# A relaxation viewpoint to Unbalanced Optimal Transport: duality, optimality
and Monge formulation

Giuseppe Savaré  Giuseppe Savaré: Department of Decision Sciences, Bocconi
University. Via Roentgen 1, 20136 Milan (Italy<EMAIL_ADDRESS>and  Giacomo Enrico Sodini  Giacomo Enrico Sodini: Institut für Mathematik -
Fakultät für Mathematik - Universität Wien, Oskar-Morgenstern-Platz 1, 1090
Wien (Austria<EMAIL_ADDRESS>
###### Abstract.

We present a general convex relaxation approach to study a wide class of
Unbalanced Optimal Transport problems for finite non-negative measures with
possibly different masses. These are obtained as the lower semicontinuous and
convex envelope of a cost for non-negative Dirac masses. New general primal-
dual formulations, optimality conditions, and metric-topological properties
are carefully studied and discussed.

###### Key words and phrases:

Unbalanced Optimal Transport, Kantorovich-Monge problem, Optimality conditions

###### 1991 Mathematics Subject Classification:

Primary: 49Q22, 28A33 ; Secondary: 49K27.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries, measures and functions on the cone
  3. 3 Convexification and duality
  4. 4 Existence of a maximazing pair for the dual problem
  5. 5 Optimality conditions
  6. A Metric and topological properties
  7. B The case of a cost function finite on an open cone

##  1\. Introduction

The problem of extending Optimal Transport methods to pairs of unbalanced non-
negative measures has been considered in a large number of works with
different techniques and different aims.

For what concerns dynamical formulations, many models inspired by the Benamou-
Brenier’s [BB00] fluid dynamic formulation of the classical Kantorovich-
Rubinstein-Wasserstein Optimal Transport metric have been proposed, see for
example [KMV16, Maa+15, LM15, PR14, PR16, Chi+18, Chi+18a, LMS18, LM19]. In
such works, the authors consider source terms in the continuity equation, thus
leading to gain/loss of mass during the evolution. The models proposed differ
in the kind of source or penalization. We refer to [Chi+18] where a more
detailed description of these models is given.

Static formulations of the Unbalanced Optimal Transport problem were proposed
already by Kantorovich and Rubinstein [KR58] and subsequently extended by
Hanin [Han99] (see also the dual norm in [Han92]). These approaches can be
thought as a classical Optimal Transport problem where a fraction of the mass
is allowed to go (or come from) a point at infinity (see also [FG10, Gui02]).
More recent approaches are given by the so called optimal partial transport
[CM10, Fig10], which was previously related to image retrieval [PW08, RGT97].

Optimal partial transport (see [Chi+18]) is in turn also related to [PR14,
PR16], since this latter works also provide a dynamic formulation of optimal
partial transport. We also mention that [PR14, PR16] are also connected to
[Ben03] where it was proposed to change the marginal constraints and to add a
penalization term.

The important case of the Hellinger-Kantorovich metric (a.k.a. Wasserstein-
Fisher-Rao), somehow interpolating the Wasserstein and the Hellinger metrics,
has been proposed independently in [KMV16, Chi+18, Chi+18a, LMS18], with
different equivalent characterizations.

A very useful one [LMS18] involves an entropic relaxation of the marginal
constraints in the classical static formulation of Optimal Transport, it
provides a paradigmatic example of the wide class of Optimal Entropy-Transport
problems, and suggests the crucial role of the so-called cone geometry.

[Chi+18a] introduces a general class of Unbalanced Optimal Transport problems
in compact subsets of some Euclidean space induced by a sublinear cost
function depending on positions and masses, using a suitable static semi-
coupling formulation and proving a general duality result, with interesting
applications to various dynamic problems.

It turns out that all these different viewpoints are specific examples of a
general construction based on convex relaxation, which we want to exploit in
the present paper.

#### A convex relaxation viewpoint.

Adopting the same approach of [SS22] used for the classical Optimal Transport
problem and extending some ideas already contained in [LMS18], the aim of this
work is to define a general class of Unbalanced Optimal Transport problems
between non-negative measures as the _convex and lower semicontinuous
envelope_ of a cost initially defined only between weighted Dirac masses, and
then to study the corresponding duality formulas, optimality conditions and
metric-topologial properties.

Let us briefly recall the result of [SS22] which is the starting point for the
present work; given a pair of completely regular spaces $\mathsf{X}_{1}$ and
$\mathsf{X}_{2}$, every proper and lower semicontinuous cost function
$\mathsf{c}:\mathsf{X}_{1}\times\mathsf{X}_{2}\to[0,+\infty]$ can be naturally
lifted to a singular cost functional
$\mathcal{F}_{\mathsf{c}}:\mathcal{M}(\mathsf{X}_{1})\times\mathcal{M}(\mathsf{X}_{2})\to[0,+\infty]$,
where $\mathcal{M}(\mathsf{X}_{i})$ denote the space of signed and finite
Radon measures in $\mathsf{X}_{i}$. $\mathcal{F}_{\mathsf{c}}$ is finite only
between balanced pairs of Dirac masses and can be defined as

$\mathcal{F}_{\mathsf{c}}(\mu_{1},\mu_{2}):=\begin{cases}r\mathsf{c}(x_{1},x_{2})\quad&\text{
if }\begin{aligned}
&\quad\mu_{1}=r\delta_{x_{1}},\,\mu_{2}=r\delta_{x_{2}},\\\ &\quad
x_{1}\in\mathsf{X}_{1},x_{2}\in\mathsf{X}_{2},\,r\geq 0,\end{aligned}\\\
+\infty\quad&\text{ elsewhere.}\end{cases}$ (1.1)

By [SS22, Theorem 4.4] the convex and lower semicontinuous (w.r.t. the product
weak topology) envelope of $\mathcal{F}_{\mathsf{c}}$ coincides with the
Optimal Transport functional

