{x}^{\prime}$, the measure $\gamma(\cdot,\boldsymbol{x}^{\prime})$
is unique [9, Lemma 10.4.3] and coincides with the disintegration [23, 452E]
of $\gamma$ that is consistent with the projection $p_{2}$.

Finally, throughout this document, we assume that (Riemannian) manifolds are
connected and $\mathcal{C}^{3}$ smooth without boundary. These assumptions
enable us to apply the results by McCann [36, Proposition 6] and Cordero-
Erausquin et al. [16]. In most propositions, we also assume that the manifolds
are complete. We always denote by $M$ such a manifold, by $d$ its intrinsic
geodesic metric, by $\exp:TM\rightarrow M$ the exponential map on its tangent
bundle, and by $\operatorname{Vol}$ the volume measure on it.

###  2.2 Construction, existence and uniqueness of Wasserstein barycenters

This subsection covers the fundamental properties of Wasserstein barycenters.
We construct them via optimal transport theory and measurable barycenter
selection maps. It is crucial to comprehend further features of these maps, as
highlighted in Section 2.3.2. Once the construction is explained, we discuss
the problem of existence and uniqueness of Wasserstein barycenters. These
properties are closely related to the law of large numbers for Wasserstein
barycenters.

We begin with the existence of measurable barycenter selection maps.

######  Lemma 2.6 (Measurable barycenter selection maps).

Let $(E,d)$ be a proper space. Given an integer $n\geq 1$, let
$\lambda_{i}>0,1\leq i\leq n$, be $n$ positive real numbers such that
$\sum_{i=1}^{n}\lambda_{i}=1$. There exists a measurable barycenter selection
map $B:E^{n}\rightarrow E$ such that $B(x_{1},\ldots,x_{n})$ is a barycenter
of $\sum_{i=1}^{n}\lambda_{i}\,\delta_{x_{i}}\in\mathcal{W}_{2}(E)$.

###### Proof.

As in 2.3, for a subset $\boldsymbol{A}\subset E^{n}$, denote by
$\operatorname{bary}(\boldsymbol{A})\subset E$ the set of barycenters of
$\sum_{i=1}^{n}\lambda_{i}\,\delta_{x_{i}}$ when
$\boldsymbol{x}=(x_{1},\ldots,x_{n})$ runs through $\boldsymbol{A}$. For the
existence of a measurable function $B:E^{n}\rightarrow E$ such that
$B(\boldsymbol{x})\in\operatorname{bary}(\\{\boldsymbol{x}\\})$, we shall
apply the Kuratowski and Ryll-Nardzewski measurable selection theorem (2.4).
By 2.3, the set $\Gamma:=\\{(\boldsymbol{x},z)\in E^{n+1}\mid
z\in\operatorname{bary}(\\{\boldsymbol{x}\\})\\}$ is closed. Let $C\subset E$
be a compact set, then

$\\{\boldsymbol{x}\,|\,\operatorname{bary}(\\{\boldsymbol{x}\\})\cap
C\neq\emptyset\\}=p_{1}(\\{(\boldsymbol{x},z)\in E^{n}\times
C\,\mid\,(\boldsymbol{x},z)\in\Gamma\\}),$

where $p_{1}:E^{n}\times C\rightarrow E^{n}$ is the canonical projection map.
Since $C$ is compact and $\Gamma$ is closed, $p_{1}(\Gamma\cap(E^{n}\times
C))$ is a closed set. For an open subset $U\subset E$, we can thus express the
complement of
$\\{\boldsymbol{x}\,|\,\operatorname{bary}(\\{\boldsymbol{x}\\})\cap
U\neq\emptyset\\}$, which is
$\\{\boldsymbol{x}\,|\,\operatorname{bary}(\\{\boldsymbol{x}\\})\cap(E\setminus
U)\neq\emptyset\\}$, as a countable union of closed sets since $E\setminus U$
is a countable union of compact sets. It follows that
$\\{\boldsymbol{x}\,|\,\operatorname{bary}(\\{\boldsymbol{x}\\})\cap
U\neq\emptyset\\}$ is measurable for any open subset $U\subset E$. Since
$\operatorname{bary}(\\{\boldsymbol{x}\\})$ is compact for $\boldsymbol{x}\in
E^{n}$ by 2.3, the assumptions of 2.4 are satisfied by the map
$\boldsymbol{x}\mapsto\operatorname{bary}(\\{\boldsymbol{x}\\})$. This proves
the lemma. ∎

To construct Wasserstein barycenters of finitely many measures, we first
recall the following particular type of multi-marginal optimal transport
plans.

######  Definition 2.7 (Multi-marginal optimal transport plans).

Let $(E,d)$ be a proper space. Given an integer $n\geq 2$, let
$\lambda_{i}>0,1\leq i\leq n$, be $n$ positive real numbers such that
$\sum_{i=1}^{n}\lambda_{i}=1$ and let $\mu_{i}\in\mathcal{W}_{2}(E),1\leq
i\leq n$, be $n$ probability measures on $E$. Denote by $\Pi$ the set of
probability measures on $E^{n}$ with marginals $\mu_{1},\ldots,\mu_{n}$ in
this order. We call $\gamma\in\Pi$ a multi-marginal optimal transport plan (of
its marginals) if

$\int_{E^{n}}W_{2}(\sum_{i=1}^{n}\lambda_{i}\,\delta_{x_{i}},E)^{2}\operatorname{d}\gamma(x_{1},\ldots,x_{n})=\min_{\theta\in\Pi}\int_{E^{n}}W_{2}(\sum_{i=1}^{n}\lambda_{i}\,\delta_{x_{i}},E)^{2}\operatorname{d}\theta(x_{1},\ldots,x_{n}).$
(2)

In what follows, the marginal measures $\mu_{i}$ and constants $\lambda_{i}$
will be clear from the context. In the proof of 2.3, it is shown that
$W_{2}(\sum_{i=1}^{n}\lambda_{i}\,\delta_{x_{i}},E)^{2}$ is continuous with
respect to $(x_{1},\ldots,x_{n})\in E^{n}$. Hence, we can conclude the
existence of a multi-marginal optimal transport plan $\gamma$ in the same way
as the classic existence of optimal couplings between two measures [49,
Theorem 4.1]. Now we are ready to construct Wasserstein barycenters.

######  Proposition 2.8 (Construction of Wasserstein barycenters of
$\sum_{i=1}^{n}\lambda_{i}\,\delta_{\mu_{i}}$).

Let $(E,d)$ be a proper space. Given an integer $n\geq 2$, let
$\lambda_{i}>0,1\leq i\leq n$, be $n$ positive real numbers such that
$\sum_{i=1}^{n}\lambda_{i}=1$. Let
$\mu_{1},\ldots,\mu_{n}\in\mathcal{W}_{2}(E)$ be $n$ probability measures and
let $\gamma$ be a multi-marginal optimal transport plan of them, i.e.,
satisfying (2). If $B:E^{n}\rightarrow E$ is a measurable map such that
$B(x_{1},\ldots,x_{n})$ is a barycenter of
$\sum_{i=1}^{n}\lambda_{i}\,\delta_{x_{i}}$, then

  1. 1.

$\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\mu}:=B_{\\#}\gamma$ is a barycenter of
$\mathbb{P}:=\sum_{i=1}^{n}\lambda_{i}\,\delta_{\mu_{i}}$;

  2. 2.

$(B,p_{i})_{\\#}\gamma$ is an optimal transport plan between $\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\mu}$ and $\mu_{i}$, where $p_{i}$ denotes the canonical projection
$(x_{1},\ldots,x_{n})\in E^{n}\mapsto x_{i}\in E$;

  3. 3.

if $X,X_{1},\ldots,X_{n}$ are $n+1$ random variables from a probability space
$(\Omega,\mathcal{B},P)$ to $(E,d)$ with law $\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\mu},\mu_{1},\ldots,\mu_{n}$, then for $P$-almost every $\omega\in\Omega$,
$X(\omega)$ is a barycenter of
$\sum_{i=1}^{n}\lambda_{i}\,\delta_{X_{i}(\omega)}$.

###### Proof.

Given an arbitrary probability measure $\nu\in\mathcal{W}_{2}(E)$, thanks to
the gluing lemma [48, Lemma 7.1], there are $n+1$ random variables
$X,X_{1},\ldots X_{n}$ valued in $E$ with laws $\nu,\mu_{1},\ldots\mu_{n}$
such that $\mathbb{E}\,d(X,X_{i})^{2}=W_{2}(\nu,\mu_{i})^{2}$. Since
$\mu_{i}={p_{i}}_{\\#}\gamma$, we have

$\displaystyle\sum_{i=1}^{n}\lambda_{i}\,W_{2}(\mu_{i},\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\mu})^{2}$
$\displaystyle\leq\sum_{i=1}^{n}\int_{E^{n}}\lambda_{i}\,d(x_{i},B(\boldsymbol{x}))^{2}\operatorname{d}\gamma(\boldsymbol{x})=\int_{E^{n}}W_{2}(\sum_{i=1}^{n}\lambda_{i}\,\delta_{x_{i}},E)^{2}\operatorname{d}\gamma(\boldsymbol{x})$
$\displaystyle\leq\mathbb{E}\,W_{2}(\sum_{i=1}^{n}\lambda_{i}\,\delta_{X_{i}},E)^{2}\leq\mathbb{E}\sum_{i=1}^{n}\lambda_{i}\,d(X_{i},X)^{2}$
$\displaystyle=\sum_{i=1}^{n}\lambda_{i}\,W_{2}(\mu_{i},\nu)^{2},$

where we sequentially applied the definitions of $\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\mu}=B_{\\#}\gamma$, $W_{2}(\mu_{i},\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\mu})$, $\gamma$, $W_{2}(\cdot,E)$ and $X,X_{1},\ldots X_{n}$. Since $\nu$
is arbitrary, it follows that $\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\mu}$ is a Wasserstein barycenter. By setting $\nu=\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\mu}$ in the above inequality, we actually obtain an equality. This shows
that the law of $(X_{1},\ldots,X_{n})$ is a multi-marginal optimal transport
plan and
$W_{2}(\sum_{i=1}^{n}\lambda_{i}\,\delta_{X_{i}(\omega)},E)^{2}=\sum_{i=1}^{n}\lambda_{i}\,d(X(\omega),X_{i}(\omega))^{2}$
for $P$-almost every $\omega\in\Omega$, which validates our last two
statements. ∎

The general existence of Wasserstein barycenters was first established in
[33]. Recall that finitely supported measures are dense in Wasserstein spaces
[49, Theorem 6.18], so the above construction of Wasserstein barycenters
together with the law of large numbers for Wasserstein barycenters (2.2)
implies the following theorem.

######  Theorem 2.9 (Existence of Wasserstein barycenters).

If $(E,d)$ is a proper space, then any
$\mathbb{P}\in\mathcal{W}_{2}(\mathcal{W}_{2}(E))$ has a barycenter.

Note that in the law of large numbers for Wasserstein barycenters (2.2), we
may need to pass to a subsequence of Wasserstein barycenters $\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\mu}_{j}$ and the limit barycenter $\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\mu}$ is not known in advance. Hence, 2.2 will be enhanced if we can
assert barycenters’ uniqueness under some additional assumptions, as follows.

######  Proposition 2.10 (Uniqueness of Wasserstein barycenters).

Let $(E,d)$ be a proper space. If a probability measure
$\mathbb{P}\in\mathcal{W}_{2}(\mathcal{W}_{2}(E))$ gives mass to a Borel
subset $\mathcal{A}\subset\mathcal{W}_{2}(E)$ such that for
$\mu\in\mathcal{A}$ and $\nu\in\mathcal{W}_{2}(E)$, any optimal transport plan
between $\mu$ and $\nu$ is induced by a measurable map $T$ pushing $\mu$
forward to $\nu$, i.e., $\nu=T_{\\#}\mu$ and
$\mathcal{W}_{2}(\mu,\nu)^{2}=\int_{E}d(x,T(x))^{2}\operatorname{d}\mu$, then
$\mathbb{P}$ has a unique barycenter in $\mathcal{W}_{2}(E)$.

###### Proof.

The uniqueness follows from the strict convexity of a distance function to a
point in $\mathcal{W}_{2}(E)$, as shown by [41, Theorem 7.19] and [31, Theorem
3.1]. We recall the proof for the sake of completeness.

Observe that any convex combination of probability measures in the space
$\mathcal{W}_{2}(E)$ is still a probability measure in it. Fix
$\mu\in\mathcal{A}$ and consider the squared Wasserstein distance function
$W_{2}(\mu,\cdot)^{2}$ with respect to this convex structure. For
$\lambda\in[0,1]$ and two different probability measures
$\nu_{1},\nu_{2}\in\mathcal{W}_{2}(E)$, by definition of Wasserstein metric we
have

$W_{2}(\mu,\lambda\,\nu_{1}+(1-\lambda)\nu_{2})^{2}\leq\lambda\,W_{2}(\mu,\nu_{1})^{2}+(1-\lambda)W_{2}(\mu,\nu_{2})^{2}.$
(3)

By our assumptions, there are two measurable maps $T_{1},T_{2}:E\rightarrow E$
such that $\gamma_{1}:=(\operatorname{Id}\times T_{1})_{\\#}\mu$ and
$\gamma_{2}:=(\operatorname{Id}\times T_{2})_{\\#}\mu$ are optimal transport
plans between $\mu$ and the two measures $\nu_{1}$ and $\nu_{2}$ respectively.
We claim that (3) cannot be an equality unless $\lambda=0$ or $\lambda=1$.
Indeed, if (3) is an equality for some $0<\lambda<1$, then by setting
$\gamma:=\lambda\,\gamma_{1}+(1-\lambda)\,\gamma_{2}$ we have

$\displaystyle\lambda\,W_{2}(\mu,\nu_{1})^{2}+(1-\lambda)W_{2}(\mu,\nu_{2})^{2}$
$\displaystyle=W_{2}(\mu,\lambda\,\nu_{1}+(1-\lambda)\nu_{2})^{2}$
$\displaystyle\leq\int_{E\times E}d(x,y)^{2}\operatorname{d}\gamma(x,y)$
$\displaystyle=\lambda\,W_{2}(\mu,\nu_{1})^{2}+(1-\lambda)W_{2}(\mu,\nu_{2})^{2},$

and thus $\gamma$ is an optimal plan between $\mu$ and
$\lambda\,\nu_{1}+(1-\lambda)\nu_{2}$. By assumptions, there exists a
measurable map $T:E\rightarrow E$ such that $\gamma=(\operatorname{Id}\times
T)_{\\#}\mu$. Denote by $\operatorname{graph}(S)\subset E^{2}$ the graph of a
map $S:E\rightarrow E$. Note that if $S$ is a measurable map, then
$\operatorname{graph}(S)=\\{(x,y)\in E^{2}\mid d(S(x),y)=0\\}$ is a Borel
subset of $E^{2}$. Since
$\gamma[\operatorname{graph}(T)]=\lambda\,\gamma_{1}[\operatorname{graph}(T)]+(1-\lambda)\gamma_{2}[\operatorname{graph}(T)]=1$
and $0<\lambda<1$, we have
$\gamma_{1}[\operatorname{graph}(T)]=\gamma_{2}[\operatorname{graph}(T)]=1$.
Hence, for $i\in\\{1,2\\}$, $\mu(\\{x\in E\mid
T_{i}(x)=T(x)\\})=\gamma_{i}[\operatorname{graph}(T)\cap\operatorname{graph}(T_{i})]=1$.
It follows that both $T_{1}$ and $T_{2}$ coincide with $T$ almost everywhere
with respect to $\mu$ and thus $\gamma_{1}=\gamma_{2}$, which is a
contradiction since $\nu_{1}\neq\nu_{2}$.

This shows that $W_{2}(\mu,\cdot)^{2}$ is strictly convex on
$\mathcal{W}_{2}(E)$ for $\mu\in\mathcal{A}$. Since
$\mathbb{P}(\mathcal{A})>0$, the map

$\nu\in\mathcal{W}_{2}(E)\mapsto\int_{\mathcal{W}_{2}(E)}W_{2}(\mu,\nu)^{2}\operatorname{d}\mathbb{P}(\mu)$

is also strictly convex on $\mathcal{W}_{2}(E)$ by the linearity and
positivity of the above integral. It follows that the Wasserstein barycenter
of $\mathbb{P}$ asserted by 2.9 is unique. ∎

######  Remark 2.11.

Under the assumptions of 2.10, the optimal transport plan between
$\mu\in\mathcal{A}$ and $\nu\in\mathcal{W}_{2}(M)$ is unique. By setting
$\nu_{1}=\nu_{2}=\nu$, (3) becomes an equality for any $\lambda\in[0,1]$. It
is shown above that any two optimal transport plans $\gamma_{1}$ and
$\gamma_{2}$ between measures $\mu$ and $\nu$ coincide.