$\mathsf{OT}_{\mathsf{c}}(\mu_{1},\mu_{2}):=\begin{cases}\displaystyle\inf\Big{\\{}\int\mathsf{c}\,\,\mathrm{d}\bm{\gamma}:\bm{\gamma}\in\Gamma(\mu_{1},\mu_{2})\Big{\\}}&\text{
if }\mu_{i}\in\mathcal{M}_{+}(X_{i}),\
\mu_{1}(\mathsf{X}_{1})=\mu_{2}(\mathsf{X}_{2}),\\\ +\infty\quad&\text{
elsewhere},\end{cases}$ (1.2)

where $\mathcal{M}_{+}(\mathsf{X}_{i})$ denotes the cone of nonnegative
(finite Radon) measures in $\mathcal{M}(\mathsf{X}_{i})$ and
$\Gamma(\mu_{1},\mu_{2})$ is the subset of couplings between $\mu_{1}$ and
$\mu_{2}$, i.e. measures in
$\mathcal{M}_{+}(\mathsf{X}_{1}\times\mathsf{X}_{2})$ with marginals $\mu_{1}$
and $\mu_{2}$ respectively. Clearly $\Gamma(\mu_{1},\mu_{2})$ is empty if
$\mu(\mathsf{X}_{1})\neq\mu_{2}(\mathsf{X}_{2}).$

In order to extend this construction to the case of non-negative measures with
possibly different masses, we represent weighted Dirac masses in $\mathsf{X}$
of the form $r\delta_{x}$, $r\geq 0$, as points $(x,r)$ of the so called
geometric cone
$\mathfrak{C}[\mathsf{X}]:=\big{(}\mathsf{X}\times[0,+\infty)\big{)}/\sim$
(see Section 2.1), where the equivalence relation $\sim$ identifies all the
points of the form $(x,0)$, $x\in\mathsf{X}$, which correspond to the null
measure in $\mathsf{X}$. A cost on weighted Dirac masses can thus be expressed
by a function

$\mathsf{H}:\mathfrak{C}[\mathsf{X}_{1}]\times\mathfrak{C}[\mathsf{X}_{2}]\to[0,+\infty],$

which we will assume to be proper, lower semicontinuous and _radially
$1$-homogeneous_, in the sense that the map

$\mathbb{R}_{2}^{+}\ni(r_{1},r_{2})\mapsto\mathsf{H}([x_{1},r_{1}],[x_{2},r_{2}])$

is $1$-homogeneous for every fixed
$(x_{1},x_{2})\in\mathsf{X}_{1}\times\mathsf{X}_{2}$. While the properness and
the lower semicontinuity assumptions are natural, the $1$-homogeneity
assumption deserves a comment: from a modeling point of view we are saying
that moving $mr_{1}\delta_{x_{1}}$ to $mr_{2}\delta_{x_{2}}$ costs exactly $m$
times moving $r_{1}\delta_{x_{1}}$ to $r_{2}\delta_{x_{2}}$. In analogy with
(1.1) we can define the unbalanced singular cost
$\mathscr{S}_{\mathsf{H}}:\mathcal{M}(\mathsf{X}_{1})\times\mathcal{M}(\mathsf{X}_{2})\to[0,+\infty]$
as

$\mathscr{S}_{\mathsf{H}}(\mu_{1},\mu_{2}):=\begin{cases}\mathsf{H}([x_{1},r_{1}],[x_{2},r_{2}])\quad&\text{
if }\begin{aligned}
&\quad\mu_{1}=r_{1}\delta_{x_{1}},\,\mu_{2}=r_{2}\delta_{x_{2}},\\\ &\quad
x_{1}\in\mathsf{X}_{1},x_{2}\in\mathsf{X}_{2},\,r_{1},r_{2}\geq
0,\end{aligned}\\\ +\infty\quad&\text{ elsewhere},\end{cases}$

and look for the largest convex and lower semicontinuous functional
$\mathscr{U}_{\mathsf{H}}$ on
$\mathcal{M}(\mathsf{X}_{1})\times\mathcal{M}(\mathsf{X}_{2})$ dominated by
$\mathscr{S}_{\mathsf{H}}$. It turns out that such a functional admits an
explicit characterization which resembles (1.2) via the formula

$\mathscr{U}_{\mathsf{H}}(\mu_{1},\mu_{2}):=\inf\left\\{\int_{\mathfrak{C}[\mathsf{X}_{1}]\times\mathfrak{C}[\mathsf{X}_{2}]}\mathsf{H}\,\mathrm{d}\bm{\alpha}\mid\bm{\alpha}\in\mathfrak{H}^{1}(\mu_{1},\mu_{2})\right\\},$
(1.3)

where $\mathfrak{H}^{1}(\mu_{1},\mu_{2})$ is the set of _homogeneous
couplings_ , i.e. measures $\bm{\alpha}$ on
$\mathfrak{C}[\mathsf{X}_{1}]\times\mathfrak{C}[\mathsf{X}_{2}]$ with _$1$
-homogeneous marginals_ $\mu_{1}$ and $\mu_{2}$ (see (2.12)): $\bm{\alpha}$
belongs to $\mathfrak{H}^{1}(\mu_{1},\mu_{2})$ if

$\left\\{\begin{aligned}
\mu_{1}(A_{1})&=\int_{A_{1}\times[0,+\infty)\times\mathfrak{C}[\mathsf{X}_{2}]}r_{1}\,\,\mathrm{d}\bm{\alpha}(x_{1},r_{1};x_{2},r_{2}),\\\
\mu_{2}(A_{2})&=\int_{\mathfrak{C}[\mathsf{X}_{1}]\times
A_{2}\times[0,+\infty)}r_{2}\,\,\mathrm{d}\bm{\alpha}(x_{1},r_{1};x_{2},r_{2}),\end{aligned}\right.\qquad\text{for
every Borel subsets $A_{i}\subset\mathsf{X}_{i}$.}$

This choice has been inspired by [LMS18], where the authors prove that the
class of Optimal Entropy-Transport problems can be formulated precisely as in
(1.3) for a suitable choice of the function $\mathsf{H}$ (see in particular
[LMS18, Definition 5.1]).

#### Structural properties of Unbalanced Optimal Transport problems

The following result collects some of the fundamental properties of the
unbalanced framework (see Theorems 3.4, 3.5 and 3.15), in parallel to similar
results of the classical Optimal Transport theory.

######  Theorem 1.1.

Let $\mathsf{X}_{1},\mathsf{X}_{2}$ be completely regular spaces and let
$\mathsf{H}:\mathfrak{C}[\mathsf{X}_{1}]\times\mathfrak{C}[\mathsf{X}_{2}]\to[0,+\infty]$
be a proper, radially $1$-homogeneous and lower semicontinuous function.

  1. (1)

For every
$(\mu_{1},\mu_{2})\in\mathcal{M}_{+}(\mathsf{X}_{1})\times\mathcal{M}_{+}(\mathsf{X}_{2})$
such that $\mathscr{U}_{\mathsf{H}}(\mu_{1},\mu_{2})<+\infty$, there exists an
optimal homogeneous coupling $\bm{\alpha}\in\mathfrak{H}^{1}(\mu_{1},\mu_{2})$
such that

$\mathscr{U}_{\mathsf{H}}(\mu_{1},\mu_{2})=\int_{\mathfrak{C}[\mathsf{X}_{1}]\times\mathfrak{C}[\mathsf{X}_{2}]}\mathsf{H}\,\mathrm{d}\bm{\alpha}.$

  2. (2)

$\mathscr{U}_{\mathsf{H}}$ is a lower semicontinuous convex function and
satisfies

$\mathscr{U}_{\mathsf{H}}(r_{1}\delta_{x_{1}},r_{2}\delta_{x_{2}})=\operatorname{\overline{\mathrm{co}}}\left(\mathsf{H}\right)([x_{1},r_{1}],[x_{2},r_{2}])\leq\mathsf{H}([x_{1},r_{1}],[x_{2},r_{2}])$

for every $(x_{1},x_{2})\in\mathsf{X}_{1}\times\mathsf{X}_{2}$ and every
$(r_{1},r_{2})\in\mathbb{R}_{+}^{2}$, where
$\operatorname{\overline{\mathrm{co}}}\left(\mathsf{H}\right)$ is the l.s.c.
convex envelope of $\mathsf{H}([x_{1},\cdot],[x_{2},\cdot])$ with respect to
the variables $(r_{1},r_{2})\in\mathbb{R}^{2}_{+}.$ If, in addition,
$\mathsf{H}$ is also radially convex (i.e. the map
$(r_{1},r_{2})\mapsto\mathsf{H}([x_{1},r_{1}],[x_{2},r_{2}])$ is convex for
every fixed $(x_{1},x_{2})\in\mathsf{X}_{1}\times\mathsf{X}_{2}$), then the
above inequality is an equality.

  3. (3)

$\mathscr{U}_{\mathsf{H}}$ is the convex l.s.c. envelope of
$\mathscr{S}_{\mathsf{H}}$ and admits the dual formulation

$\displaystyle\mathscr{U}_{\mathsf{H}}(\mu_{1},\mu_{2})=\mathscr{U}_{\operatorname{\overline{\mathrm{co}}}\left(\mathsf{H}\right)}(\mu_{1},\mu_{2})$
$\displaystyle=\operatorname{\overline{\mathrm{co}}}\left(\mathscr{S}_{\mathsf{H}}\right)(\mu_{1},\mu_{2})$
(1.4)
$\displaystyle=\sup\Big{\\{}\int_{\mathsf{X}_{1}}\varphi_{1}\,\,\mathrm{d}\mu_{1}+\int_{\mathsf{X}_{2}}\varphi_{2}\,\,\mathrm{d}\mu_{2}\mid(\varphi_{1},\varphi_{2})\in\Phi_{\mathsf{H}}\Big{\\}}$

for every
$(\mu_{1},\mu_{2})\in\mathcal{M}_{+}(\mathsf{X}_{1})\times\mathcal{M}_{+}(\mathsf{X}_{2})$,
where

$\displaystyle\Phi_{\mathsf{H}}:=\Big{\\{}(\varphi_{1},\varphi_{2})\in\mathrm{C}_{b}(\mathsf{X}_{1})\times\mathrm{C}_{b}(\mathsf{X}_{2}):{}$
$\displaystyle\varphi_{1}(x_{1})r_{1}+\varphi_{2}(x_{2})r_{2}\leq\mathsf{H}([x_{1},r_{1}],[x_{2},r_{2}])$
(1.5) $\displaystyle\text{ for every
}(x_{1},x_{2})\in\mathsf{X}_{1}\times\mathsf{X}_{2},\,r_{1},r_{2}\geq
0\Big{\\}}.$

Notice that, if the cost function $\mathsf{H}$ is given by

$\mathsf{H}([x_{1},r_{1}],[x_{2},r_{2}]):=\begin{cases}r\mathsf{c}(x_{1},x_{2})\quad&\text{
if }r_{1}=r_{2}=r\geq 0,\\\ +\infty\quad&\text{ elsewhere,}\end{cases}$

for some proper and lower semicontinuous function
$\mathsf{c}:\mathsf{X}_{1}\times\mathsf{X}_{2}\to[0,+\infty]$, then
$\mathscr{U}_{\mathsf{H}}=\mathsf{OT}_{\mathsf{c}}$,
$\mathscr{S}_{\mathsf{H}}=\mathcal{F}_{\mathsf{c}}$, and we recover precisely
the analogous results for the classical Optimal Transport theory (see in
particular [SS22, Theorems 3.5 and 4.4]). The relaxation formula (1.4) extends
the corresponding one of [Chi+18a, Proposition 3.11], showing in particular
that the primal formulation (1.3) in terms of homogeneous couplings is
equivalent to the semi-coupling approach of [Chi+18a]. (1.4) has also nice
applications to the structure of the Hellinger-Kantorovich metric, see Remark
3.18.

Along the same lines of the classical Optimal Transport theory, it is natural
to investigate the _existence of optimal potentials maximizing ( 1.4) and thus
solving the dual problem._ We first focus on the class of continuous
functions, when $\mathsf{X}_{1},\mathsf{X}_{2}$ are compact and metrizable and
$\mathsf{H}$ is finite and continuous on the whole product cone
$\mathfrak{C}[\mathsf{X}_{1}]\times\mathfrak{C}[\mathsf{X}_{2}]$, radially
$1$-homogeneous and convex. In this case a sufficient condition relies on the
singular behaviour of the normal derivative of $\mathsf{H}$ at the boundary of
the product cone as in the fundamental example

$\mathsf{H}([x_{1},r_{1}],[x_{2},r_{2}]):=r_{1}+r_{2}-2\sqrt{r_{1}r_{2}}\,\mathrm{e}^{-|x_{1}-x_{2}|^{2}/2},\quad
x_{i}\in\mathbb{R}^{d},\,r_{i}\geq 0.$ (1.6)

We refer to Section 4 for a detailed discussion of these hypotheses and
Appendix B for an alternative set of assumptions (i.e. when $\mathsf{H}$ is
finite and continuous on an open conical domain). In both these situations it
is possible to define the analogue of the celebrated $\mathsf{c}$-transform
for a pair $(\varphi_{1},\varphi_{2})\in\Phi_{\mathsf{H}}$ as

$\displaystyle\varphi_{1}^{\mathsf{H}}(x_{2})$
$\displaystyle:=\inf_{x_{1}\in\mathsf{X}_{1}}\inf_{\alpha\geq
0}\biggl{\\{}\mathsf{H}([x_{1},\alpha],[x_{2},1])-\alpha\varphi_{1}(x_{1})\biggr{\\}},\quad
x_{2}\in\mathsf{X}_{2},$
$\displaystyle\varphi_{1}^{\mathsf{H}\mathsf{H}}(x_{1})$
$\displaystyle:=\inf_{x_{2}\in\mathsf{X}_{2}}\inf_{\alpha\geq
0}\biggl{\\{}\mathsf{H}([x_{1},1],[x_{2},\alpha])-\alpha\varphi_{1}^{\mathsf{H}}(x_{2})\biggr{\\}},\quad
x_{1}\in\mathsf{X}_{1},$

and prove that the transformed potentials are equicontinuous so that the use
of a compactness argument in the space
$\mathrm{C}(\mathsf{X}_{1})\times\mathrm{C}(\mathsf{X}_{2})$ is possible. This
will produce an optimal pair of potentials, see Theorems 4.7 and B.3.