There are many setups in which we can apply 2.10. We typically choose
$\mathcal{A}$ as the set of absolutely continuous measures with respect to
some given reference measure. The following lemma ensures that $\mathcal{A}$
is a Borel set of $(\mathcal{W}_{2}(E),W_{2})$.

######  Lemma 2.12.

Let $E$ be a metric space with a $\sigma$-finite Borel measure $\mu$ on $E$.
Assume that $\mu$ is outer regular, i.e., for any Borel set
$N\in\mathcal{B}(E)$, $\mu(N)=\inf\\{\mu(O)\mid O\text{ open neighborhood of N
}\\}$. Denote by $\mathcal{A}$ the set of probability measures in
$\mathcal{W}_{2}(E)$ that are absolutely continuous with respect to $\mu$. For
$\epsilon,\delta>0$, define the set

$\mathcal{E}_{\epsilon,\delta}:=\left\\{\nu\in\mathcal{W}_{2}(E)\mid\forall\,N\in\mathcal{B}(E),\,\mu(N)<\delta\implies\nu(N)\leq\epsilon\right\\}.$

It is a closed set with respect to the weak convergence topology of
$\mathcal{W}_{2}(E)$, and we have

$\mathcal{A}=\bigcap_{k\in\mathbb{N}}\bigcup_{l\in\mathbb{N}}\mathcal{E}_{2^{-k},2^{-l}}.$

In particular, if $E$ is a proper space and $\mu$ is a locally finite Borel
measure, i.e., $\mu$ gives finite mass to some open neighborhood of every
point in $E$, then for the Wasserstein space topology,
$\mathcal{E}_{\epsilon,\delta}$ is a closed set and $\mathcal{A}$ is a Borel
set.

###### Proof.

Our proof is based on [31, Proposition 2.1, Remark 2.2] though we use
different assumptions.

Suppose that $\nu_{j}\in\mathcal{E}_{\epsilon,\delta}$ converges weakly to
$\nu\in\mathcal{W}_{2}(E)$. For any $N\in\mathcal{B}(E)$ such that
$\mu(N)<\delta$, there exists an open set $O$ such that $N\subset O$ and
$\mu(O)<\delta$ since $\mu$ is outer regular. By the characterization of weak
convergence of probability measures on metric spaces [9, Corollary 8.2.10], we
have

$\nu(N)\leq\nu(O)\leq\liminf_{j\rightarrow\infty}\nu_{j}(O)\leq\epsilon$

and thus $\mathcal{E}_{\epsilon,\delta}$ is closed with respect to weak
convergence topology on $\mathcal{W}_{2}(E)$.

The inclusion
$\mathcal{A}\supset\bigcap_{k\in\mathbb{N}}\bigcup_{l\in\mathbb{N}}\mathcal{E}_{2^{-k},2^{-l}}$
follows from the definition of a measure $\nu$ being absolutely continuous
with respect to $\mu$:
$\forall\,N\in\mathcal{B}(E),\,\mu(N)=0\implies\nu(N)=0$. Fix a measure
$\nu\in\mathcal{A}$. Since $\mu$ is $\sigma$-finite, we can apply the Radon-
Nikodym theorem to write $\nu=f\cdot\mu$. The reverse inclusion
$\mathcal{A}\subset\bigcap_{k\in\mathbb{N}}\bigcup_{l\in\mathbb{N}}\mathcal{E}_{2^{-k},2^{-l}}$
follows from the absolute continuity of Lebesgue integral [9, Theorem 2.5.7,
Proposition 2.6.4].

Given a proper space $E$ and a locally finite Borel measure $\mu$, $\mu$ gives
finite mass to compact sets, and every open subset of $E$ is $\sigma$-compact.
It follows that $\mu$ is outer regular [45, Theorem 6 of §2.7] and also
$\sigma$-finite. Since Wasserstein convergence implies weak convergence, the
set $\mathcal{W}_{2}(E)\cap\mathcal{E}_{\epsilon,\delta}$ is closed for the
Wasserstein metric. It follows that $\mathcal{A}$ is a Borel set of
$\mathcal{W}_{2}(E)$. ∎

######  Remark 2.13.

On a metric space, any finite Borel measure is outer regular, see [9,
Definition 7.1.5, Theorem 7.1.7] or [8, Theorem 1.1]. However, this is not
true for $\sigma$-finite Borel measures. For example, define the Borel measure
$\mu$ on $\mathbb{R}$ such that for $N\in\mathcal{B}(\mathbb{R})$, $\mu$
counts the number of rational points in $N$. This measure is $\sigma$-finite
but not outer regular since $\mu$ never gives finite mass to open sets. As for
the assumption regarding the $\sigma$-compactness of open sets in the above
cited theorem [45, Theorem 6 of §2.7], for metric spaces it can be replaced by
assuming that $\mu$ gives finite mass to a sequence of open sets $O_{i},i\geq
1$ such that $E=\cup_{i\geq 1}O_{i}$. We also mention that there exists a
$\sigma$-finite and locally finite but not outer regular Borel measure on a
locally compact Hausdorff space [10, problem 5 of Exercise §1, INT IV.119].

Thanks to 2.12, the Wasserstein barycenter of $\mathbb{P}$ is unique for the
following spaces, provided that $\mathbb{P}$ gives mass to the set of
absolutely continuous measures with respect to the corresponding canonical
reference measure:

  1. 1.

complete Riemannian manifolds, see Villani [49, Theorem 10.41] or Gigli [25,
Theorem 7.4];

  2. 2.

compact finite dimensional Alexandrov spaces, see Bertrand [7, Theorem 1.1];

  3. 3.

for $K\in\mathbb{R}$ and $N\geq 1$, non-branching $\operatorname{CD}(K,N)$
spaces, see Gigli [26, Theorem 3.3];

  4. 4.

for $K\in\mathbb{R}$ and $N\geq 1$, $\operatorname{RCD^{*}}(K,N)$ spaces, see
Gigli, Rajala and Sturm [27, Theorem 1.1];

  5. 5.

for $K\in\mathbb{R}$ and $N\geq 1$, essentially non-branching
$\operatorname{MCP}(K,N)$ spaces, see Cavalletti and Mondino [14, Theorem
1.1];

  6. 6.

($2$-)essentially non-branching spaces with qualitatively non-degenerate
reference measures, see Kell [29, Theorem 5.8].

The above spaces are listed in (nearly) ascending order of generality. For the
metric measure spaces, we assume that the metric space is proper and the
reference measure is locally finite. The references cited above demonstrate
that the unique optimal transport plan (2.11) between an absolutely continuous
probability measure and a given probability measure is induced by a measurable
map, allowing us to apply 2.10.

Since the existence and uniqueness of Wasserstein barycenters (under mild
assumptions) on Riemannian manifolds $M$ are established, we are ready to
prove the absolute continuity of Wasserstein barycenters with respect to
$\operatorname{Vol}$. In the next subsection, we consider Wasserstein spaces
$\mathcal{W}_{2}(M)$ over Riemannian manifolds and show that the unique
Wasserstein barycenter of finitely many measures is absolutely continuous if
one of those measures is so.

###  2.3 The absolute continuity of Wasserstein barycenters of finitely many
measures

Let $M$ be a complete Riemannian manifold and let
$\mathbb{P}=\sum_{i=1}^{n}\lambda_{i}\,\delta_{\mu_{i}}$ be a probability
measure on $\mathcal{W}_{2}(M)$ with positive real numbers $\lambda_{i}$ and
compactly supported measures $\mu_{i}$ in $\mathcal{W}_{2}(M)$. Assuming that
$M$ is compact and $\mu_{1}$ is absolutely continuous (with respect to
$\operatorname{Vol}$), Kim and Pass [31, Theorem 5.1] proved that
$\mathbb{P}$’s unique barycenter $\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\mu}$ is absolutely continuous. For completeness and also for a rigorous
foundation of our later arguments, we provide a proof for general non-compact
manifolds.

The proof strategy is to investigate different cases for measures
$\mu_{i},2\leq i\leq n$, step by step. In the simplest case when
$\mu_{i}=\delta_{x_{i}},2\leq i\leq n$, are Dirac measures, the unique
barycenter $\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\mu}$ is the push-forward of
$\mu_{1}\otimes\delta_{x_{2}}\otimes\ldots\otimes\delta_{x_{n}}$ by a
measurable barycenter selection map $B$ (2.8). To deduce more properties of
$B$, we shortly review $c$-concave functions.

####  2.3.1 $c$-concave functions

For $x,y\in M$, we define the function $c(x,y):=\frac{1}{2}d(x,y)^{2}$ as the
half of the squared distance between $x$ and $y$ in $M$. Also, we define
$d_{y}^{2}(\cdot):=d(\cdot,y)^{2}$ to avoid ambiguity when fixing the point
$y$.

######  Definition 2.14 ($c$-transforms and $c$-concave functions).

Let $M$ be a Riemannian manifold. Let $X$ and $Y$ be two non-empty compact
subsets of $M$. A function $\phi$ : $X\rightarrow\mathbb{R}$ is $c$-concave if
there exists a function $\psi:Y\rightarrow\mathbb{R}$ such that

$\phi(x)=\inf_{y\in Y}c(x,y)-\psi(y),\quad\forall x\in X.$ (4)

We write it as $\phi=\psi^{c}$ and call $\phi$ the $c$-transform of $\psi$.
The set of all $c$-concave functions with respect to $X$ and $Y$ is denoted by
$\mathcal{I}^{c}(X,Y)$.

As shown in the following theorem by McCann [36], $c$-concave functions are
fundamental in the optimal transport theory on manifolds. Recall that given a
$c$-concave function $\phi$ on a compact set $\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\mathcal{X}}$ with $\mathcal{X}\subset M$ open, its gradient $\nabla\phi$
exists on $\mathcal{X}$ almost everywhere with respect to $\operatorname{Vol}$
since $\phi$ is Lipschitz [36, Lemma 4].

######  Theorem 2.15 (Optimal transport on manifolds, [16, Theorem 3.2]).

Let $M$ be a complete Riemannian manifold. Fix two measures $\mu,\nu\in
W_{2}(M)$ with compact support such that $\mu$ is absolutely continuous. Given
two bounded open subsets $\mathcal{X},\mathcal{Y}\subset M$ containing the
support of $\mu$ and $\nu$ respectively, there exists
$\phi\in\mathcal{I}^{c}(\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\mathcal{X}},\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\mathcal{Y}})$ such that $(\operatorname{Id},F)_{\\#}\mu$ is the unique
optimal transport plan between $\mu$ and $\nu$, where the function
$F:=\exp(-\nabla\phi)$ is $\mu$-almost everywhere well-defined.

The following lemma shows that the definition of barycenters for measures
$\sum_{i=1}^{n}\lambda_{i}\,\delta_{x_{i}}$ on $M$ involves $c$-concave
functions.

######  Lemma 2.16.

Let $M$ be a complete Riemannian manifold. Given an integer $n\geq 2$, let
$\lambda_{i}>0,1\leq i\leq n$, be $n$ positive real numbers such that
$\sum_{i=1}^{n}\lambda_{i}=1$. We define

$f:(x_{1},x_{2},\ldots,x_{n})\in M^{n}\mapsto\min_{w\in
M}\sum_{i=1}^{n}\lambda_{i}\,c(w,x_{i})=\frac{1}{2}W_{2}(\sum_{i=1}^{n}\lambda_{i}\,\delta_{x_{i}},M)^{2}.$
(5)

Fix a non-empty compact subset $X\subset M$ and $n-1$ points $x_{i}\in M,2\leq
i\leq n$. Denote by $Y$ the set of all barycenters of
$\sum_{i=1}^{n}\lambda_{i}\,\delta_{x_{i}}$ when $x_{1}$ runs through $X$.
Define $f_{1}:x_{1}\in X\mapsto f(x_{1},\ldots,x_{n})/\lambda_{1}$ and
$g_{1}:y\in Y\mapsto-1/\lambda_{1}\sum_{i=2}^{n}\lambda_{i}\,c(y,x_{i})$, then
$f_{1}=g_{1}^{c}\in\mathcal{I}^{c}(X,Y)$ and
$g_{1}=f_{1}^{c}\in\mathcal{I}^{c}(Y,X)$.

###### Proof.

The set $Y\subset M$ is compact by 2.3. Using the given definition of $Y$, we
can replace the minimum over $M$ in (5) by the minimum over $Y$, which shows
the equality $f_{1}=g_{1}^{c}\in\mathcal{I}^{c}(X,Y)$.

Since $f_{1}(x)+g_{1}(y)\leq c(x,y)$ for any $(x,y)\in X\times Y$, we have

$g_{1}(y)\leq f_{1}^{c}(y):=\inf_{x\in X}c(x,y)-f_{1}(x).$ (6)

Fix an arbitrary point $y\in Y$. Our definition of $Y$ implies the existence
of $x_{1}\in X$ such that $y$ is a barycenter of
$\sum_{i=1}^{n}\lambda_{i}\,\delta_{x_{i}}$. For such a pair $(x_{1},y)\in
X\times Y$, $f_{1}(x_{1})+g_{1}(y)=c(x_{1},y)$ by the definitions of $f_{1}$
and $g_{1}$. It follows from the inequalities $f_{1}(x_{1})+f_{1}^{c}(y)\leq
c(x_{1},y)=f_{1}(x_{1})+g_{1}(y)$ and (6) that $g_{1}(y)=f_{1}^{c}(y)$. Since
$y$ is arbitrarily chosen, we conclude that
$g_{1}=f_{1}^{c}\in\mathcal{I}^{c}(Y,X)$. ∎

The $c$-concave function $g_{1}\in\mathcal{I}^{c}(Y,X)$ defined in 2.16 has
simple expression unlike its $c$-transform $f_{1}$. Furthermore, thanks to the
following lemma by Kim and Pass [30, Lemma 3.1], we conclude that $g_{1}$ is
$\mathcal{C}^{3}$ smooth since $M$ is a $\mathcal{C}^{3}$ smooth manifold.
This differential property of $g_{1}$ (to be used in 2.20) is crucial to prove
Wasserstein barycenters’ absolute continuity.

######  Lemma 2.17 (Barycenters and cut loci, [30, Lemma 3.1 and proof of
Theorem 6.1]).

Let $M$ be a complete Riemannian manifold. Given an integer $n\geq 1$, let
$\lambda_{i}>0,1\leq i\leq n$, be $n$ positive real numbers such that
$\sum_{i=1}^{n}\lambda_{i}=1$ and let $x_{i}\in M,1\leq i\leq n$, be $n$
points of $M$. For $1\leq i\leq n$, $x_{i}$ is out of the cut locus of any
barycenters of $\sum_{i=1}^{n}\lambda_{i}\,\delta_{x_{i}}$.

It will be shown in 2.20 that the map $\exp(-\nabla g_{1})$ is an optimal
transport map $F$ as stated in 2.15. Given the above 2.17, the following lemma
further illustrates how to differentiate such $\mathcal{C}^{2}$ maps.

######  Lemma 2.18.

Let $M$ be a complete Riemannian manifold. Fix an open set $U\subset M$, a
point $x\in U$, and a $\mathcal{C}^{2}$ smooth function $\phi$ defined on $U$.
Define $F:=\exp(-\nabla\phi)$ on $U$. Assume that the (fixed) point $y:=F(x)$
is out of the cut locus of $x$. If the two functions, $\phi$ and
$d^{2}_{y}/2$, have the same gradient at $x$, then

$\operatorname{d}F(x)=\operatorname{d}\exp_{x}|_{-\nabla\phi}\circ(\operatorname{Hess}_{x}d^{2}_{y}/2-\operatorname{Hess}_{x}\phi).$
(7)

In the above formula,

  1. 1.

$\operatorname{Hess}_{x}$ denotes the Hessian operator at $x$ and its values
are maps from $T_{x}M$ to $T_{x}M$;

  2. 2.

$\operatorname{d}\exp_{x}|_{-\nabla\phi}:T_{-\nabla\phi(x)}T_{x}M\rightarrow
T_{y}M$ denotes the differential of the exponential map
$\exp_{x}:T_{x}M\rightarrow M$ at $-\nabla\phi(x)$;

  3. 3.

the composition is valid since $T_{-\nabla\phi(x)}T_{x}M$ can be canonically
identified with $T_{x}M$.