#### Optimality conditions and relaxed duality

In this unbalanced setting, general optimality conditions still involve the
notion of cyclical monotonicity: a subset
$\Gamma\subset\mathfrak{C}[\mathsf{X}_{1}]\times\mathfrak{C}[\mathsf{X}_{2}]$
is $\mathsf{H}$-cyclically monotone if for every finite family of points
$\\{(\mathfrak{y}_{1}^{i},\mathfrak{y}_{2}^{i})\\}_{i=1}^{N}\subset\Gamma$ and
every permutation $\sigma$ of $\\{1,\dots,N\\}$ it holds

$\sum_{i=1}^{N}\mathsf{H}(\mathfrak{y}_{1}^{i},\mathfrak{y}_{2}^{i})\leq\sum_{i=1}^{N}\mathsf{H}(\mathfrak{y}_{1}^{i},\mathfrak{y}_{2}^{\sigma(i)}).$

Similarly to the classical Optimal Transport case, _every optimal homogeneous
coupling is concentrated on a $\mathsf{H}$-cyclically monotone set $\Gamma$_
(see Proposition 5.2) which in addition is _a radial convex cone_ in the sense
that

$([x_{1},r_{1}^{i}],[x_{2},r_{2}^{i}])\in\Gamma,\,\lambda_{i}\geq
0,\,i=1,2\,\Rightarrow\Big{(}\big{[}x_{1},\sum_{i=1}^{2}\lambda_{i}r_{1}^{i}],[x_{2},\sum_{i=1}^{2}\lambda_{i}r_{2}^{i}]\Big{)}\in\Gamma.$

This further property comes from the radial homogeneity and convexity
assumption on $\mathsf{H}$.

Formulating a converse statement to the above proposition, thus involving
sufficient optimality conditions, requires additional assumptions on the
compatibility between the radial cone $\Gamma$, on which an admissible
homogeneous coupling $\bm{\alpha}$ between measures
$\mu_{i}\in\mathcal{M}_{+}(\mathsf{X}_{i})$ is concentrated, and the cost
function $\mathsf{H}$: in Section 5.1 we study the natural directed graph
structures induced by $\mathsf{H}$ and $\Gamma$, similarly to what has been
done in the classical Optimal Transport theory [Bei+09, BC10] for possibly
infinite costs. This leads to the notion of $\mathsf{H}$-connectedness:
$\Gamma$ is $\mathsf{H}$-connected if, whenever
$\mathfrak{y}_{1},\mathfrak{y}^{\prime}_{1}\in\pi^{1}(\Gamma)$ (the projection
of $\Gamma$ on the first cone $\mathfrak{C}[\mathsf{X}_{1}]$), we can find a
sequence of points
$\\{\mathfrak{y}_{2}^{1},\mathfrak{y}_{1}^{2},\mathfrak{y}_{2}^{2},\dots,\mathfrak{y}_{1}^{N},\mathfrak{y}_{2}^{N}\\}$
such that

$(\mathfrak{y}_{1},\mathfrak{y}_{2}^{1}),(\mathfrak{y}_{1}^{i},\mathfrak{y}_{2}^{i})\in\Gamma,\,i=2,\dots,N,\quad\mathsf{H}(\mathfrak{y}_{1}^{\prime},\mathfrak{y}_{2}^{N}),\mathsf{H}(\mathfrak{y}_{1}^{i+1},\mathfrak{y}_{2}^{i})<+\infty,\,i=1,\dots,N-1.$

In other words, the sequence
$\\{\mathfrak{y}_{2}^{1},\mathfrak{y}_{1}^{2},\mathfrak{y}_{2}^{2},\dots,\mathfrak{y}_{1}^{N},\mathfrak{y}_{2}^{N}\\}$
“connects” $\mathfrak{y}_{1}$ to $\mathfrak{y}^{\prime}_{1}$ keeping the cost
finite when moving from a point in $\mathfrak{C}[\mathsf{X}_{2}]$ to a point
in $\mathfrak{C}[\mathsf{X}_{1}]$ and imposing that the considered pair is in
$\Gamma$, when moving from a point in $\mathfrak{C}[\mathsf{X}_{1}]$ to a
point in $\mathfrak{C}[\mathsf{X}_{2}]$. For example, if $\mathsf{H}$ is
everywhere finite, then any $\Gamma$ is $\mathsf{H}$-connected (see Theorem
5.8 for simple conditions implying $\mathsf{H}$-connectedness).

Upon assuming that $\Gamma$ is a $\mathsf{H}$-cyclically monotone and
$\mathsf{H}$-connected subset of the effective domain of $\mathsf{H}$ also
intersecting the interior of such domain, it is possible to prove the
existence of _relaxed optimal potentials_ : these are Borel functions
$\varphi_{i}:\mathsf{X}_{i}\to\mathbb{R}\cup\\{\pm\infty\\}$ satisfying

$\displaystyle\varphi_{1}(x_{1})r_{1}+_{o}\varphi_{2}(x_{2})r_{2}$
$\displaystyle\leq\mathsf{H}([x_{1},r_{1}],[x_{2},r_{2}])\quad\text{ for every
}x_{i}\in\mathsf{X}_{i},\,r_{i}\geq 0,$ (1.7)
$\displaystyle\varphi_{1}(x_{1})r_{1}+\varphi_{2}(x_{2})r_{2}$
$\displaystyle=\mathsf{H}([x_{1},r_{1}],[x_{2},r_{2}])\quad\text{ if
}([x_{1},r_{1}],[x_{2},r_{2}])\in\Gamma,$ (1.8)

where the inequality in (1.7) corresponds to the constraint as in (1.5) and
the notation $+_{o}$ means that, whenever an ambiguity $\pm\infty\mp\infty$
arises, the sum is set equal to $0$. This existence result, together with some
finiteness conditions relating $\mu_{i}$ and $\mathsf{H}$ (see
(LABEL:ss22:cond1)), imply that
$\varphi_{i}\in\mathcal{L}^{1}(\mathsf{X}_{i},\mu_{i})$ solve the dual problem

$\int_{\mathsf{X}_{1}}\varphi_{1}\,\mathrm{d}\mu_{1}+\int_{\mathsf{X}_{2}}\varphi_{2}\,\mathrm{d}\mu_{2}=\mathscr{U}_{\mathsf{H}}(\mu_{1},\mu_{2}),$
(1.9)

and yields optimality for the homogeneous plan $\bm{\alpha}$. We refer to
Theorem 5.10 for the detailed sufficiency result.

#### A Monge-like formulation via transport-growth maps.

As in classical Optimal Transport problems, Unbalanced Optimal Transport
admits a more restrictive Monge-like formulation in terms of _transport-
growth_ pairs [LMS23]: they are maps $(\mathsf{T},\mathsf{g})$ from
$\mathsf{X}_{1}$ to $\mathsf{X}_{2}\times[0,+\infty)$ acting on measures
$\mu_{1}\in\mathcal{M}(\mathsf{X}_{1})$ via the formula

$(\mathsf{T},\mathsf{g})_{\star}\mu_{1}=\mathsf{T}_{\sharp}(\mathsf{g}\mu_{1}),\quad\mu_{2}=(\mathsf{T},\mathsf{g})_{\star}\mu_{1}\
\Leftrightarrow\
\mu_{2}(B)=\int_{\mathsf{T}^{-1}(B)}\mathsf{g}\,\,\mathrm{d}\mu_{1}\quad\text{for
every Borel set }B\subset\mathsf{X}_{2}.$ (1.10)

Whenever $\mathsf{g}\in L^{1}(\mathsf{X}_{1},\mu_{1})$, a transport-growth
pair induces a homogeneous coupling
$\bm{\alpha}\in\mathfrak{H}^{1}(\mu_{1},\mu_{2})$,
$\mu_{2}=(\mathsf{T},\mathsf{g})_{\star}\mu_{1}$, via the formula

$\bm{\alpha}=([\text{id}_{\mathsf{X}_{1}},1],[\mathsf{T},\mathsf{g}])_{\sharp}\mu_{1},\quad\int\mathsf{H}\,\,\mathrm{d}\bm{\alpha}=\int\mathsf{H}([x,1],[\mathsf{T}(x),\mathsf{g}(x)])\,\,\mathrm{d}\mu_{1}(x),$
(1.11)

and one can study the corresponding Monge formulation of (1.3):

$\mathscr{M}_{\mathsf{H}}(\mu_{1},\mu_{2}):=\inf\Big{\\{}\int_{\mathsf{X}_{1}}\mathsf{H}([x,1],[\mathsf{T}(x),\mathsf{g}(x)])\,\,\mathrm{d}\mu_{1}(x):\mu_{2}=(\mathsf{T},\mathsf{g})_{\star}\mu_{1}\Big{\\}}.$
(1.12)

Clearly, $\mathscr{U}_{\mathsf{H}}\leq\mathscr{M}_{\mathsf{H}}$; notice also
that, whenever an optimal plan $\bm{\alpha}$ between $\mu_{1}$ and $\mu_{2}$
charges the set $\\{\mathfrak{y}_{1}=\mathfrak{o}_{1}\\}$, then it is
impossible that such $\bm{\alpha}$ can be induced by a transport-growth map
$(\mathsf{T},\mathsf{g})$ as in (1.11). This corresponds to the fact that some
of the mass of $\mu_{2}$ does not come from $\mu_{1}$ but it is “created”.
Under the same hypotheses of the classical balanced case [Pra07] (i.e. the
cost function is continuous and $\mu_{1}$ is atomless),

######  Theorem 1.2.

Let $\mathsf{X}_{i}$ be Polish spaces, let
$\mathsf{H}:\mathfrak{C}[\mathsf{X}_{1}]\times\mathfrak{C}[\mathsf{X}_{2}]\to[0,+\infty]$
be a proper, radially $1$-homogeneous and continuous function,
$\mu_{i}\in\mathcal{M}_{+}(\mathsf{X}_{i})$ with $\mu_{1}$ diffuse (i.e.
$\mu_{1}(\\{x\\})=0$ for every $x\in\mathsf{X}_{1}$) and such that
$\mu_{1}(\mathsf{X}_{1})>0$. Then

$\mathscr{U}_{\mathsf{H}}(\mu_{1},\mu_{2})=\mathscr{M}_{\mathsf{H}}(\mu_{1},\mu_{2}).$

In general, the infimum in the definition of
$\mathscr{M}_{\mathsf{H}}(\mu_{1},\mu_{2})$ is not attained; however, as a
consequence of the existence of relaxed optimal potentials as in (1.9) and
(1.7), when $\mathsf{X}_{1}=\mathsf{X}_{2}=\mathbb{R}^{d}$, $\mu_{1}$ is
absolutely continuous with respect to the Lebesgue measure in $\mathbb{R}^{d}$
and the differential of $\mathsf{H}$ satisfies suitable assumptions, we also
obtain the existence of an optimal transport-growth map attaining the minimum
in (1.12). This is the case, for example, of the cost function in (1.6)
associated with the Gaussian Hellinger-Kantorovich metric.

######  Theorem 1.3.

Let
$\mathsf{H}:\mathfrak{C}[\mathbb{R}^{d}]\times\mathfrak{C}[\mathbb{R}^{d}]\to\mathbb{R}$
be given by

$\mathsf{H}([x_{1},r_{1}],[x_{2},r_{2}]):=r_{1}+r_{2}-2\sqrt{r_{1}r_{2}}\,\mathrm{e}^{-|x_{1}-x_{2}|^{2}/2},\quad
x_{i}\in\mathbb{R}^{d},\,r_{i}\geq 0.$

If $\mu_{i}\in\mathcal{M}_{+}(\mathbb{R}^{d})$ with
$\mu_{i}(\mathbb{R}^{d})>0$ for $i=1,2$ and $\mu_{1}\ll\mathcal{L}^{d}$, then
there exists a transport-growth map
$(\mathsf{T},\mathsf{g}):\mathbb{R}^{d}\to\mathbb{R}^{d}\times[0,+\infty)$
such that

$\mu_{2}=(\mathsf{T},\mathsf{g})_{\star}\mu_{1}=\mathsf{T}_{\sharp}(\mathsf{g}\mu_{1}),\quad\int_{\mathbb{R}^{d}}\mathsf{H}([x,1],[\mathsf{T}(x),\mathsf{g}(x)])\,\mathrm{d}\mu_{1}(x)=\mathscr{U}_{\mathsf{H}}(\mu_{1},\mu_{2}).$

The above theorem holds also for large class of cost functions $\mathsf{H}$,
we refer to Theorem 5.13 for the general result.