###### Proof.

The formula (7) is already proven in [16, Proposition 4.1], whose proof can be
simplified thanks to our assumptions. Define $y:=F(x)$. By the assumption that
$y$ is not in the cut locus of $x$, $\operatorname{Hess}_{x}d^{2}_{y}/2$ is
well-defined. Shrink the neighborhood $U$ of $x$ if necessary so that for
$(w,z)\in U\times U$, $w$ is not in the cut loci of $y$ and $z$. Define the
following function $g$ on $U\times U$,

$g(w,z):=\exp_{w}\left(-\nabla d_{y}^{2}(w)/2+\Pi_{z\rightarrow w}\left[\nabla
d_{y}^{2}(z)/2-\nabla\phi(z)\right]\right),$

where $\Pi_{z\rightarrow w}:T_{z}M\rightarrow T_{w}M$ denotes the parallel
transport of tangent vectors along the minimal geodesic from $z$ to $w$. For
$z\in U$, since $\Pi_{z\rightarrow z}$ is the identity map on $T_{z}M$,
$g(z,z)=F(z)$. For $w\in U$, $g(w,x)=\exp_{w}(-\nabla d_{y}^{2}(w)/2)\equiv y$
is a constant, where we used the assumption $\nabla d^{2}_{y}(x)/2=\nabla(x)$
for the first equality and used that $w$ is not in the cut locus of $y$ for
the second one. It follows that

$\displaystyle\operatorname{d}F(x)$
$\displaystyle=\partial_{w}g(x,x)+\partial_{z}g(x,x)=\partial_{z}g(x,x)$
$\displaystyle=\operatorname{d}\exp_{x}|_{-\nabla\phi}\circ(\operatorname{Hess}_{x}d^{2}_{y}/2-\operatorname{Hess}_{x}\phi),$

where we applied the chain rule and the definition of Hessian in the last
equality. ∎

####  2.3.2 Lipschitz continuous optimal transport maps of Wasserstein
barycenters

To better illustrate our approach towards the absolute continuity of
Wasserstein barycenters of finitely many measures, we recall the following
result corresponding to the case of two measures.

######  Proposition 2.19 (Regularity of displacement interpolations, [49,
Theorem 8.5, Theorem 8.7]).

Let $M$ be a complete Riemannian manifold. Let
$t\in[0,1]\mapsto\mu_{t}\in\mathcal{W}_{2}(M)$ be a minimal geodesic in the
Wasserstein space $\mathcal{W}_{2}(M)$ such that both $\mu_{0}$ and $\mu_{1}$
have compact support. For any $0<\lambda<1$, $\mu_{\lambda}$ is the barycenter
of $(1-\lambda)\delta_{\mu_{0}}+\lambda\,\delta_{\mu_{1}}$. The optimal
transport map from $\mu_{\lambda}$ to $\mu_{0}$ is Lipschitz continuous, and
it follows that $\mu_{\lambda}$ is absolutely continuous provided that
$\mu_{0}$ is absolutely continuous.

The Lipschitz continuity in 2.19 can be shown as a consequence of Mather’s
shortening lemma [49, Chapter 8]. Since Lipschitz maps send Lebesgue
negligible sets to Lebesgue negligible sets, the last statement on absolute
continuity follows. Another approach to the Lipschitz continuity is given by
Bernard and Buffoni [6] using the Hamilton-Jacobi equation, which is
generalized to non-compact settings by Fathi and Figalli [19]. For the case
when both $\mu_{0}$ and $\mu_{1}$ are absolutely continuous measures on
Euclidean spaces, McCann [35, Proposition 1.3] presented a concise proof of
the Lipschitz continuity. See relevant references in Villani [49,
Bibliographical notes of Chapter 8]. The goal of this subsection is to
generalize 2.19.

We deduce the following Lipschitz continuity from the $c$-concave functions
defined in 2.16, which are related to barycenter selection maps and thus
barycenters of
$\lambda_{1}\mu_{1}+\sum_{i=2}^{n}\lambda_{i}\,\delta_{\delta_{x_{i}}}$.
Recall that a measurable barycenter selection map $B:M^{n}\rightarrow M$ (2.4)
sends $(x_{1},\ldots,x_{n})\in M^{n}$ to a barycenter of
$\sum_{i=1}^{n}\lambda_{i}\,\delta_{x_{i}}$. In the following propositions,
the constant $\lambda_{i},1\leq i\leq n$ for $B$ are given in the context.

######  Lemma 2.20 (Lipschitz continuous maps $F=\exp(-\nabla g_{1})$).

Let $M$ be a complete Riemannian manifold. Given an integer $n\geq 2$, let
$\lambda_{i}>0,1\leq i\leq n$, be $n$ positive real numbers such that
$\sum_{i=1}^{n}\lambda_{i}=1$. Fix a non-empty compact subset $X\subset M$ and
a point $\boldsymbol{x}^{\prime}=(x_{2},\ldots,x_{n})\in M^{n-1}$. Denote by
$Y$ the compact set of all barycenters of
$\sum_{i=1}^{n}\lambda_{i}\,\delta_{x_{i}}$ when $x_{1}$ runs through $X$.
Define the function $g_{1}:y\in
M\mapsto-1/\lambda_{1}\sum_{i=2}^{n}\lambda_{i}\,c(y,x_{i})$. It is
$\mathcal{C}^{3}$ smooth in a neighborhood of $Y$ and thus $F:=\exp(-\nabla
g_{1}):Y\rightarrow M$ is a well-defined Lipschitz continuous function. We
have $F(Y)=X$ and the following characterization of $F$:

$z\in Y\text{ and }x_{1}=F(z)\iff x_{1}\in X\text{ and }z\text{ is a
barycenter of }\sum_{i=1}^{n}\lambda_{i}\,\delta_{x_{i}}.$ (8)

Given a measure $\mu_{1}\in\mathcal{W}_{2}(M)$ with support $X$ and a
measurable barycenter selection map $B:M^{n}\rightarrow M$, $\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\mu}:=B_{\\#}(\mu_{1}\otimes\delta_{x_{2}}\otimes\cdots\otimes\delta_{x_{n}})$
is a barycenter of
$\lambda_{1}\,\delta_{\mu_{1}}+\sum_{i=2}^{n}\lambda_{i}\,\delta_{\delta_{x_{i}}}$
and $(\operatorname{Id},F)_{\\#}\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\mu}$ is an optimal transport plan between $\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\mu}$ and $\mu_{1}$.

###### Proof.

2.17 implies the differential property of $g_{1}$ and thus the Lipschitz
continuity of $F$. Since $g_{1}$ restricted to $Y$ is a $c$-concave function
(2.16) and $\nabla g_{1}$ exists on $Y$, by defining $g_{1}^{c}:x\in
X\mapsto\min_{y\in Y}\\{c(x,y)-g_{1}(y)\\}$, a well-known property of
$c$-concave functions proven by McCann [36, Lemma 7] shows that

$z\in Y\text{ and }x_{1}=\exp(-\nabla g_{1})(z)=F(z)\iff(x_{1},z)\in X\times
Y\text{ and }g_{1}^{c}(x_{1})+g_{1}(z)=c(x_{1},z).$

Note that though McCann’s lemma is proven for compact manifolds, the arguments
of its proof only depend on the existence of gradient $\nabla g_{1}$ and the
compactness of $X$ and $Y$. For $x_{1}\in X$, we have
$g_{1}^{c}(x_{1})=1/\lambda_{1}\inf_{w\in
M}\sum_{i=1}^{n}\lambda_{i}\,c(w,x_{i})^{2}$ (2.16) and thus

$z\in Y\text{ and
}g_{1}^{c}(x_{1})+g_{1}(z)=c(x_{1},z)\iff\sum_{i=1}^{n}\lambda_{i}\,d(z,x_{i})^{2}=\inf_{w\in
M}\sum_{i=1}^{n}\lambda_{i}\,d(w,x_{i})^{2},$

which implies the characterization (8). $F(Y)=X$ follows from (8) and the
definition of $Y$.

Since $\gamma:=\mu_{1}\otimes\delta_{x_{2}}\otimes\cdots\otimes\delta_{x_{n}}$
is the only measure on $M^{n}$ with marginals
$\mu_{1},\delta_{x_{2}},\ldots,\delta_{x_{n}}$ in this order, it is the
(unique) multi-marginal optimal transport plan of its marginals. 2.8 shows
that $\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\mu}=B_{\\#}\gamma$ is a Wasserstein barycenter. Denote by $p_{1}:M\times
M^{n-1}\rightarrow M$ the canonical projection map. Since
$p_{1}(x_{1},\boldsymbol{x}^{\prime})=x_{1}=F(B(x_{1},\boldsymbol{x}^{\prime}))$
for $x_{1}\in X$ by the characterization (8), 2.8 shows that
$(B,p_{1})_{\\#}\gamma=(B,F\circ
B)_{\\#}\gamma=(\operatorname{Id},F)_{\\#}\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\mu}$ is an optimal transport plan between $\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\mu}$ and $\mu_{1}$. ∎

2.20 implies that any barycenter selection map on
$X\times\\{\boldsymbol{x}^{\prime}\\}$ is injective. The following lemma by
Kim and Pass [30, Lemma 3.5] generalizes this injectivity, and it will help us
to generalize 2.20.

######  Lemma 2.21.

Let $M$ be a complete Riemannian manifold. Given an integer $n\geq 2$, let
$\lambda_{i}>0,1\leq i\leq n$, be $n$ positive real numbers such that
$\sum_{i=1}^{n}\lambda_{i}=1$ and let $\mu_{i}\in\mathcal{W}_{2}(M),1\leq
i\leq n$, be $n$ measures with compact support. If $\gamma$ is a multi-
marginal optimal transport plan with marginals $\mu_{1},\ldots,\mu_{n}$, then

$\boldsymbol{x},\boldsymbol{y}\in\operatorname{supp}(\gamma),\quad\boldsymbol{x}\neq\boldsymbol{y}\implies\operatorname{bary}(\\{\boldsymbol{x}\\})\cap\operatorname{bary}(\\{\boldsymbol{y}\\})=\emptyset,$

where $\operatorname{supp}(\gamma)$ is the support of $\gamma$ and
$\operatorname{bary}(\\{x_{1},\ldots,x_{n}\\})$ is the set of barycenters of
$\sum_{i=1}^{n}\lambda_{i}\,\delta_{x_{i}}$.

To avoid being lengthy, we skip the proof of above lemma [30, Lemma 3.5],
which is based on $c$-cyclical monotonicity and 2.17. Though the proof in the
given reference is for the case when $\lambda_{1}=\cdots=\lambda_{n}=1/n$,
there is no essential difficulty to apply it to the stated case [30, proof of
Theorem 6.1]. The following proposition constructs an optimal transport map
from $\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\mu}:=B_{\\#}\gamma$ to $\mu_{1}$ when $\mu_{i},2\leq i\leq n$ are
discrete measures and thus generalizes 2.20. The optimal transport map may
fail to be a Lipschitz map, but it is a disjoint union of Lipschitz maps.
Recall that given (at most) countably many disjoint subsets $Y_{j}\subset
M,j\in J\subset\mathbb{N}$ with functions $F_{j}:Y_{j}\rightarrow M$, the
_disjoint union_ $F$ of $F_{j},j\in J$ is the function defined on $\cup_{j\in
J}F_{j}$ such that $F|_{Y_{j}}=F_{j}$. We shall use conditional measures (2.5)
to deduce further conclusions from $F_{j}$’s Lipschitz continuity.

######  Proposition 2.22.

Let $M$ be an $m$-dimensional complete Riemannian manifold. Given an integer
$n\geq 2$, let $\lambda_{i}>0,1\leq i\leq n$, be $n$ positive real numbers
such that $\sum_{i=1}^{n}\lambda_{i}=1$. Let $\mu_{1}\in\mathcal{W}_{2}(M)$ be
a measure with compact support and let $\mu_{i}\in\mathcal{W}_{2}(M),2\leq
i\leq n$, be $n-1$ discrete measures, i.e., measures concentrated on at most
countably many points. Given a multi-marginal optimal transport plan $\gamma$
of $\mu_{1},\ldots,\mu_{n}$ in this order and a measurable barycenter
selection map $B:M^{n}\rightarrow M$, the measure $\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\mu}:=B_{\\#}\gamma$ is a barycenter of
$\sum_{i=1}^{n}\lambda_{i}\,\delta_{\mu_{i}}$. The support of
$\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\mu}$ is contained in a disjoint union of at most countably many compact
sets, and on each of them 2.20 defines a Lipschitz continuous map with compact
subset $X\subset M$ and point $\boldsymbol{x^{\prime}}\in M^{n-1}$ such that
$X\times\\{\boldsymbol{x}^{\prime}\\}$ is contained in the support of
$\gamma$. Denote by $F$ the disjoint union of the Lipschitz maps.
$(\operatorname{Id},F)_{\\#}\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\mu}$ is an optimal transport plan between $\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\mu}$ and $\mu_{1}$.

For positive real numbers $\delta,\epsilon>0$, we define the set

$\mathcal{E}_{\epsilon,\delta}:=\left\\{\mu\in\mathcal{W}_{2}(M)\mid\forall\,N\in\mathcal{B}(M),\,\operatorname{Vol}(N)<\delta\implies\mu(N)\leq\epsilon\right\\}.$

If there is a common Lipschitz constant $C$ of the Lipschitz maps, then
$\mu_{1}\in\mathcal{E}_{\epsilon,\delta}\implies\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\mu}\in\mathcal{E}_{\epsilon,\delta/C^{m}}$.

###### Proof.

2.8 shows that $\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\mu}$ is a Wasserstein barycenter. Let us reveal more details of $\gamma$.
Denote by $p_{1}$ and $p_{2}$ the canonical projections sending
$\boldsymbol{x}=(x_{1},\boldsymbol{x}^{\prime})\in M\times M^{n-1}$ to
$x_{1}\in M$ and $\boldsymbol{x}^{\prime}\in M^{n-1}$ respectively. The
measure $\pi:={p_{2}}_{\\#}\gamma$ on $M^{n-1}$ is discrete since its
marginals $\mu_{2},\ldots,\mu_{n}$ are so. Denote by
$\\{\boldsymbol{x}_{j}^{\prime}\\}_{j\in J}$ the support of $\pi$, where
$J\subset\mathbb{N}$ is an at most countable index set. For each $j\in J$, we
introduce the following definitions. Define
$\pi_{j}:=\pi(\\{\boldsymbol{x}^{\prime}_{j}\\})\geq 0$. Denote by
$\operatorname{supp}\gamma$ the support of $\gamma$ and define
$X_{j}:=p_{1}(\operatorname{supp}\gamma\cap(M\times\\{\boldsymbol{x}_{j}^{\prime}\\}))$.
Applying 2.20 to $X_{j}$ and $\boldsymbol{x}_{j}^{\prime}\in M^{n-1}$, we get
a compact set $Y_{j}$ and a Lipschitz continuous map $F_{j}:Y_{j}\rightarrow
M$ such that $F_{j}(Y_{j})=X_{j}$.

We claim that $Y_{i}\cap Y_{k}=\emptyset$ for two different indices $i,k\in
J$. Indeed, if $z\in Y_{i}\cap Y_{k}$ for $i,k\in J$, then by the
characterization of $F_{i},F_{k}$ in 2.20,
$z\in\operatorname{bary}(\\{(F_{i}(z),\boldsymbol{x}_{i}^{\prime})\\})\cap\operatorname{bary}(\\{(F_{k}(z),\boldsymbol{x}_{k}^{\prime})\\})$,
where $\operatorname{bary}(\\{(x_{1},\ldots,x_{n})\\})$ denotes the set of
barycenters of $\sum_{l=1}^{n}\lambda_{l}\,\delta_{x_{l}}$. Since
$\operatorname{supp}\gamma=\bigcup_{j\in
J}X_{j}\times\\{\boldsymbol{x}_{j}^{\prime}\\}$ and $F_{j}(Y_{j})=X_{j}$, 2.21
forces that $\boldsymbol{x}^{\prime}_{i}=\boldsymbol{x}^{\prime}_{k}$ and thus
$i=k$. Define $F$ as the disjoint union of $F_{j},j\in J$, i.e.,
$F|_{Y_{j}}=F_{j}$. Since
$p_{1}(x,\boldsymbol{x}^{\prime}_{j})=x=F(B(x,\boldsymbol{x}^{\prime}_{j}))$
for $x\in X_{j}$, 2.8 implies that $(B,p_{1})_{\\#}\gamma=(B,F\circ
B)_{\\#}\gamma=(\operatorname{Id},F)_{\\#}\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\mu}$ is an optimal transport plan between $\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\mu}$ and $\mu_{1}$. Since $\bigcup_{j\in
J}Y_{j}=\operatorname{bary}(\bigcup_{j\in
J}X_{j}\times\\{\boldsymbol{x}_{j}^{\prime}\\})=\operatorname{bary}(\operatorname{supp}\gamma)$
is a compact set by our definitions of $X_{j},Y_{j}$ and 2.3, it contains the
support of $\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\mu}$. This shows our description of the support of $\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\mu}$.

Define the index set $J^{\prime}:=\\{j\in
J\mid\pi_{j}:=\pi(\\{\boldsymbol{x}^{\prime}\\})>0\\}$, which is not equal to
$J$ if $\bigcup_{j\in J^{\prime}}\boldsymbol{x}^{\prime}_{j}$ is not closed.
We claim that $\mu_{1}(X_{i}\cap X_{k})=0$ for two different indices $i,k\in
J^{\prime}$. Consider the conditional measure such that
$\operatorname{d}\gamma(\boldsymbol{x})=\gamma(\operatorname{d}\boldsymbol{x},\boldsymbol{x}^{\prime})\operatorname{d}\pi(\boldsymbol{x}^{\prime})$.
For $j\in J^{\prime}$, define $\nu_{j}:=\frac{1}{\pi_{j}}\mu_{1}|_{X_{j}}$ and
$\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\nu}_{j}:=B_{\\#}\gamma(\cdot,\boldsymbol{x}_{j}^{\prime})$. Note that for
$j\in J^{\prime}$ and $\boldsymbol{R}\in\mathcal{B}(M^{n})$,
$\gamma[\boldsymbol{R}\cap(M\times\\{\boldsymbol{x}_{j}^{\prime}\\})]=\gamma(\boldsymbol{R},\boldsymbol{x}_{j}^{\prime})\,\pi_{j}$
by 2.5, so $\gamma(\cdot,\boldsymbol{x}_{j}^{\prime})$ is concentrated on
$X_{j}\times\\{\boldsymbol{x}_{j}^{\prime}\\}$ and its first marginal is
$\nu_{j}$. Furthermore, for a measurable map $f:M^{n}\rightarrow M$,

$\forall
N\in\mathcal{B}(M),\quad[f_{\\#}\gamma](N)=\gamma(f^{-1}(N))=\sum_{j\in
J}\gamma(f^{-1}(N),\boldsymbol{x}_{j}^{\prime})\,\pi_{j}=\sum_{j\in
J^{\prime}}[f_{\\#}\gamma(\cdot,\boldsymbol{x}_{j}^{\prime})](N)\,\pi_{j}.$

Taking $f=p_{1}$ and $f=B$, we get $\mu_{1}=\sum_{j\in
J^{\prime}}\pi_{j}\,\nu_{j}$ and $\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\mu}=\sum_{j\in J^{\prime}}\pi_{j}\,\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\nu}_{j}$. Hence, given $i\in J^{\prime}$, $\mu_{1}(X_{i})=\sum_{j\in
J^{\prime}}\mu_{1}|_{X_{j}}(X_{i})$ and thus $\mu_{1}(X_{i}\cap X_{k})=0$ for
$k\in J^{\prime}$ different from $i$.

Assume that $C$ is a common Lipschitz constant of all $F_{j},j\in J$. For any
Borel set $N\in\mathcal{B}(M)$, there exist Borel sets $N_{j},j\in J$ such
that $F_{j}(N\cap Y_{j})\subset N_{j}\subset X_{j}$ and
$\operatorname{Vol}(N_{j})\leq C^{m}\operatorname{Vol}(N\cap Y_{j})$ [46,
Proposition 12.6, Proposition 12.12, Remark after Proposition 12.12] (c.f.
[49, Proof of Theorem 8.7]). For $j\in J^{\prime}$, since
$\gamma(\cdot,\boldsymbol{x}^{\prime}_{j})$ is the product measure of its
marginals, 2.20 shows that ${F_{j}}_{\\#}\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\nu}_{j}=\nu_{j}$. It follows that $\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\nu}_{j}(N\cap Y_{j})\leq\nu_{j}(N_{j})$ for $j\in J^{\prime}$ and thus

$\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\mu}(N)=\sum_{j\in J^{\prime}}\pi_{j}\,\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\nu}_{j}(N\cap Y_{j})\leq\sum_{j\in
J^{\prime}}\pi_{j}\frac{1}{\pi_{j}}\mu_{1}|_{X_{j}}(N_{j})=\sum_{j\in
J^{\prime}}\mu_{1}(N_{j})=\mu_{1}(\bigcup_{j\in J^{\prime}}N_{j}),$ (9)

where for the equalities we used $N_{j}\subset X_{j}$ and $\mu_{1}(X_{i}\cap
X_{k})=0$ if $i,k$ are two different indices in $J^{\prime}$. Since
$Y_{j},j\in J$ are disjoint, $\operatorname{Vol}(\bigcup_{j\in
J^{\prime}}N_{j})\leq C^{m}\sum_{j\in J^{\prime}}\operatorname{Vol}(N\cap
Y_{j})\leq C^{m}\operatorname{Vol}(N)$, which concludes the proof thanks to
(9). ∎

######  Remark 2.23.

Figuratively speaking, the sets $X_{j},j\in J$ create a tiling of the support
of $\mu_{1}$ and the points $\boldsymbol{x}^{\prime}_{j},j\in J$ pull them
apart (via barycenter selection maps) into disjoint sets $Y_{j},j\in J$, which
contain different pieces of the support of $\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\mu}$ separately.

####  2.3.3 Proof of absolute continuity

Consider the probability measure
$\mathbb{P}=\sum_{i=1}^{n}\lambda_{i}\,\delta_{\mu_{i}}$ with positive real
numbers $\lambda_{i}$ and compactly supported measures
$\mu_{i}\in\mathcal{W}_{2}(M)$. We can approximate each $\mu_{i}$ for $2\leq
i\leq n$ with discrete measures to apply 2.22. If $\mu_{1}$ is absolutely
continuous, then $\mathbb{P}$ has a unique barycenter $\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\mu}$, which is approximated by the barycenters of the approximating
sequence (2.2). Recall that the sets $\mathcal{E}_{\epsilon,\delta}$ (2.12)
fully characterize absolutely continuous measures and are closed with respect
to weak convergence. Hence, to prove the absolute continuity of
$\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\mu}$, it remains to find a common Lipschitz constant $C$ for $F$ defined
as in 2.20 valid for any element of the whole approximating sequence. Then we
get the result thanks to 2.22. Note that the domain $Y$ of $F$ is at least
varying along the approximating sequence, so the existence of $C$ is not
simply a direct consequence of compactness. More precisely, we shall prove:

######  Theorem 2.24 (Absolute continuity of the barycenter of
$\sum_{i=1}^{n}\lambda_{i}\,\delta_{\mu_{i}}$).

Let $M$ be a complete Riemannian manifold. Given an integer $n\geq 2$, let
$\lambda_{i}>0,1\leq i\leq n$, be $n$ positive real numbers such that
$\sum_{i=1}^{n}\lambda_{i}=1$ and let $\mu_{i}\in\mathcal{W}_{2}(M),1\leq
i\leq n$, be $n$ probability measures with compact support. If $\mu_{1}$ is
absolutely continuous, then the unique barycenter $\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\mu}$ of $\sum_{i=1}^{n}\lambda_{i}\,\delta_{\mu_{i}}$ is absolutely
continuous with compact support.

###### Proof.

The uniqueness and compact support of $\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\mu}$ follow from Section 2.2 and 2.3. We approximate each $\mu_{i}$ for
$2\leq i\leq n$ in $(\mathcal{W}_{2}(M),W_{2})$ by a sequence of discrete
measures $\\{\mu_{i}^{j}\\}_{j\geq 1}$ whose supports are contained in the
compact support of $\mu_{i}$. Then
$\mathbb{P}_{j}:=\lambda_{1}\delta_{\mu_{1}}+\sum_{i=2}^{n}\lambda_{i}\,\delta_{\mu_{i}^{j}}$
converges to $\mathbb{P}$ in $\mathcal{W}_{2}(\mathcal{W}_{2}(M))$. By the law
of large numbers for Wasserstein barycenters (2.2), the unique barycenter
$\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\mu}_{j}$ of $\mathbb{P}_{j}$ converges in $(\mathcal{W}_{2}(M),W_{2})$ to
the unique barycenter $\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\mu}$ of $\mathbb{P}$.

Denote by $\gamma_{j}$ a multi-marginal optimal transport plan of marginal
measures $\mu_{1},\mu_{2}^{j},\ldots,\mu_{n}^{j}$ in this order. Fix an index
$j$, a non-empty compact subset $X\subset M$ and a point
$\boldsymbol{x}^{\prime}:=(x_{2},\ldots,x_{n})\in M^{n-1}$ such that
$X\times\\{\boldsymbol{x}^{\prime}\\}\subset\operatorname{supp}\gamma_{j}$,
where $\operatorname{supp}\gamma_{j}$ denotes the support of $\gamma_{j}$.
Applying 2.20 to $X$ and $\boldsymbol{x}^{\prime}$, we obtain a Lipschitz
continuous function $F=\exp(-\nabla g_{1})$ on a compact set $Y$. We claim
that there exists a Lipschitz constant $C$ of $F$ on $Y$ independent of $j,X$
and $\boldsymbol{x}^{\prime}$. Recall that
$g_{1}(y):=-1/\lambda_{1}\sum_{i=2}^{n}\lambda_{i}\,c(y,x_{i})$ is
$\mathcal{C}^{3}$ smooth in a neighborhood of $Y$. Given $z\in Y$, since $z$
is a barycenter of $\sum_{i=1}^{n}\lambda_{i}\,\delta_{x_{i}}$ (2.20) with
$x_{1}:=F(z)$, we have $\sum_{i=1}^{n}\nabla d^{2}_{x_{i}}(z)=0$ thanks to
2.17 and thus $\nabla d^{2}_{x_{1}}/2(z)=\nabla g_{1}(z)$. Moreover, 2.17
enables us to apply 2.18, which implies

$\displaystyle\operatorname{d}F(z)=\operatorname{d}\exp(-\nabla g_{1})(z)$
$\displaystyle=\operatorname{d}\exp_{z}|_{-\nabla
g_{1}(z)}\circ(\operatorname{Hess}_{z}d^{2}_{x_{1}}/2-\operatorname{Hess}_{z}g_{1})$
$\displaystyle=\operatorname{d}\exp_{z}|_{-\nabla
g_{1}(z)}\circ\frac{1}{2\lambda_{1}}\sum_{i=1}^{n}\lambda_{i}\operatorname{Hess}_{z}d^{2}_{x_{i}}.$
(10)

In (10), $\sum_{i=1}^{n}\lambda_{i}\,\operatorname{Hess}_{z}d^{2}_{x_{i}}$ is
positive semi-definite since $z$ reaches the global minimum of $w\in
M\mapsto\sum_{i=1}^{n}\lambda_{i}\,d(w,x_{i})^{2}$. We now bound (10) using
compactness as follows. By 2.3 and our construction of $\mathbb{P}_{j}$, the
union of the supports of $\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\mu}$, $\mu_{i}$, $\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\mu}_{j}$ and $\mu_{i}^{j}$ for $1\leq i\leq n$ and $j\geq 1$ is compact.
Hence, independent of $z$, $j$ and $\boldsymbol{x}^{\prime}$,
$\operatorname{d}\exp_{z}|_{-\nabla g_{1}}$ is uniformly bounded (in norm) and
$\sum_{i=1}^{n}\lambda_{i}\,\operatorname{Hess}_{z}d^{2}_{x_{i}}$ is uniformly
bounded from above by the Rauch comparison theorem for Hessians of distance
functions, which is applicable here and gives a constant upper bound thanks to
the compactness, see [16, Lemma 3.12 and Corollary 3.13] or [39, Theorem
6.4.3]. This shows the existence of the claimed Lipschitz constant $C$. We
remark that the absolute continuity of $\mu_{1}$ is not needed for the
existence of $C$.

Applying 2.22 to measures $\mu_{1},\mu_{2}^{j},\ldots,\mu_{n}^{j}$, we have
for $\epsilon,\delta>0$,
$\mu_{1}\in\mathcal{E}_{\epsilon,\delta}\implies\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\mu}_{j}\in\mathcal{E}_{\epsilon,\delta/C^{m}}$ since $\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\mu}_{j}$ is the unique barycenter of $\mathbb{P}_{j}$. As
$\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\mu}_{j}$ converges to $\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\mu}$ weakly, 2.12 shows that all measures $\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\mu}_{j}$ for $j\geq 1$ and $\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\mu}$ are absolutely continuous since $\mu_{1}$ is so. ∎

##  3 Hessian equality for Wasserstein barycenters

In this section, we prove the Hessian equality for Wasserstein barycenters of
finitely many measures (3.13). A similar property is named as the _second
order balance_ (inequality) by Kim and Pass [31, Theorem 4.4], but being an
equality instead of an inequality is crucial for our proof of 4.2. Let us take
a special case to illustrate this equality. Consider the reduced case in 2.20.
Namely, take $n$ positive numbers $\lambda_{i}>0$ such that
$\sum_{i=1}^{n}\lambda_{i}=1$ and denote by $\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\mu}$ the barycenter of $\sum_{i=1}^{n}\lambda_{i}\,\delta_{\mu_{i}}$,
where $\mu_{1}$ is absolutely continuous with compact support and
$\mu_{i}=\delta_{x_{i}},2\leq i\leq n$, are Dirac measures. Let us set
$\phi_{1}(z):=g_{1}(z):=-1/\lambda_{1}\sum_{i=2}^{n}\lambda_{i}\,c(z,x_{i})$
and $\phi_{i}(z):=c(z,x_{i}),2\leq i\leq n$. Thanks to 2.3 and 2.17, if $z$ is
in the support of $\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\mu}$, then $z$ is not in the cut locus of any $x_{i}$, which implies
$\exp(-\nabla\phi_{i})_{\\#}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\mu}}=\mu_{i}$ for $2\leq i\leq n$. Besides, by definition of the
$\phi_{i}$’s, $\sum_{i=1}^{n}\lambda_{i}\,\phi_{i}\equiv 0$; therefore
$\sum_{i=1}^{n}\lambda_{i}\nabla\phi_{i}(z)=0$. Consequently we get
$\sum_{i=1}^{n}\lambda_{i}\operatorname{Hess}_{z}\phi_{i}=0$, which is the
Hessian equality we are referring to.

###  3.1 Local semi-concavity

The c-concave functions we meet when dealing with optimal transport maps are
locally semi-concave functions. The weak second-order regularity of such maps
is discussed in this part.

######  Definition 3.1 (Semi-concavity).

Let $M$ be a Riemannian manifold. Fix an open subset $O\subset M$. A function
$\phi:O\rightarrow\mathbb{R}$ is semi-concave at $x\in O$ if there exists a
geodesically convex ball $B(x)$ centered at $x$ and a smooth function
$V:B(x)\rightarrow\mathbb{R}$ such that $\phi+V$ is geodesically concave
throughout $B(x)$. The function $\phi$ is locally semi-concave on $O$ if it is
semi-concave at each point of $O$.

It was proven by Bangert [5, (2.3) Satz] that the notion of local semi-
concavity does not depend on the Riemannian metric. This property also follows
from the following characterization of locally semi-concave functions. We
remark that in the Euclidean case, this characterization is already given in
[47, Proposition 4.3, Proposition 4.8] and [15, Theorem 5.1]. In [19, Appendix
A], it is taken as the definition of local semi-concavity.

######  Proposition 3.2 (Characterization of local semi-concavity, [49,
Proposition 10.12]).

Let $M$ be a Riemannian manifold. Fix an open subset $O$ of $M$. A function
$f:O\rightarrow\mathbb{R}$ is locally semi-concave if and only if for each
$x\in O$, $(\varphi,U)$ chart defined around $x$, there exist a linear form
$\omega:\mathbb{R}^{m}\rightarrow\mathbb{R}$, such that for all
$\tilde{x}\in\varphi(U)$ and $u\in\mathbb{R}^{m}$ we have

$(f\circ\varphi^{-1})\left(\tilde{x}+u\right)\leq(f\circ\varphi^{-1})(\tilde{x})+\omega(u)+o(\|u\|_{2}).$

Hence, a function is locally semi-concave if and only if it is so when
expressed in local charts [19, discussion after Lemma A.9]. This property
yields some weak second-order regularity as explained in the next part.

###  3.2 Approximate differentiability

We start by recalling the definition of _density point_ on a Riemannian
manifold, then we compare it to its usual Euclidean counterpart.