#### Metric and topological properties.

Finally we frame in our general setting some of the results of [LMS18,
Chi+18a, De ̵20, DM22] concerning the metric properties of Unbalanced Optimal
Transport functionals. In particular, we prove that, in case $\mathsf{H}$ is
(the $p$-th power of) a distance on $\mathfrak{C}[\mathsf{X}]$, then the
resulting cost $\mathscr{U}_{\mathsf{H}}$ is itself (the $p$-th power of) a
distance on an appropriate subset of $\mathcal{M}_{+}(\mathsf{X})$ metrizing
the weak convergence of measures (see Theorems A.4 and A.5), precisely as it
is for the standard Optimal Transport problem [AGS08, Proposition 7.1.5].

##### Plan of the paper.

Section 2 is devoted to establish the general setting and a few technical
tools that will be used in the sequel.

Section 3 contains the core of our results: the convexification approach, some
structural properties and the comparison with the semi-coupling approach of
[Chi+18a], the duality, and the Monge formulation.

Section 4 treats the dual attainment in spaces of continuous potentials under
additional regularity assumptions on $\mathsf{H}$ and on the spaces
$\mathsf{X}_{i}$.

In Section 5 we present the general optimality conditions, the relaxed duality
result and, as a consequence of the latter, the existence of optimal
transport-growth maps.

Appendix A contains a few remarks on the metric and topological properties of
$\mathscr{U}_{\mathsf{H}}$ in case $\mathsf{H}$ is (the $p$-th power of) a
distance. Finally Appendix B reproduces the results of Section 4 under a
different set of additional assumptions on $\mathsf{H}$.

##### Acknowledgments.

The authors gratefully acknowledge the support of the Institute for Advanced
Study of the Technical University of Munich, funded by the German Excellence
Initiative. G.S. has also been supported by IMATI-CNR, Pavia and by the MIUR-
PRIN 202244A7YL project Gradient Flows and Non-Smooth Geometric Structures
with Applications to Optimization and Machine Learning.

##  2\. Preliminaries, measures and functions on the cone

In this section, $\mathsf{X}$ is a completely regular space (i.e. it is
Hausdorff and for every closed set $C$ and point $x\in\mathsf{X}\setminus C$
there exists a continuous function $f:\mathsf{X}\to[0,1]$ s.t. $f(x)=0$ and
$f(C)=\\{1\\}$). We denote by $\mathscr{B}(\mathsf{X})$ the Borel
$\sigma$-algebra on $\mathsf{X}$ and by $\mathcal{M}(\mathsf{X})$ (resp.
$\mathcal{M}_{+}(\mathsf{X})$) the vector space of real valued (resp. the cone
of non-negative) Radon measures on $\mathsf{X}$ i.e. the countably additive
set functions $\mu:\mathscr{B}(\mathsf{X})\to\mathbb{R}$ (resp.
$\mu:\mathscr{B}(\mathsf{X})\to[0,+\infty)$) s.t.

$\text{ for every }B\in\mathscr{B}(\mathsf{X})\text{ and every
}\varepsilon>0\text{ there exists }K\subset B\text{ compact s.t.
}\left|\mu\right|(\mathsf{X}\setminus K)<\varepsilon,$

where $|\mu|$ denotes the total variation measure of $\mu$. We say that
$\mu\in\mathcal{M}_{+}(\mathsf{X})$ is concentrated on a set
$\Gamma\subset\mathsf{X}$ if for every $\varepsilon>0$ there exists a Borel
set $\Gamma_{\varepsilon}\subset\Gamma$ such that
$\mu(\mathsf{X}\setminus\Gamma_{\varepsilon})<\varepsilon$. The Radon property
ensures that if $\mu$ is concentrated on $\Gamma$ then it is concentrated on a
$\sigma$-compact subset $\Gamma^{\prime}\subset\Gamma$.

We denote by $\mathcal{P}(\mathsf{X})$ the set of probability measures on
$\mathsf{X}$ i.e. the elements $\mu$ of $\mathcal{M}_{+}(\mathsf{X})$ s.t.
$\mu(\mathsf{X})=1$. When $\mathsf{X}$ is a Polish space (i.e. its topology is
induced by a metric $\mathsf{d}$ such that $(\mathsf{X},\mathsf{d})$ is
complete and separable), then every finite Borel measure is Radon so that
$\mathcal{M}(\mathsf{X})$ coincides with the space of real valued Borel
measures.

The zero measure in $\mathsf{X}$ is denoted by $\bm{0}_{\mathsf{X}}$. We
define the sets of non-negative discrete measures and non-negative weighted
Dirac masses respectively as

$\displaystyle\operatorname{\mathcal{M}_{+,f}}(\mathsf{X})$
$\displaystyle:=\left\\{\sum_{j=1}^{m}c_{i}\delta_{x_{j}}:\\{c_{j}\\}_{j=1}^{m}\subset[0,+\infty)\,,\,\\{x_{j}\\}_{j=1}^{m}\subset\mathsf{X}\,,\,m\in\mathbb{N}\right\\},$
$\displaystyle\Delta_{+}(\mathsf{X})$
$\displaystyle:=\left\\{r\delta_{x}:r\in[0,+\infty)\,,\,x\in\mathsf{X}\right\\}.$

Notice that
$\operatorname{\mathcal{M}_{+,f}}(\mathsf{X}),\Delta_{+}(\mathsf{X})\subset\mathcal{M}_{+}(\mathsf{X})$.
If $\mathsf{X},\mathsf{Y}$ are two completely regular spaces and
$f:\mathsf{X}\to\mathsf{Y}$ is a Borel function we denote by
$f_{\sharp}:\mathcal{M}(\mathsf{X})\to\mathcal{M}(\mathsf{Y})$ the push
forward operator defined, for every $\mu\in\mathcal{M}(\mathsf{X})$, as

$f_{\sharp}\mu(B)=\mu(f^{-1}(B))\quad\text{ for every
}\,B\in\mathscr{B}(\mathsf{Y}).$

We denote by $\mathrm{C}_{b}(\mathsf{X})$ the vector space of continuous and
bounded real functions. There is a natural duality pairing
$\langle\cdot,\cdot\rangle$ between $\mathcal{M}(\mathsf{X})$ and
$\mathrm{C}_{b}(\mathsf{X})$

$\langle\mu,\varphi\rangle:=\int_{\mathsf{X}}\varphi\,\mathrm{d}\mu\quad\text{for
every }\mu\in\mathcal{M}(\mathsf{X}),\ \varphi\in\mathrm{C}_{b}(\mathsf{X}).$
(2.1)

(2.1) defines a real nondegenerate bilinear form in
$\mathcal{M}(\mathsf{X})\times\mathrm{C}_{b}(\mathsf{X})$, for if a Radon
measure $\mu\in\mathcal{M}(\mathsf{X})$ satisfies
$\int_{\mathsf{X}}\varphi\,\mathrm{d}\mu=0$ for every
$\varphi\in\mathrm{C}_{b}(\mathsf{X})$, then $|\mu|(B)=0$ for every
$B\in\mathscr{B}(\mathsf{X})$ (e.g. by the approximation result [Bog06, Lemma
7.2.8]) so that $\mu$ is the null measure. Hence we can endow
$\mathcal{M}(\mathsf{X})$ with the weak Hausdorff topology
$\sigma(\mathcal{M}(\mathsf{X}),\mathrm{C}_{b}(\mathsf{X}))$: the coarsest
topology on $\mathcal{M}(\mathsf{X})$ for which the maps
$\mu\mapsto\int_{\mathsf{X}}\varphi\,\mathrm{d}\mu$ are continuous for every
$\varphi\in\mathrm{C}_{b}(\mathsf{X})$. Since in general
$\mathcal{M}(\mathsf{X})$ is not first-countable (but
$\mathcal{M}_{+}(\mathsf{X})$ is Polish, thus metrizable, if $\mathsf{X}$ is
Polish), we will mostly deal with general nets
$(\mu_{\lambda})_{\lambda\in\mathbb{L}}$, i.e. maps $\lambda\to\mu_{\lambda}$
defined in a directed set $\mathbb{L}$ with values in
$\mathcal{M}(\mathsf{X})$, see e.g. [[]§4.3]Folland99. By definition, a net
$\left(\mu_{\lambda}\right)_{\lambda\in\mathbb{L}}\subset\mathcal{M}(\mathsf{X})$
converges to $\mu\in\mathcal{M}(\mathsf{X})$ in the weak topology if

$\lim_{\lambda\in\mathbb{L}}\int_{\mathsf{X}}\varphi\,\mathrm{d}\mu_{\lambda}=\int_{\mathsf{X}}\varphi\,\mathrm{d}\mu\quad\text{
for every }\,\varphi\in\mathrm{C}_{b}(\mathsf{X}).$

If $f:\mathsf{X}\to(-\infty,+\infty]$ is a function, we denote by $D(f)$ its
effective domain, defined as

$D(f):=\left\\{x\in\mathsf{X}\mid f(x)<+\infty\right\\}$

and by $\bar{f}$ its lower semicontinuous envelope i.e.

$\bar{f}(x):=\inf\left\\{\liminf_{\lambda}f(x_{\lambda}):\left(x_{\lambda}\right)_{\lambda\in\mathbb{L}}\subset\mathsf{X}\,,\,\,x_{\lambda}\to
x\right\\},\quad x\in\mathsf{X};$ (2.2)

$\bar{f}$ is the largest lower semicontinuous function below $f$. If
$\mathsf{X}$ is also a topological vector space and $A\subset\mathsf{X}$, we
denote by $\operatorname{\mathrm{co}}\left(A\right)$ (resp. by
$\operatorname{\overline{\mathrm{co}}}\left(A\right)$) the convex (resp.
closed and convex) envelope of $A$. If $g:\mathsf{X}\to(-\infty,+\infty]$ is a
function, we denote its convex envelope and its closed convex envelope by
$\operatorname{\mathrm{co}}\left(g\right)$ and
$\operatorname{\overline{\mathrm{co}}}\left(g\right)$, defined by

$\displaystyle\operatorname{\mathrm{co}}\left(g\right)(x)$
$\displaystyle:=\inf\left\\{\sum_{i=1}^{n}\alpha_{i}g(x_{i}):\\{x_{i}\\}_{i=1}^{n}\subset\mathsf{X},\,\\{\alpha_{i}\\}_{i=1}^{n}\in
S_{n},\,\sum_{i=1}^{n}\alpha_{i}x_{i}=x,\,n\geq 1\right\\},\ x\in\mathsf{X}$
(2.3) $\displaystyle\operatorname{\overline{\mathrm{co}}}\left(g\right)$
$\displaystyle:=\overline{\operatorname{\mathrm{co}}\left(g\right)},$ (2.4)

where

$S_{n}:=\left\\{\\{\alpha_{i}\\}_{i=1}^{n}:\alpha_{i}\in[0,1],\,\sum_{i=1}^{n}\alpha_{i}=1\right\\}.$

$\operatorname{\mathrm{co}}\left(g\right)$ is the largest convex function
below $g$ and $\operatorname{\overline{\mathrm{co}}}\left(g\right)$ is the
largest lower semicontinuous and convex function below $g$. The following
proposition is a simple density result whose proof easily follows adapting
[Bog06, Example 8.1.6].

######  Proposition 2.1.

Let $\mathsf{X}$ be a completely regular space. Then for every $c\geq 0$ we
have

$\overline{\\{\mu\in\operatorname{\mathcal{M}_{+,f}}(\mathsf{X}):\mu(\mathsf{X})=c\\}}=\\{\mu\in\mathcal{M}_{+}(\mathsf{X}):\mu(\mathsf{X})=c\\},\quad\overline{\operatorname{\mathcal{M}_{+,f}}(\mathsf{X})}=\mathcal{M}_{+}(\mathsf{X}).$

######  Remark 2.2.

It holds

$\operatorname{\overline{\mathrm{co}}}\left(\Delta_{+}(\mathsf{X})\right)=\mathcal{M}_{+}(\mathsf{X}).$

This is an immediate consequence of the fact that
$\operatorname{\mathrm{co}}\left(\Delta_{+}(\mathsf{X})\right)=\operatorname{\mathcal{M}_{+,f}}(\mathsf{X})$
and Proposition 2.1.