######  Lemma 3.3 (Density points on manifolds).

Let $M$ be a Riemannian manifold and let $A$ be a Borel subset of $M$. We call
$x\in M$ a density point of $A$ (with respect to $\operatorname{Vol}$) if

$\lim_{r\downarrow 0}\frac{\operatorname{Vol}[B(x,r)\setminus
A]}{\operatorname{Vol}[B(x,r)]}=0,$

where $B(x,r)\subset M$ is the closed ball centered at $x$ with radius $r>0$.
This definition is equivalent to the standard one with respect to the Lebesgue
measure after pulling $x$ and $A$ back to the Euclidean space through an
arbitrary chart around $x$. In particular, almost every point of $A$ is a
density point of $A$ with respect to $\operatorname{Vol}$.

###### Proof.

Denote by $m$ the dimension of $M$. In a local chart $(\psi,U)$ with $U$ a
small enough neighborhood of $x\in M$, the metric of $M$ is bounded (from both
sides) by the metric of $\mathbb{R}^{m}$ with constant scales $0<c_{1}<c_{2}$.
It follows that
$c_{1}^{m}\operatorname{Leb}(\psi(N))\leq\operatorname{Vol}(N)\leq
c_{2}^{m}\operatorname{Leb}(\psi(N))$ for any measurable subset $N\subset U$
[46, Proposition 12.6 and 12.7], where $\operatorname{Leb}$ denotes the
Lebesgue measure on $\mathbb{R}^{m}$. Hence, $x$ is a density point of $A$ if
and only if

$\lim_{r\downarrow
0}\frac{\operatorname{Leb}[\psi(B(x,r))\setminus\psi(A)]}{\operatorname{Leb}[\psi(B(x,r))]}=0.$
(11)

Applying again the relation between the metric of $M$ and the metric of
$\mathbb{R}^{m}$, for any $r>0$, we have
$B(\psi(x),c_{1}\,r)\subset\psi(B(x,r))\subset B(\psi(x),c_{2}\,r)$, where we
use $B$ to denote closed balls in $\mathbb{R}^{m}$ as well. Therefore, (11) is
equivalent to that $\psi(x)$ is a density point of $\psi(A)$ with respect to
$\operatorname{Leb}$. ∎

In the next definition, we recall the definition of _approximate derivatives_
first on Euclidean space (see [9, 5.8(v)] and [20, 3.1.2] for more detailed
discussions) and then on manifolds.

######  Definition 3.4 (Approximate derivatives on Euclidean spaces).

Let $m,n\geq 1$ be two positive integers. Fix a Borel subset $A$ of
$\mathbb{R}^{m}$ and a measurable function $f:A\rightarrow\mathbb{R}^{n}$. We
call $l$ an approximate limit of $f$ at a point $x\in M$, written
$l=\operatorname{ap}\lim_{y\rightarrow x}f(y)$, if there exists a Borel set
$A_{x}\subset A$ such that $x$ is a density point of $A_{x}$ and
$\displaystyle\lim_{y\in A_{x},y\rightarrow x}f(y)=l$. A linear map
$L:\mathbb{R}^{m}\rightarrow\mathbb{R}^{n}$ is called the approximate
derivative of $f$ at a point $x\in A$ (denoted by $\operatorname{ap}D_{x}f$)
if

$\operatorname{ap}\lim_{y\rightarrow x}\frac{|f(y)-f(x)-L(y-x)|}{|y-x|}=0.$
(12)

The previous definition can be extended to the Riemannian setting as follows:

######  Lemma 3.5 (Approximate derivatives on manifolds).

Let $M$ be an $m$-dimensional Riemannian manifold $M$ and let
$\tilde{f}:\tilde{A}\rightarrow\mathbb{R}^{n}$ be a measurable function with
$\tilde{A}$ a Borel subset of $M$. Given an arbitrary local chart $(\psi,U)$
around a point $x\in\tilde{A}$, $\tilde{f}$ is said to be approximately
differentiable at $x$ if the approximate derivative
$\operatorname{ap}D_{\psi(x)}[\tilde{f}\circ\psi^{-1}|_{\psi(\tilde{A}\cap
U)}]$ exists. The approximate derivative of $\tilde{f}$ at $x$ is then defined
as

$\operatorname{ap}D_{x}\tilde{f}:=\operatorname{ap}D_{x}[\tilde{f}\circ\psi^{-1}|_{\psi(\tilde{A}\cap
U)}]\circ\operatorname{d}_{x}\psi.$

In particular, a constant function has null approximate derivative at density
points located in its domain.

###### Proof.

In Euclidean space, approximate derivatives are unique when they exist [18,
Theorem 6.3]. Since density points are well-defined for manifolds by 3.3 and
coordinate changes for $M$ are ${C}^{1}$-diffeomorphisms, this together with
(12) imply that the existence of approximate derivative at a given point is
independent of the choice of the chart and the change of variables rule
applies. To show our last statement, note that $L=0$ satisfies (12) whenever
$f$ is a constant function. ∎

###  3.3 Hessian of semi-concave function

Aleksandrov proved that a semi-concave function on $\mathbb{R}^{m}$ admits a
Hessian defined almost everywhere, see [18, Chapter 6.4] for a proof.

######  Theorem 3.6 (Aleksandrov theorem).

Let $f:U\subset\mathbb{R}^{m}\rightarrow\mathbb{R}$ be a semi-concave
function. Then the Euclidean gradient $\nabla^{E}f$ of $f$ is defined
Lebesgue-almost everywhere on $U$:

$\nabla^{E}f:\tilde{U}\longrightarrow\mathbb{R}^{m},$

and the function $\nabla^{E}f$ is approximately differentiable Lebesgue-almost
everywhere on $\tilde{U}$ and its approximate derivatives
$(\partial_{ij}^{2}f)$ form a symmetric matrix. Besides, at every point $x$
where the approximate derivative of $\nabla^{E}f$ exists, $f$ admits a second-
order Taylor expansion:

$f(z)=f(x)+\langle\nabla^{E}f(x),z-x\rangle+\frac{1}{2}\langle\operatorname{ap}D_{x}\nabla^{E}f(z-x),z-x\rangle+o(\|z-x\|_{2}^{2}).$

######  Remark 3.7.

Aleksandrov actually proved a stronger statement on the differentiability
almost everywhere of the subdifferential $\partial f$ of $f$ as a multivalued
map.

In order to get a similar result in the Riemannian setting, we provide a brief
reminder on Riemannian Hessian. Thanks to the Riemannian metric, the Hessian
of a $\mathcal{C}^{2}$ function at a point $x\in M$ can either be seen as a
self-adjoint linear map from $T_{x}M$ to itself or as a symmetric bilinear
form on $T_{x}M\times T_{x}M$. The identification of the two Hessians is made
by duality through the Riemannian metric at $x$. While we shall adopt the
former point of view in the rest of the paper, we shall however use the latter
one in the next two paragraphs. This is due to the fact that the expression of
the Hessian of a function read in a chart is simpler for that choice.

In what follows, the Hessian of a $\mathcal{C}^{2}$ function on a Riemannian
manifold is a particular instance of a $(2,0)$-tensor $S$. Namely, for any
$x\in M$, and any charts $\varphi,\psi$ defined around $x$; there exist two
bilinear forms $S_{\varphi}$ and $S_{\psi}$ whose coefficients are continuous
functions such that $\forall\tilde{x}\in\varphi(U),\,\forall
u,v\in\mathbb{R}^{m},$

$S_{\varphi}{\scriptstyle(\tilde{x})}\,(u,v)=S_{\psi}{\scriptstyle(T(\tilde{x}))}(\operatorname{d}_{\tilde{x}}T(u),\operatorname{d}_{\tilde{x}}T(v)),$
(13)

where $T=\psi\circ\varphi^{-1}$ is assumed to be a smooth map defined on
$\varphi(U)\subset\mathbb{R}^{m}$ and $x\in U$. In the case of the Hessian of
a $\mathcal{C}^{2}$ function $f$, its expression in a chart $\varphi$ is given
by

$\operatorname{Hess}_{\tilde{x}}(f\circ\varphi^{-1})(\partial_{i},\partial_{j})=\partial^{2}_{ij}(f\circ\varphi^{-1})(\tilde{x})-\sum_{k=1}^{m}\Gamma_{ij}^{k}(\tilde{x})\,\partial_{k}(f\circ\varphi^{-1})(\tilde{x}),$

where $\partial_{i}$ are the canonical vectors of the related coordinate
system, see [39, Chapter 2] for more details.

In the particular case of a chart $\varphi$ inducing a normal coordinate
system at $x_{0}$, namely $\varphi^{-1}(u)=\exp_{x_{0}}(u)$, the matrix made
with the metric components is the identity at $\tilde{x}_{0}=\varphi(x_{0})$,
and all its first order partial derivatives vanish at $\tilde{x}_{0}$ [24,
2.89 bis]. Hence, the above formula at the point $\tilde{x}_{0}$ turns into
the simpler one

$\operatorname{Hess}_{\tilde{x}_{0}}(f\circ\varphi^{-1})(\partial_{i},\partial_{j})=\partial^{2}_{ij}(f\circ\varphi^{-1})(\tilde{x}_{0}).$

In other terms, when considered as a linear map from $\mathbb{R}^{m}$ to
itself, $\operatorname{Hess}_{\tilde{x}_{0}}(f\circ\varphi^{-1})$ coincides
with the derivative of $\nabla^{E}(f\circ\varphi^{-1})$ at $\tilde{x}_{0}$
(here we use the fact that the matrix induced by the metric at $\tilde{x}_{0}$
is the identity).

As a consequence, we are led to the following definition of Hessian for semi-
concave functions on a Riemannian manifold.

######  Definition 3.8 (Hessian of a semi-concave function).

Let $M$ be an $m$-dimensional Riemannian manifold, $f:O\rightarrow\mathbb{R}$
be a semi-concave function defined on an open subset $O$, and $A\subset O$ be
the subset of points where $f$ is differentiable.

The function $f$ is said to have an _approximate Hessian_ or simply a
_Hessian_ at a point $x\in A$ if there exists a chart $(\varphi,U)$ inducing a
_normal_ coordinate system around $x$, and such that
$\nabla^{E}(f\circ\varphi^{-1})$ is approximately differentiable at
$\varphi(x)$. Then the Hessian of $f$ at $x$ is the function
$\operatorname{Hess}_{x}f$ from $T_{x}M$ to $T_{x}M$ defined by

$\operatorname{Hess}_{x}f(u):=(\operatorname{d}_{x}\varphi)^{-1}\circ\operatorname{ap}D_{\varphi(x)}\nabla^{E}[f\circ\varphi^{-1}]\circ\operatorname{d}_{x}\varphi(u),\quad\forall\,u\in
T_{x}M.$ (14)

######  Remark 3.9.

First note that if $(\psi,V)$ is another chart defined in a neighborhood of
$x$, then $\nabla^{E}(f\circ\varphi^{-1})$ is approximately differentiable at
$\varphi(x)$ if and only if $\nabla^{E}(f\circ\psi^{-1})$ is approximately
differentiable at $\psi(x)$; indeed both vector fields are related by the
formula

${}^{t}(\operatorname{d}T)(\nabla^{E}(f\circ\varphi^{-1})(\varphi(z))=\nabla^{E}(f\circ\psi^{-1})(\psi(z)),$

where $z$ is close to $x$ and $T=\varphi\circ\psi^{-1}$ is a diffeomorphism
around $\psi(x)$. See the proof of Lemma 3.5 for a similar argument.

Since the Riemannian metric is assumed to be $\mathcal{C}^{2}$ and our
definition is pointwise, the fact that the Hessian of a $\mathcal{C}^{2}$
function is a tensor guarantees that our definition does not depend on the
choice of a chart (inducing a normal coordinate system).

To summarize the content of this part, we have obtained the analog of
Aleksandrov’s theorem for semi-concave functions locally defined on a
Riemannian manifold.

######  Proposition 3.10.

Let $(M,g)$ be a Riemannian manifold. Fix an open subset $O\subset M$ and a
locally semi-concave function $f:O\rightarrow\mathbb{R}$. For
$\operatorname{Vol}$-almost every $x\in O$, there exists a function
$\operatorname{Hess}_{x}f:T_{x}M\rightarrow T_{x}M$, called the Hessian of $f$
at $x$, such that

  * •

$\operatorname{Hess}_{x}f$ is a self-adjoint operator on $T_{x}M$;

  * •

the function $f$ satisfies the following second-order expansion at $x$,

$f(\exp_{x}u)=f(x)+\operatorname{d}_{x}f(u)+\frac{1}{2}g_{x}(\operatorname{Hess}_{x}f(u),u)+o(\|u\|^{2}),$
(15)

for $u\in T_{x}M$.

###  3.4 Differentiating optimal transport maps

In this part, we collect properties on optimal transport maps between
absolutely continuous measures on a Riemannian manifold taken from [16,
Sections 4 & 5]. These properties will be used in Section 4. We first recall
the definition of the (weak) differential of an optimal map under these
assumptions, then we state the change of variable formula.

######  Proposition 3.11 (Differentiating optimal transport maps, [16,
Proposition 4.1]).

Let $M$ be a complete Riemannian manifold. Given a $c$-concave function $\phi$
defined on $\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\mathcal{X}}\subset M$ with $\mathcal{X}$ a bounded open set, we set
$F:=\exp(-\nabla\phi)$, which is $\operatorname{Vol}$-almost everywhere well-
defined on $\mathcal{X}$. Fix a point $x\in\mathcal{X}$ such that
$\operatorname{Hess}_{x}\phi$ exists (14). Then the point $y:=F(x)$ is not in
the cut locus of $x$, $\nabla\phi(x)=\nabla d^{2}_{y}/2(x)$, and
$\operatorname{Hess}_{x}d^{2}_{y}/2-\operatorname{Hess}_{x}\phi$ is positive
semi-definite. Define the differential $\operatorname{d}F(x):T_{x}M\rightarrow
T_{y}M$ as

$\operatorname{d}F(x):=\operatorname{d}\exp_{x}|_{-\nabla\phi(x)}\circ(\operatorname{Hess}_{x}d^{2}_{y}/2-\operatorname{Hess}_{x}\phi),$
(16)

and $\operatorname{Jac}F(x):=\det\operatorname{d}F(x)$ as the determinant of
$\operatorname{d}F(x)$.

######  Proposition 3.12 (Interpolation and change of variable formula).

Let $M$ be a complete Riemannian manifold. Fix two absolutely continuous
measures $\mu,\nu\in\mathcal{W}_{2}(M)$ with supports contained in two bounded
open sets $\mathcal{X}$ and $\mathcal{Y}$ respectively. Let
$F:=\exp(-\nabla\phi)$ be the optimal transport map that pushes $\mu$ forward
to $\nu$, where $\phi\in\mathcal{I}^{c}(\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\mathcal{X}},\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\mathcal{Y}})$ is a $c$-concave function given by 2.15.

Denote by $\phi^{c}\in\mathcal{I}^{c}(\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\mathcal{Y}},\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\mathcal{X}})$ the $c$-conjugate of $\phi$. The set

$\Omega:=\left\\{x\in\mathcal{X}\mid
F(x)\in\mathcal{Y},\,\operatorname{Hess}_{x}\phi\text{ and
}\operatorname{Hess}_{F(x)}\phi^{c}\text{ exist }\right\\}$

satisfies the following properties:

  1. 1.

$\mu(\Omega)=1$;

  2. 2.

defining $F^{t}:=\exp(-t\nabla\phi)$ for $0\leq t\leq 1$, we have
$\operatorname{Jac}F^{t}>0$ on $\Omega$;

  3. 3.

denote by $f$ and $g$ the density functions of $\mu$ and $\nu$ respectively;
there exists a measurable subset $N\subset\Omega$ depending on these two
density functions such that $\mu(N)=1$ and for $x\in N$,

$f(x)=g(F(x))\operatorname{Jac}F(x)>0;$

  4. 4.

for any Borel function $A$ on $[0,+\infty)$ with $A(0)=0$, with the set $N$
given in 3,

$\int_{M}A(g)\operatorname{d}\operatorname{Vol}=\int_{N}A\left(\frac{f}{\operatorname{Jac}F}\right)\operatorname{Jac}F\operatorname{d}\operatorname{Vol}.$
(17)

(Either both integrals are undefined or both take the same value in
$\mathbb{R}\cup\\{+\infty,-\infty\\}$.)

###### Proof.

All the statements follow from [16, Claim 4.4, Theorem 4.2, Corollary 4.7]
except Property 2 for $t\in(0,1)$. Recall that for any $c$-concave function
$\phi$, we always have $\det[\operatorname{d}\exp_{x}|_{-t\nabla\phi(x)}]>0$
since $\exp_{x}(-t\nabla\phi(x))$ is not in the cut locus of $x$. Note that
$t\phi$ is $c$-concave for $0<t<1$, thus it suffices to show that

$\det[\operatorname{Hess}_{x}d^{2}_{F(x)}/2-\operatorname{Hess}_{x}\phi]>0\implies\det[\operatorname{Hess}_{x}d^{2}_{F^{t}(x)}/2-t\operatorname{Hess}_{x}\phi]>0,\quad\forall\,0<t<1.$
(18)

But
$\operatorname{Hess}_{x}d^{2}_{F^{t}(x)}/2-t\operatorname{Hess}_{x}d^{2}_{F(x)}/2$
is positive semi-definite for $0<t<1$ [16, Lemma 2.3], thus (18) follows from
Minkowski’s determinant inequality [48, (5.23)]. ∎

###  3.5 Proof of Hessian equality

The Hessian equality (19) to prove is a second-order relation. We first
demonstrate a first-order counterpart of this equality using the conclusion of
2.8 that relates barycenters in manifolds to Wasserstein barycenters.

######  Theorem 3.13 (Hessian equality for Wasserstein barycenters).

Let $M$ be a complete Riemannian manifold. Given an integer $n\geq 2$, let
$\lambda_{i}>0,1\leq i\leq n$, be $n$ positive real numbers such that
$\sum_{i=1}^{n}\lambda_{i}=1$ and let $\mu_{i}\in\mathcal{W}_{2}(M),1\leq
i\leq n$, be $n$ probability measures with compact support. We assume that
$\mu_{1}$ is absolutely continuous. The unique barycenter $\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\mu}$ of $\mathbb{P}:=\sum_{i=1}^{n}\lambda_{i}\,\delta_{\mu_{i}}$ is
absolutely continuous with compact support. For $1\leq i\leq n$, let
$F_{i}=\exp(-\nabla\phi_{i})$ be the optimal transport map pushing
$\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\mu}$ forward to $\mu_{i}$, where $\phi_{i}$ is a $c$-concave function
given by 2.15.

For $\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\mu}$-almost every $x\in M$, $x$ is a barycenter of
$\sum_{i=1}^{n}\lambda_{i}\,\delta_{F_{i}(x)}$, and we have the Hessian
equality

$\displaystyle\sum_{i=1}^{n}\lambda_{i}\operatorname{Hess}_{x}\phi_{i}$
$\displaystyle=0.$ (19)

###### Proof.

By 2.24, $\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\mu}$ is absolutely continuous with compact support. We now apply 2.8 to
$\mathbb{P}$. Since $\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\mu}$ is the unique barycenter of $\mathbb{P}$, it coincides with the
barycenter constructed in 2.8. Consider the identity map
$\operatorname{Id}:(M,\mathcal{B}(M),\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\mu})\rightarrow M$ as a random variable taking values in $M$. It has law
$\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\mu}$, and the random variable $F_{i}=F_{i}\circ\operatorname{Id}$ has law
$\mu_{i}$ for $1\leq i\leq n$. 2.8 implies that for $\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\mu}$-almost every $x\in M$, $x$ is a barycenter of
$\sum_{i=1}^{n}\lambda_{i}\,\delta_{F_{i}(x)}$.

Let $\Omega$ be a Borel subset of $M$ with $\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\mu}(\Omega)=1$ such that for $x\in\Omega$, $\nabla\phi_{i}(x)$ exists for
$1\leq i\leq n$ and $x$ is a barycenter of
$\sum_{i=1}^{n}\lambda_{i}\,\delta_{F_{i}(x)}$. Fix a point $x\in\Omega$. By
definition, $x$ reaches the minimum of the function

$h:w\in M\mapsto
W_{2}(\delta_{w},\sum_{i=1}^{n}\lambda_{i}\,\delta_{F_{i}(x)})^{2}=\sum_{i=1}^{n}\lambda_{i}\,d(w,F_{i}(x))^{2}.$

By 2.17, the fixed point $x$ is out of the cut locus of any point $F_{i}(x)$
for $1\leq i\leq n$. We can thus differentiate $h$ at $w=x$ and get $\nabla
h|_{w=x}=0$. Since $\nabla\phi_{i}(x)=\frac{1}{2}\nabla
d^{2}_{F_{i}(x)}|_{w=x}$ holds as both gradients exist [16, Lemma 3.3], it
follows that $\sum_{i=1}^{n}\lambda_{i}\nabla\phi_{i}(x)=\frac{1}{2}\nabla
h|_{w=x}=0$.

Define $f:=\sum_{i=1}^{n}\lambda_{i}\,\phi_{i}$ on a neighborhood of $\Omega$
that is a common domain for $\phi_{i},1\leq i\leq n$. The function $f$ is
locally semi-concave as each $\phi_{i}$ is so, and for $x\in\Omega$, $\nabla
f(x)=\sum_{i=1}^{n}\lambda_{i}\nabla\phi_{i}(x)=0\in T_{x}M$ by the previous
arguments. Let $\Omega_{1}\subset\Omega$ be the set where the Hessians of $f$
and $\phi_{i},1\leq i\leq n$, all exist. Let $\Omega_{2}$ be the set of
density points of $\Omega$. We have
$\operatorname{Vol}(\Omega\setminus\Omega_{1})=0$ by 3.10, and
$\operatorname{Vol}(\Omega\setminus\Omega_{2})=0$ by [18, Theorem 1.35].

For $x\in\Omega_{1}$, using the linearity of the Hessian operator, we get
$\operatorname{Hess}_{x}f=\sum_{i=1}^{n}\lambda_{i}\,\operatorname{Hess}_{x}\phi_{i}$
by (14). Besides, noting that $\nabla f$ is constant on $\Omega$, we infer
from the last statement of Lemma 3.5 that for $x\in\Omega_{2}\cap\Omega$,
$\operatorname{Hess}_{x}f=0$. It follows that for
$x\in\Omega_{1}\cap\Omega_{2}$,
$\sum_{i=1}^{n}\lambda_{i}\,\operatorname{Hess}_{x}\phi_{i}=0$. This proves
the theorem since $\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\mu}(\Omega_{1}\cap\Omega_{2})=1$ thanks to the absolute continuity of
$\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\mu}$. ∎

##  4 Lower Ricci curvature bounds and displacement functionals

In this section, we introduce a class of displacement functionals exploiting
the Hessian equality in 3.13. This is one of the primary difference between
our approach and the one proposed by Kim and Pass [31] regarding the absolute
continuity of the barycenter.

In Section 3, the notion of Hessian plays a central role in differentiating
optimal transport maps. There is also the following widely used connection
between $\operatorname{Hess}_{x}\phi$ and Jacobi equations involving
$\exp(-\nabla\phi)$, which is demonstrated in various works including Sturm
[42], Lott and Villani [34, §7], Cordero-Erausquin et al. [17] and Villani
[49, Chapter 14]. The function $J(t)$ defined below is actually
$\operatorname{d}\exp(-\nabla t\,\phi)(x)$ using (16). By convention, for a
function $f$ with variable $t$, we denote by $\dot{f}$ its derivative with
respect to $t$.

######  Proposition 4.1.

Let $M$ be an $m$-dimensional complete Riemannian manifold and let $\phi$ be a
$c$-concave function defined on $\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\mathcal{X}}\subset M$ with $\mathcal{X}$ a bounded open set. Fix a point
$x\in\mathcal{X}$ such that $\operatorname{Hess}_{x}\phi$ (3.10) exists. Then
$t\in[0,1]\mapsto\gamma(t)=\exp(-t\nabla\phi)(x)$ is a minimal geodesic.
Define

$J:t\in[0,1]\mapsto\operatorname{d}\exp_{x}|_{-t\nabla\phi(x)}\cdot(\operatorname{Hess}_{x}d^{2}_{\gamma(t)}/2-t\operatorname{Hess}_{x}\phi).$

Denote by $\Delta\phi(x)$ the trace of $\operatorname{Hess}_{x}\phi$ and by
$\det J(t),0\leq t\leq 1$ the determinant of $J(t)$ calculated in coordinates
using orthonormal bases of $T_{x}M$ and $T_{\gamma(t)}M$. If $-K\in\mathbb{R}$
is a lower Ricci curvature bound of $M$ along $\gamma$ and $\det J>0$, then
$\ell:=-\log\det J$ defined on $[0,1]$ satisfies

$\ddot{\ell}\geq\dot{\ell}^{2}/m-K\|\nabla\phi(x)\|^{2}$

with $\ell(0)=0$ and $\dot{\ell}(0)=\Delta\phi(x)$. In particular,

$l\geq\Delta\phi(x)-K\|\nabla\phi(x)\|^{2}/2,$

where we define $l:=\ell(1)=-\log\det J(1)$.

The following displacement functionals
$f\operatorname{d}\operatorname{Vol}\in\mathcal{W}_{2}(M)\mapsto\int
G(f)\operatorname{d}\operatorname{Vol}$ are inspired by the entropy
functional, where $G(x):=x\log x$. To uniformly bound (from above) their
values of the sequence of barycenter measures in the law of large numbers for
Wasserstein barycenters, we add the assumption of bounded derivatives.
Examples of $G$ can be constructed according to 5.9.

######  Proposition 4.2 (Displacement functionals).

Let $M$ be an $m$-dimensional complete Riemannian manifold with a lower Ricci
curvature bound $-K\leq 0$. Given an integer $n\geq 2$, let
$\lambda_{i}>0,1\leq i\leq n$, be $n$ positive real numbers such that
$\sum_{i=1}^{n}\lambda_{i}=1$ and let $\mu_{i}\in\mathcal{W}_{2}(M),1\leq
i\leq n$, be $n$ probability measures with compact support. Assume that there
is an integer $1\leq k\leq n$ such that for any index $1\leq i\leq k$,
$\mu_{i}$ is absolutely continuous with density function $g_{i}$. Denote by
$\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\mu}$ the unique Wasserstein barycenter of
$\mathbb{P}:=\sum_{i=1}^{n}\lambda_{i}\,\delta_{\mu_{i}}\in(\mathcal{W}_{2}(\mathcal{W}_{2}(M)),\mathbb{W}_{2})$,
which is absolutely continuous, and we denote by $f$ its density function.

Let $G$ be a function on $[0,\infty)$ such that $G(0)=0$, and the function
$H:x\in\mathbb{R}\mapsto{G(e^{x})}\,{e^{-x}}$ is continuously differentiable
with non-negative derivative bounded above by some constant $L_{H}>0$. The
following inequality holds,

$\int_{M}G(f)\operatorname{d}\operatorname{Vol}\leq\sum_{i=1}^{k}\frac{\lambda_{i}}{\Lambda}\int_{M}G(g_{i})\operatorname{d}\operatorname{Vol}+\frac{L_{H}K}{2\Lambda}\mathbb{W}_{2}(\mathbb{P},\delta_{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\mu}})^{2}+\frac{L_{H}}{2\Lambda}({m^{2}}+2m),$ (20)

where we define the constant $\Lambda:=\sum_{i=1}^{k}\lambda_{i}$.

######  Remark 4.3.

The following example helps to understand (20). Take
$\mathbb{P}=\lambda\,\delta_{\mu_{1}}+(1-\lambda)\delta_{\mu_{2}}$ with
$0<\lambda<1$ and absolutely continuous measures
$\mu_{1},\mu_{2}\in\mathcal{W}_{2}(M)$. Set $G(x):=x\log x$. Since $H(x)=x$,
we choose $L_{H}=1$. Define
$\operatorname{Ent}(f\cdot\operatorname{Vol}):=\int_{M}G(f)\operatorname{d}\operatorname{Vol}$.
The inequality (20) becomes

$\operatorname{Ent}(\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\mu})\leq\lambda\operatorname{Ent}(\mu_{1})+(1-\lambda)\operatorname{Ent}(\mu_{2})+\frac{K}{2}\lambda(1-\lambda)W_{2}(\mu_{1},\mu_{2})^{2}+\frac{m^{2}}{2}+m,$

which has exactly one additional term ${L_{H}}(m^{2}+2m)/(2\Lambda)$ compared
to the $\lambda$-convexity expression of $\operatorname{Ent}$ used to define
lower Ricci curvature bound $-K$ for metric measure spaces in [43, §4,2] and
[34, Definition 0.7].

###### Proof of 4.2.

For $1\leq i\leq n$, let $F_{i}:=\exp(-\nabla\phi_{i})$ be the optimal
transport map from $\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\mu}$ to $\mu_{i}$ with $\phi_{i}$ a $c$-concave function given by 2.15.
According to 3.13 and 3.12, there exists a Borel set $\Omega\subset M$ with
$\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\mu}(\Omega)=1$ such that
$\sum_{i=1}^{n}\lambda_{i}\operatorname{Hess}_{x}\phi_{i}=0$ for $x\in\Omega$,
$\operatorname{Jac}\exp(-t\nabla\phi_{i})>0$ on $\Omega$ for $t\in[0,1]$ and
$1\leq i\leq k$, and

$\int_{M}G\left(g_{i}\right)\operatorname{d}\operatorname{Vol}=\int_{N_{i}}G\left(\frac{f}{\operatorname{Jac}F_{i}}\right)\operatorname{Jac}F_{i}\operatorname{d}\operatorname{Vol},\quad
1\leq i\leq k,$ (21)

where $N_{i}\subset\Omega$ for $1\leq i\leq k$ are Borel sets such that
$\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\mu}(N_{i})=1$ and $f=g_{i}(F_{i})\operatorname{Jac}F_{i}>0$ on $N_{i}$.
Hence, $\log f$ is well-defined on $\cup_{i=1}^{k}N_{i}$. Define
$l_{i}(x):=-\log\operatorname{Jac}F_{i}(x)$ on $\Omega$. It follows from (21)
that

$\int_{M}G(g_{i})\operatorname{d}\operatorname{Vol}=\int_{N_{i}}H(\log
f+l_{i})\operatorname{d}\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\mu},\quad 1\leq i\leq k.$ (22)

Applying 4.1 to $\phi_{i}$ for $1\leq i\leq k$, we have on $\Omega$,

$l_{i}\geq\Delta\phi_{i}-K\|\nabla\phi_{i}\|^{2}/2,\quad 1\leq i\leq k.$ (23)

For $x\in\Omega$ and $1\leq i\leq n$, since
$\operatorname{Hess}_{x}d^{2}_{F_{i}(x)}/2-\operatorname{Hess}_{x}\phi_{i}$ is
positive semi-definite (3.11), we can also bound $\Delta\phi_{i}(x)$ from
above using the upper bound of the Laplacian of distance functions observed by
Kim and Pass [31, Lemmma 2.7]:

$\displaystyle\Delta\phi_{i}(x)\leq\Delta d^{2}_{F_{i}(x)}/2$
$\displaystyle\leq
m\frac{\sqrt{K}d(x,F_{i}(x))}{\tanh(\sqrt{K}d(x,F_{i}(x)))}$
$\displaystyle\leq m(1+\sqrt{K}d(x,F_{i}(x)))\leq
m+m^{2}/2+K\,\|\nabla\phi_{i}(x)\|^{2}/2,$ (24)

where we used $d(x,F_{i}(x))=\|\nabla\phi_{i}(x)\|$ for $x\in\Omega$. With our
assumptions on $H$, (23) and (4) imply that for $1\leq i\leq k$, on the set
$\cup_{i=1}^{k}N_{i}$ (where $\log f$ is well-defined),

$\displaystyle H(\log f+l_{i})-H(\log f)=H^{\prime}(\xi)\,l_{i}$
$\displaystyle\geq H^{\prime}(\xi)[\Delta\phi_{i}-K\|\nabla\phi_{i}\|^{2}/2]$
$\displaystyle\geq
H^{\prime}(\xi)[\Delta\phi_{i}-K\|\nabla\phi_{i}\|^{2}/2-m-{m^{2}}/{2}]$
$\displaystyle\geq
L_{H}(\Delta\phi_{i}-K\|\nabla\phi_{i}\|^{2}/2)-L_{H}(m+m^{2}/2),$ (25)

where we applied the mean value theorem to $H$ that gave the real number $\xi$
between $\log f+l_{i}$ and $\log f$. Sum up $k$ inequalities as (25) with
coefficients $\lambda_{i}/\Lambda$ on the set $\cup_{i=1}^{k}N_{i}$,

$\displaystyle H(\log f)$
$\displaystyle\leq\sum_{i=1}^{k}\frac{\lambda_{i}}{\Lambda}H(\log
f+l_{i})-\frac{L_{H}}{\Lambda}\sum_{i=1}^{k}\lambda_{i}(\Delta\phi_{i}-K\|\nabla\phi_{i}\|^{2}/2)+L_{H}(m+m^{2}/2)$
$\displaystyle=\sum_{i=1}^{k}\frac{\lambda_{i}}{\Lambda}H(\log
f+l_{i})+\frac{L_{H}}{\Lambda}\sum_{i>k}^{n}\lambda_{i}\,\Delta\phi_{i}+\frac{L_{H}K}{2\Lambda}\sum_{i=1}^{k}\lambda_{i}\,\|\nabla\phi_{i}\|^{2}+L_{H}(m+m^{2}/2)$
$\displaystyle\leq\sum_{i=1}^{k}\frac{\lambda_{i}}{\Lambda}H(\log
f+l_{i})+\frac{L_{H}K}{2\Lambda}\sum_{i=1}^{n}\lambda_{i}\,\|\nabla\phi_{i}\|^{2}+\frac{L_{H}}{2\Lambda}(m^{2}+2m),$
(26)

where we used $\sum_{i=1}^{n}\lambda_{i}\,\Delta\phi_{i}=0$ derived from the
Hessian equality for the first equality and used (4) for the last inequality.
Finally, (20) follows from (22) after integrating (26) over
$N_{1}\cap\ldots\cap N_{k}$ against $\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\mu}$ since $\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\mu}(N_{i})=1$ for $1\leq i\leq k$ and $W_{2}(\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\mu},\mu_{i})^{2}=\int_{M}\|\nabla\phi_{i}\|^{2}\operatorname{d}\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\mu}$ for $1\leq i\leq n$. ∎

##  5 Proof of our main result

In this section, we prove our main result, i.e., the following theorem.

######  Theorem 5.1.

Let $M$ be a complete Riemannian manifold with a lower Ricci curvature bound.
If a probability measure $\mathbb{P}\in\mathcal{W}_{2}(\mathcal{W}_{2}(M))$
gives mass to the set of absolutely continuous probability measures on $M$,
then its unique Wasserstein barycenter is absolutely continuous.

New auxiliary results in this section no longer require Riemannian structure,
so we usually consider a Polish space equipped with a $\sigma$-finite Borel
measure.

###  5.1 Wasserstein barycenters’ absolute continuity by approximation

We first deduce an intermediate result by applying the law of large numbers
for Wasserstein barycenters to the displacement functionals introduced in 4.2.

The following lemma, taken from Santambrogio [41, Proposition 7.7, Remak 7.8],
originates from Buttazzo and Freddi [13, Theorem 2.2], which was slightly
generalized later in [3, Theorem 2.34]. One can find another slightly
generalized version by Ambrosio et al. [2, Theorem 15.8, Theorem 15.9] with a
proof for the case of Euclidean spaces.

######  Lemma 5.2.

Let $E$ be a Polish space with a $\sigma$-finite Borel measure $\mu$. Let $G$
be a function on $[0,\infty)$ such that

  1. 1.

$G(x)\geq 0$;

  2. 2.

$G$ is lower-semi continuous and convex;

  3. 3.

$\displaystyle\lim_{x\rightarrow\infty}{G(x)}/{x}=\infty$.

With respect to the reference measure $\mu$, if there is a sequence of
absolutely continuous probability measures
$\nu_{i}=f_{i}\operatorname{d}\mu,\,i\geq 1$ converging weakly to a
probability measure $\nu$ such that
$\displaystyle\liminf_{i\rightarrow\infty}\int_{E}G(f_{i})\operatorname{d}\mu$
is finite, then $\nu$ is also absolutely continuous and

$\int_{E}G(f)\operatorname{d}\mu\leq\liminf_{i\rightarrow\infty}\int_{E}G(f_{i})\operatorname{d}\mu<\infty,$
(27)

where $f$ is the density of $\nu$.

Since convergence in Wasserstein metric implies weak convergence, 5.2 ensures
that the set below is closed in $\mathcal{W}_{2}(E)$.

######  Definition 5.3 ($\operatorname{B}(G,L)$ sets).

Let $E$ be Polish space with a $\sigma$-finite Borel measure $\mu$. Let $G$ be
a function on $[0,\infty)$ such that

  1. 1.

$G(x)\geq 0$ and $G(0)=0$;

  2. 2.

$G$ is lower-semi continuous and convex;

  3. 3.

$\displaystyle\lim_{x\rightarrow\infty}{G(x)}/{x}=\infty$;

  4. 4.

the function $H(x):=G(e^{x})/e^{x}$ has continuous non-negative bounded
derivative.

Given $L>0$, the following set of measures,

$\operatorname{B}(G,L):=\left\\{\nu\in\mathcal{W}_{2}(E)\mid\nu=f\cdot\mu,\,\int_{M}G(f)\operatorname{d}\mu\leq
L\right\\},$

is a closed subset of $\mathcal{W}_{2}(E)$.

The function $\widehat{G}:x\mapsto x\,\log x$ on $[0,+\infty)$ is not always
positive, so it fails to meet the above assumptions. Since
$\widehat{G}(e^{-1})=-e^{-1}$ is the minimum value of $\widehat{G}$, we can
consider the function that equals $0$ on $[0,e^{-1}]$ and equals
$\widehat{G}+e^{-1}$ on $[e^{-1},+\infty)$, which is a valid example. As these
assumptions include the ones for constructing displacement functionals in 4.2,
we obtain the following intermediate result.

######  Proposition 5.4.

Let $M$ be a complete Riemannian manifold with a lower Ricci curvature bound.
If $\mathbb{P}\in\mathcal{W}_{2}(\mathcal{W}_{2}(M))$ gives mass to some
closed set $\operatorname{B}(G,L)$ defined in 5.3, i.e.,
$\mathbb{P}(\operatorname{B}(G,L))>0$, then the unique barycenter of
$\mathbb{P}$ is absolutely continuous.

###### Proof.

Write
$\mathbb{P}=\mathbb{P}(\operatorname{B}(G,L))\,\mathbb{P}^{1}+(1-\mathbb{P}(\operatorname{B}(G,L))\,\mathbb{P}^{2}$
with $\mathbb{P}^{1},\mathbb{P}^{2}\in\mathcal{W}_{2}(\mathcal{W}_{2}(M))$
such that $\mathbb{P}^{1}$ is concentrated on $\operatorname{B}(G,L)$. We
approximate $\mathbb{P}$ in the Wasserstein metric $\mathbb{W}_{2}$ with
finitely supported measures
$\mathbb{P}_{j}\in\mathcal{W}_{2}(\mathcal{W}_{2}(M))$ by approximating
$\mathbb{P}^{1}$ and $\mathbb{P}^{2}$ as follows.

Since $\operatorname{B}(G,L)$ equipped with the Wasserstein metric
$\mathbb{W}_{2}$ is a non-empty closed subspace of $\mathcal{W}_{2}(M)$, we
can construct the Wasserstein space $\mathcal{W}_{2}(\operatorname{B}(G,L))$
and treat $\mathbb{P}^{1}$ as an element in it. Since the set of finitely
supported measures is dense in Wasserstein spaces [49, Theorem 6.18], there
are two sequences of finitely supported probability measures
$\\{\mathbb{P}^{1}_{j}\\}_{j\geq 1}$ and $\\{\mathbb{P}^{2}_{j}\\}_{j\geq 1}$
on $\mathcal{W}_{2}(M)$ such that each $\mathbb{P}^{1}_{j}$ is concentrated on
$\operatorname{B}(G,L)$ and
$\mathbb{W}_{2}(\mathbb{P}_{j}^{1},\mathbb{P}^{1})\rightarrow 0$,
$\mathbb{W}_{2}(\mathbb{P}_{j}^{2},\mathbb{P}^{2})\rightarrow 0$ when
$j\rightarrow\infty$. Furthermore, we require that all
$\mathbb{P}_{j}^{1},\mathbb{P}_{j}^{2}$ for $j\geq 1$ only give mass to
measures with compact support. Define
$\mathbb{P}_{j}:=\mathbb{P}(\operatorname{B}(G,L))\,\mathbb{P}_{j}^{1}+(1-\mathbb{P}(\operatorname{B}(G,L))\,\mathbb{P}_{j}^{2}$.
It follows that $\mathbb{W}_{2}(\mathbb{P}_{j},\mathbb{P})\rightarrow 0$ as
$j\rightarrow\infty$.

Consider the displacement functional
$f\cdot\operatorname{Vol}\mapsto\int_{M}G(f)\operatorname{d}\operatorname{Vol}$.
4.2 implies that its value at the barycenter $\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\mu}_{j}$ of $\mathbb{P}_{j}$ can be bounded from above using its values
on the support of $\mathbb{P}_{j}^{1}$,
$\Lambda:=\mathbb{P}(\operatorname{B}(G,L))$,
$\mathbb{W}_{2}(\mathbb{P}_{j},\delta_{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\mu}_{j}})$ and some other constants. Denote by $\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\mu}$ the unique barycenter of $\mathbb{P}$, the law of large numbers for
Wasserstein barycenters (2.2) implies that $W_{2}(\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\mu}_{j},\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\mu})\rightarrow 0$ and thus
$\mathbb{W}_{2}(\mathbb{P}_{j},\delta_{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\mu}_{j}})\rightarrow\mathbb{W}_{2}(\mathbb{P},\delta_{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\mu}})$ as $j\rightarrow\infty$. Since the support of $\mathbb{P}_{j}^{1}$
is a subset of $\operatorname{B}(G,L)$ and
$\mathbb{W}_{2}(\mathbb{P}_{j},\delta_{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\mu}_{j}})$ is bounded for $j\geq 1$, there exists $L^{\prime}>0$ such
that $\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\mu}_{j}\in\operatorname{B}(G,L^{\prime})$ for all $j\geq 1$. It follows
from 5.2 that $\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\mu}$ is absolutely continuous. ∎

We replace the assumption $\mathbb{P}(\operatorname{B}(G,L))>0$ by a more
natural one in the next subsection.

###  5.2 Compactness using Souslin space theory

The last step towards our main result is to show that the closed subset
$\operatorname{B}(G,L)$ needed in 5.4 always exists if $\mathbb{P}$ gives mass
to the set of absolutely continuous measures. Our inspiration is the criterion
of uniform integrability by Charles-Jean de la Vallée Poussin. This criterion
[9, Theorem 4.5.9] constructs a functional $f\mapsto\int
G(f)\operatorname{d}\mu$ that is uniformly bounded for a family of uniformly
integrable functions. We have enough freedom in its construction to impose the
properties required by 5.3 on the function $G$. Pre-compact sets of measures
with respect to the topology $\tau$ defined below are closely related to
uniformly integrable families.

######  Definition 5.5 (The set $\mathbb{A}$ and four topologies
$\tau_{w},\tau_{W},\tau,\tau_{L}$).

Let $E$ be a Polish space with a $\sigma$-finite reference measure $\mu$. Pick
a point $x_{0}\in E$ and define the following set of measurable functions on
$E$,

$\mathbb{A}:=\left\\{f\in L^{1}(\mu)\,\bigg{|}\,f\geq
0,\,\int_{E}f\operatorname{d}\mu=1,\,\int_{E}d(x_{0},x)^{2}f(x)\operatorname{d}\mu(x)<\infty\right\\},$
(28)

which is independent of the chosen point $x_{0}$. The set $\mathbb{A}$ is
naturally identified via $f\leftrightarrow f\cdot\mu$ with the set of
probability measures in $\mathcal{W}_{2}(E)$ that are absolutely continuous
with respect to $\mu$. We introduce the following four topologies. Denote by
$\tau_{w}$ the topology on $\mathcal{W}_{2}(E)$ with respect to the weak
convergence, denote by $\tau_{W}$ the topology of the Wasserstein space
$\mathcal{W}_{2}(E)$, denote by $\tau$ the weak topology on $L^{1}(\mu)$
induced by its dual space $L^{\infty}(\mu)$ [9, Theorem 4.4.1] and denote by
$\tau_{L}$ the topology of the Lebesgue space $L^{1}(\mu)$. By definition,
$\tau_{w}\subset\tau_{W}$ and $\tau\subset\tau_{L}$. Denote by
$(\mathbb{A},\tau_{w}),(\mathbb{A},\tau_{W}),(\mathbb{A},\tau)$ and
$(\mathbb{A},\tau_{L})$ the four topological subspaces induced by these
topologies on the set $\mathbb{A}$.

Consider the case when $E$ is a complete Riemannian manifold and $\mu$ is the
volume measure on $E$. By 2.12, $\mathbb{A}$ is a Borel set for the topology
$\tau_{W}$. Given a probability measure
$\mathbb{P}\in\mathcal{W}_{2}(\mathcal{W}_{2}(E))$ such that
$\mathbb{P}(\mathbb{A})>0$, our goal is to find a compact subset $\mathcal{F}$
in $(\mathbb{A},\tau)$ with $\mathbb{P}(\mathcal{F})>0$. If we can accomplish
this, then $\mathcal{F}$ forms a family of uniformly integrable functions by
the Dunford-Pettis theorem (5.8), bringing us closer to the main result. To
find such an $\mathcal{F}$, a direct but problematic approach is to argue that
$\mathbb{P}$ is a Radon measure. However, this argument overlooks that crucial
point that $\mathbb{P}$ (restricted on $\mathbb{A}$) must be a Borel measure
with respect to the Borel sets of $(\mathbb{A},\tau)$. To address this issue,
we revisit the Souslin space theory. Our main reference is Bogachev [9,
Section 6.6, Section 6.7, Section 7.4].

######  Definition 5.6 (Souslin space).

A set in a Hausdorff space is called Souslin if it is the image of a complete
separable metric space under a continuous map. A Souslin space is a Hausdorff
space that is a Souslin set. The empty set is Souslin as well.

By definition, Polish spaces are Souslin. Here are some properties of Souslin
spaces:

  1. 1.

Every Borel subset of a Souslin space is a Souslin space [9, Theorem 6.6.7];

  2. 2.

Let $E$ and $F$ be Souslin spaces and let $f:E\mapsto F$ be a measurable map.
If $f$ is bijective, then $E$ and $F$ share the same Borel sets, see [23,
Proposition 423F] or [9, Theorem 6.7.3];

  3. 3.

If $E$ is a Souslin space, then every finite Borel measure $\mu$ on $E$ is
Radon [9, Theorem 7.4.3].

These properties are used in the following lemma to justify the previous
arguments with Radon measures.

######  Lemma 5.7.

Let $(E,d)$ be a Polish space with an outer regular and $\sigma$-finite Borel
measure $\mu$ on $E$. Let $\mathbb{A}$ be as (28). The four topological
subspaces, $(\mathbb{A},\tau_{w}),(\mathbb{A},\tau_{W}),(\mathbb{A},\tau)$,
and $(\mathbb{A},\tau_{L})$ share the same Borel sets.

In particular, if $\mathbb{P}\in\mathcal{W}_{2}(\mathcal{W}_{2}(E))$ gives
mass to the set $\mathbb{A}$, then it gives mass to a compact subset of
$(\mathbb{A},\tau)$.

###### Proof.

For spaces $(\mathbb{A},\tau_{w})$ and $(\mathbb{A},\tau_{W})$, the first
statement is already proven in [38, Lemma 2.4.2], and we recall its arguments
here. By 2.12, $\mathbb{A}$ is a Borel set for both $\tau_{w}$ and $\tau_{W}$.
Since $(\mathcal{W}_{2}(E),W_{2})$ is a Polish space, $(\mathbb{A},\tau_{W})$
is then a Souslin space as a Borel subset of $(\mathcal{W}_{2}(E),W_{2})$.
Consider the identity map
$\operatorname{Id}:(\mathbb{A},\tau_{W})\rightarrow(\mathbb{A},\tau_{w})$, it
is continuous and bijective. By definition, $(\mathbb{A},\tau_{w})$ is a
Souslin space as the image of the Souslin space $(\mathbb{A},\tau_{W})$ under
the continuous map $\operatorname{Id}$. Moreover, $(\mathbb{A},\tau_{W})$ and
$(\mathbb{A},\tau_{w})$ share the same Borel sets since the measurable map
$\operatorname{Id}$ is bijective.

We claim that $(\mathbb{A},\tau_{L})$ is also a Souslin space. We first prove
that the Lebesgue space $L^{1}(\mu)$ is complete and separable using the
assumption that $E$ is Polish. $L^{1}(\mu)$ is complete for any measurable
space $E$ [9, Theorem 4.1.3]. Its separability is asserted in Brézis [12,
Theorem 4.13] and Bogachev [9, Section 1.12(iii), Corollary 4.2.2, Exercise
4.7.63] but only proven for the case of Euclidean spaces. Here is a brief
proof of it. Every Polish space is homeomorphic to a closed subspace of
$\mathbb{R}^{\infty}$ [9, Theorem 6.1.12]. Moreover, one can show that
$L^{1}(\mu)$ is separable when $E=\mathbb{R}^{\infty}$ using the same
arguments for Euclidean spaces. It follows that $L^{1}(\mu)$ is a Polish
space. We then prove that $\mathbb{A}$ is a Borel set for the topology
$\tau_{L}$. Fix a point $x_{0}\in E$. Define the following sets for integers
$k,j\geq 1$,

$A_{k,j}:=\left\\{f\in L^{1}(\mu)\,\bigg{|}\,f\geq
0,\,\int_{E}f\operatorname{d}\mu=1,\,\int_{E}\min\\{d(x_{0},x)^{2},k\\}f(x)\operatorname{d}\mu(x)\leq
j\right\\}.$

Fix two integers $k,j\geq 1$. We show that the set $A_{k,j}$ is a closed
subset of $L^{1}(\mu)$. Let $\\{f_{i}\\}_{i\geq 1}\subset A_{k,j}$ be a
sequence converging to $f\in L^{1}(\mu)$ in $L^{1}(\mu)$. Since
$\\{f_{i}\\}_{i\geq 1}$ has a subsequence converging almost everywhere to $f$,
$f$ is non-negative ($\mu$-almost everywhere). It follows that
$\int_{E}f\operatorname{d}\mu=\|f\|_{L^{1}(\mu)}=\lim_{i\rightarrow\infty}\|f_{i}\|_{L^{1}(\mu)}=1$.
Noting that as $i\rightarrow\infty$,

$\|\min\\{d(x_{0},\cdot)^{2},k\\}f_{i}-\min\\{d(x_{0},\cdot)^{2},k\\}f\|_{L^{1}(\mu)}\leq
k\|f_{i}-f\|_{L^{1}(\mu)}\rightarrow 0,$

which implies that $f\in A_{k,j}$. Hence, $A_{k,j}$ is a closed subset of
$L^{1}(\mu)$. By the monotone convergence theorem, we have
$\mathbb{A}=\cup_{j\geq 1}\cap_{k\geq 1}A_{k,j}$, which proves that
$\mathbb{A}$ is a Borel set. Finally, $(\mathbb{A},\tau_{L})$ is a Souslin
space as $\mathbb{A}$ is a Borel set of the Polish space $L^{1}(\mu)$.

By definition of $\tau_{w}$ and $\tau$, we have the topological inclusions
$(\mathbb{A},\tau_{w})\subset(\mathbb{A},\tau)\subset(\mathbb{A},\tau_{L})$.
Using the identity map as before, we conclude that the three topological
spaces, $(\mathbb{A},\tau_{w})$, $(\mathbb{A},\tau)$ and
$(\mathbb{A},\tau_{L})$, share the same Borel sets since
$(\mathbb{A},\tau_{L})$ is a Souslin space.

$\mathbb{P}$, restricted on $\mathbb{A}$, is then a Radon measure with respect
to the common Borel sets for the four topological subspaces since finite Borel
measures on Souslin spaces are Radon. Hence, $\mathbb{P}(\mathbb{A})>0$ can be
approximated by the $\mathbb{P}$ measure of compact subsets of
$(\mathbb{A},\tau)$. ∎

We prove the following slightly generalized Dunford-Pettis theorem that
connects uniform integrability and the weak topology $\tau$.

######  Proposition 5.8 (Dunford-Pettis theorem).

Let $(E,\mathcal{B})$ be a measurable space with a $\sigma$-finite Borel
measure $\mu$ on it. Let $\mathcal{F}\subset L^{1}(\mu)$ be a set of
$\mu$-integrable functions. If $\mathcal{F}$ has compact closure in the weak
topology induced by the dual space $L^{\infty}(\mu)$ of $L^{1}(\mu)$, then
$\mathcal{F}$ is _uniformly integrable_ , i.e.,

$\lim_{C\rightarrow\infty}\sup_{f\in\mathcal{F}}\int_{\\{|f|>C\\}}|f|\operatorname{d}\mu=0.$

###### Proof.

We need the assumption of $\mu$ being $\sigma$-finite to ensure that
$L^{\infty}(\mu)$ is the dual space of $L^{1}(\mu)$, see [9, Theorem 4.4.1]
and [40, Exercise 6.12]. The above definition of uniform integrability is
taken from Bogachev [9, Definition 4.5.1]. When $\mu$ is finite, the
equivalence between pre-compactness in the weak topology and uniform
integrability is already proven by Bogachev [9, Theorem 4.7.18]. The following
arguments for the general case are based on his proof.

We prove our statement for $\sigma$-finite measures by contradiction. Suppose
that $\mathcal{F}$ has compact closure in the weak topology, but is not
uniformly integrable. Then, there are $\epsilon>0$ and a sequence
$\\{f_{n}\\}_{n\geq 1}\subset\mathcal{F}$ such that

$\inf_{n\geq 1}\int_{\\{|f_{n}|>n\\}}|f_{n}|\operatorname{d}\mu\geq\epsilon.$
(29)

Applying the Eberlein–Šmulian theorem [9, Theorem 4.7.10] to $\\{f_{n}\\}$ and
the Banach space $L^{1}(\mu)$, we get a subsequence $\\{f_{n_{k}}\\}_{k\geq
1}$ convergent to some function $f\in L^{1}(\mu)$ in the weak topology. In
particular, for every Borel set $A\in\mathcal{B}$ we have

$\lim_{k\rightarrow\infty}\int_{A}f_{n_{k}}\operatorname{d}\mu=\int_{A}f\operatorname{d}\mu.$
(30)

According to Bogachev [9, Theorem 4.5.6], (30) implies that the sequence
$\\{f_{n_{k}}\\}$ is bounded in $L^{1}(\mu)$ and has uniformly absolutely
continuous integrals, i.e., for every $\epsilon>0$, there exists $\delta>0$
such that

$\mu(A)<\delta\implies\sup_{k\geq
1}\int_{A}|f_{n_{k}}|\operatorname{d}\mu<\epsilon.$ (31)

Set $C:=\sup_{k\geq 1}\|f_{n_{k}}\|_{L^{1}(\mu)}<\infty$. Take the $\delta$
given by (31) for the $\epsilon$ in (29), and define $n:=[C/\delta]+1$. Then
by Chebyshev’s inequality,

$\sup_{k\geq 1}\mu(\\{|f_{n_{k}}|>n\\})\leq\frac{1}{n}\sup_{k\geq
1}\|f_{n_{k}}\|_{L^{1}(\mu)}<\delta,$

which leads to a contradiction between (29) and (31). ∎

We also generalize the de la Vallée Poussin criterion to construct the
function $G$ in 5.3. In the following proposition, the $\sigma$-finiteness of
$\mu$ allows us to apply Fubini’s theorem.

######  Theorem 5.9 (De la Vallée Poussin criterion).

Let $(E,\mathcal{B})$ be a measurable space with a $\sigma$-finite Borel
measure $\mu$ on it. A subset $\mathcal{F}\subset L^{1}(\mu)$ is uniformly
integrable, i.e.,

$\lim_{C\rightarrow\infty}\sup_{f\in\mathcal{F}}\int_{\\{|f|>C\\}}|f|\operatorname{d}\mu=0$

if and only if there exists a function $G$ defined on $[0,+\infty)$ such that

  1. 1.

$G(x)=0$ for $0\leq x\leq 1$;

  2. 2.

$G$ is a non-decreasing and convex function that is smooth on $(0,+\infty)$;

  3. 3.

$\sup_{f\in\mathcal{F}}\int_{E}G(|f|)\operatorname{d}\mu\leq 1$;

  4. 4.

if we define the function $H(x):=G(e^{x})e^{-x}$ on $\mathbb{R}$, then
$\displaystyle\lim_{x\rightarrow+\infty}H(x)=+\infty$, and its derivative
${H}^{\prime}$ is smooth with $0\leq H^{\prime}(x)\leq 1$.

###### Proof.

If we have the asserted function $G$ for some subset $\mathcal{F}\subset
L^{1}(\mu)$, then for every $\epsilon>0$, we can find a real number $C>0$ such
that $G(t)/t\geq 2/\epsilon$ for any $t>C$. It implies that
$|f(x)|\leq\epsilon\,G(|f(x)|)/2$ for all $f\in\mathcal{F}$ when $|f(x)|>C$.
Hence,

$\int_{\\{|f|>C\\}}|f|\operatorname{d}\mu\leq\frac{\varepsilon}{2}\int_{\\{|f|>C\\}}\,G\circ|f|\operatorname{d}\mu\leq\epsilon,$

which shows that $\mathcal{F}$ is uniformly integrable.

Now assume that we are given a uniformly integrable subset $\mathcal{F}\subset
L^{1}(\mu)$. To better motivate our construction of $G$, we postpone the
definition of a smooth function $H$ with $H(x)=0,x\leq 0$ to (35) but use it
here to define $G(x):=H(\log x)\,x$. Differentiate this equation twice, we
obtain ${G}^{\prime\prime}(x)=[H^{\prime}(\log x)+H^{\prime\prime}(\log
x)]/x$. By our requirements on $H$, $G(x)=0$ for $0\leq x\leq 1$. Hence, we
have
$G(x)=\int_{0}^{x}\int_{0}^{s}G^{\prime\prime}(t)\operatorname{d}t\operatorname{d}s$
for $x>0$ and thus

$\displaystyle\int_{E}G(|f|)\operatorname{d}\mu$
$\displaystyle=\int_{E}\int_{0}^{|f|}\int_{0}^{s}G^{\prime\prime}(t)\operatorname{d}t\operatorname{d}s\operatorname{d}\mu=\int_{E}\int_{\mathbb{R}}\int_{\mathbb{R}}{G}^{\prime\prime}(t)\cdot\mathbbm{1}_{0<t<s<|f|}\operatorname{d}t\operatorname{d}s\operatorname{d}\mu$
$\displaystyle=\int_{\mathbb{R}}\int_{\mathbb{R}}{G}^{\prime\prime}(t)\cdot\mathbbm{1}_{0<t<s}\cdot\mu(|f|>s)\operatorname{d}t\operatorname{d}s$
$\displaystyle=\int_{\mathbb{R}}{G}^{\prime\prime}(t)\cdot\mathbbm{1}_{t>0}\int_{t}^{\infty}\mu(|f|>s)\operatorname{d}s\operatorname{d}t$
$\displaystyle=\int_{0}^{\infty}\frac{H^{\prime}(\log t)+H^{\prime\prime}(\log
t)}{t}\int_{t}^{\infty}\mu(|f|>s)\operatorname{d}s\operatorname{d}t$
$\displaystyle=\int_{\mathbb{R}}[{H}^{\prime}(y)+{H}^{\prime\prime}(y)]\int_{e^{y}}^{\infty}\mu(|f|>s)\operatorname{d}s\operatorname{d}y,$
(32)

where we applied Fubini’s theorem twice and a change of variable $y:=\log t$.
According to (32), we need to control ${H}^{\prime}+{H}^{\prime\prime}$ and
the integral of $\mu(|f|>s)$ at the same time. For the integral, note that by
Fubini’s theorem again, we have for $t>0$ and $f\in L^{1}(\mu)$ that

$\displaystyle\int_{\\{|f|>t\\}}|f|\operatorname{d}\mu$
$\displaystyle=\int_{\\{|f|>t\\}}\int_{\mathbb{R}}\mathbbm{1}_{0<s<|f|}\operatorname{d}s\operatorname{d}\mu=\int_{\mathbb{R}}\int_{E}\mathbbm{1}_{|f|>t}\cdot\mathbbm{1}_{0<s<|f|}\operatorname{d}\mu\operatorname{d}s$
$\displaystyle=\int_{\mathbb{R}}\int_{E}\mathbbm{1}_{0<s<t<|f|}+\mathbbm{1}_{0<t\leq
s<|f|}\operatorname{d}\mu\operatorname{d}s=t\,\mu(|f|>t)+\int_{t}^{\infty}\mu(|f|>s)\operatorname{d}s.$
(33)

Let $\alpha:\mathbb{N}\rightarrow\mathbb{N}$ be a strictly increasing function
such that $\alpha(0)\geq 0$ and

$\sup_{f\in\mathcal{F}}\int_{e^{\alpha(n)}}^{\infty}\mu(|f|>s)\operatorname{d}s\leq\sup_{f\in\mathcal{F}}\int_{\\{|f|>e^{\alpha(n)}\\}}|f|\operatorname{d}\mu\leq
2^{-(n+1)},$

where we used (33) for the first inequality and the uniform integrability of
$\mathcal{F}$ for the second one. It follows that

$\sup_{f\in\mathcal{F}}\sum_{n\geq
0}\int_{e^{\alpha(n)}}^{\infty}\mu(|f|>s)\operatorname{d}s\leq 1.$ (34)

For the term ${H}^{\prime}+{H}^{\prime\prime}$ in (32), we bound it from above
with a function that is non-zero only on selected intervals based on our
choice of $\alpha(n)$, allowing us to convert the integral of
$\int_{e^{y}}^{\infty}\mu(|f|>s)\operatorname{d}s$ into the series summation
(34). To achieve this, we first select a smooth function
$\gamma:\mathbb{R}\rightarrow[0,1]$ such that $\gamma(x)=1$ for
$x\in[\alpha(n)+1/3,\alpha(n)+2/3]$ and $\gamma(x)=0$ for
$x\notin(\alpha(n),\alpha(n)+1)$. Then we define

$H(x):=\begin{cases}\int_{0}^{x}e^{-s}\int_{0}^{s}\gamma(t)e^{t}\operatorname{d}t\operatorname{d}s,&x>0\\\
0,&x\leq 0\end{cases}.$ (35)

In this way, we have ${H}^{\prime\prime}(x)+{H}^{\prime}(x)=\gamma(x)$. Using
this construction, (32) and (34) imply that

$\sup_{f\in\mathcal{F}}\int_{E}G(|f|)\operatorname{d}\mu=\sup_{f\in\mathcal{F}}\sum_{n\geq
0}\int_{\alpha(n)}^{\alpha(n)+1}\gamma(y)\int_{e^{y}}^{\infty}\mu(|f|>s)\operatorname{d}s\operatorname{d}y\leq\sup_{f\in\mathcal{F}}\sum_{n\geq
0}\int_{e^{\alpha(n)}}^{\infty}\mu(|f|>s)\operatorname{d}s\leq 1.$

For the first derivative of $H$, we have

$0\leq{H}^{\prime}(x)=e^{-x}\int_{0}^{x}\gamma(t)e^{t}\operatorname{d}t\leq
e^{-x}(e^{x}-1)\leq 1.$

And by direct calculation we have that the difference

$H(\alpha(n)+1)-H(\alpha(n))>\int_{\alpha(n)+\frac{2}{3}}^{\alpha(n)+1}e^{-s}\int^{\alpha(n)+\frac{2}{3}}_{\alpha(n)+\frac{1}{3}}e^{t}\operatorname{d}t\operatorname{d}s=(1-e^{-\frac{1}{3}})^{2}$

is bigger than a constant independent of $n$, which implies that
$\displaystyle\lim_{x\rightarrow+\infty}H(x)=+\infty$ since $H$ is non-
decreasing. It follows from $0\leq\gamma\leq 1$ that $G$ is non-decreasing and
convex as ${G}^{\prime\prime}(x)=\gamma(\log x)/x\geq 0$ for $x>1$ and
$G(x)=0$ for $0\leq x\leq 1$. ∎

###  5.3 Final step of the proof

To prove 5.1, it remains to combine the previous auxiliary propositions to
replace the assumption in 5.4 that $\mathbb{P}(\operatorname{B}(G,L))>0$ for
some set $\operatorname{B}(G,L)$ (5.3).

As in 5.5, we denote by $\mathbb{A}$ the set of absolutely continuous measures
in $\mathcal{W}_{2}(M)$. If $\mathbb{P}(\mathbb{A})>0$, then 5.7 provides a
compact subset $\mathcal{F}$ of $(\mathbb{A},\tau)$ such that
$\mathbb{P}(\mathcal{F})>0$. Applying the Dunford-Pettis theorem (5.8) to
$\mathcal{F}$ with $\mu:=\operatorname{Vol}$, we see that $\mathcal{F}$ is
uniformly integrable. Then the de la Vallée Poussin criterion (5.9) asserts
the existence of a smooth function $G$ such that
$\mathcal{F}\subset\operatorname{B}(G,1)\subset\mathbb{A}$. Therefore, our
theorem follows from 5.4 and the property
$\mathbb{P}(\operatorname{B}(G,1))\geq\mathbb{P}(\mathcal{F})>0$.

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