The following Lemma is a refinement of Proposition 2.1 showing that, given a
Borel function $f$ and a non-negative measure $\alpha$, we can construct an
approximating sequence of discrete measures for which we have convergence also
of the integral of $f$.

######  Lemma 2.3.

Let $\mathsf{X}$ be a completely regular space and let
$\alpha\in\mathcal{M}_{+}(\mathsf{X})$. Let $f:\mathsf{X}\to[0,+\infty]$ be a
Borel function. Then there exists a net
$\left(\bm{\gamma}_{\lambda}\right)_{\lambda\in\mathbb{L}}\subset\\{\mu\in\operatorname{\mathcal{M}_{+,f}}(\mathsf{X}):\mu(\mathsf{X})=\alpha(\mathsf{X})\\}$
s.t.

$\lim_{\lambda\in\mathbb{L}}\bm{\gamma}_{\lambda}=\alpha,\quad\lim_{\lambda\in\mathbb{L}}\int_{\mathsf{X}}f\,\mathrm{d}\bm{\gamma}_{\lambda}=\int_{\mathsf{X}}f\,\mathrm{d}\alpha.$

###### Proof.

By Lusin’s theorem, we can find an increasing sequence of compact sets such
that

$\mathsf{X}_{k}\subset\mathsf{X}_{k+1}\quad\text{ for every }\,k\geq
1,\quad\alpha(\mathsf{X}\setminus\mathsf{X}_{k})\leq\frac{1}{k},\quad
f\raisebox{-1.29167pt}{$|$}_{\mathsf{X}_{k}}\text{ is bounded and
continuous.}$

Consider now the family of measures $\\{\alpha_{k}\\}_{k\geq
1}\subset\\{\mu\in\mathcal{M}_{+}(\mathsf{X}):\mu(\mathsf{X})=\alpha(\mathsf{X})\\}$
defined as

$\alpha_{k}:=\frac{\alpha(\mathsf{X})}{\alpha(\mathsf{X}_{k})}\alpha\raisebox{-1.29167pt}{$|$}_{\mathsf{X}_{k}}\quad\text{
for every }\,k\geq 1.$

We can easily observe that

$\lim_{k\to+\infty}\alpha_{k}=\alpha$

indeed, if $\varphi\in\mathrm{C}_{b}(\mathsf{X})$, we have

$\lim_{k\to+\infty}\int_{\mathsf{X}}\varphi\,\mathrm{d}\alpha_{k}=\lim_{k\to+\infty}\frac{\alpha(\mathsf{X})}{\alpha(\mathsf{X}_{k})}\int_{\mathsf{X}}\varphi{\raise
1.29167pt\hbox{$\chi$}}_{\mathsf{X}_{k}}\,\mathrm{d}\alpha=\int_{\mathsf{X}}\varphi\,\mathrm{d}\alpha$

by monotone convergence. The same argument shows that we also have

$\lim_{k\to+\infty}\int_{\mathsf{X}}f\,\mathrm{d}\alpha_{k}=\int_{\mathsf{X}}f\,\mathrm{d}\alpha.$
(2.5)

By Proposition 2.1, for every $k\geq 1$, we can find a net
$\\{\bm{\gamma}^{k}_{\lambda}\\}_{\lambda\in\mathbb{L}_{k}}\subset\operatorname{\mathcal{M}_{+,f}}(\mathsf{X}_{k})\cap\\{\mu\in\mathcal{M}_{+}(\mathsf{X}):\mu(\mathsf{X})=\alpha(\mathsf{X})\\}$,
such that

$\lim_{\lambda\in\mathbb{L}_{k}}\bm{\gamma}^{k}_{\lambda}=\alpha_{k}.$

Moreover, since $f\raisebox{-1.29167pt}{$|$}_{\mathsf{X}_{k}}$ is bounded and
continuous, it holds

$\lim_{\lambda\in\mathbb{L}_{k}}\int_{\mathsf{X}}f\,\mathrm{d}\bm{\gamma}^{k}_{\lambda}=\lim_{\lambda\in\mathbb{L}_{k}}\int_{\mathsf{X}_{k}}f\,\mathrm{d}\bm{\gamma}^{k}_{\lambda}=\int_{\mathsf{X}_{k}}f\,\mathrm{d}\alpha_{k}=\int_{\mathsf{X}}f\,\mathrm{d}\alpha_{k}.$

This allows us to find, for every $k\geq 1$, some
$\bar{m}(k)\in\mathbb{L}_{k}$ s.t.

$\left|\int_{\mathsf{X}}f\,\mathrm{d}\bm{\gamma}^{k}_{\lambda}-\int_{\mathsf{X}}f\,\mathrm{d}\alpha_{k}\right|\leq\frac{1}{k}\quad\text{
for every }\,\lambda\geq\bar{m}(k).$

Hence we can consider, for every $k\geq 1$, the directed sets
$\mathbb{E}_{k}:=\\{\lambda\in\mathbb{L}_{k}:\lambda_{k}\geq\bar{m}(k)\\}$ and
the corresponding new sequence of nets
$\\{\bm{\gamma}^{k}_{\lambda}\\}_{\lambda\in\mathbb{E}_{k}}$. Obviously it
holds

$\lim_{\lambda\in\mathbb{E}_{k}}\bm{\gamma}^{k}_{\lambda}=\alpha_{k},\quad\left|\int_{\mathsf{X}}f\,\mathrm{d}\bm{\gamma}^{k}_{\lambda}-\int_{\mathsf{X}}f\,\mathrm{d}\alpha_{k}\right|\leq\frac{1}{k}\quad\text{
for every }\,\lambda\in\mathbb{E}_{k}.$ (2.6)

Define now the directed set

$\mathbb{N}\otimes\mathbb{E}_{k}:=\\{(k,\lambda):\lambda\in\mathbb{E}_{k}\\}\text{
with order }(k,\lambda)\leq(k^{\prime},\lambda^{\prime})\iff
k<k^{\prime}\text{ or }(k=k^{\prime}\land\lambda\leq\lambda^{\prime}).$

By the diagonal principle for nets, we can find a directed set $\mathbb{B}$
and a monotone final function

$h:\mathbb{B}\to\mathbb{N}\otimes\mathbb{E}_{k},\quad
h(\beta)=(h_{1}(\beta),h_{2}(\beta))\text{ with
}h_{2}(\beta)\in\mathbb{E}_{h_{1}(\beta)}\quad\text{ for every
}\,\beta\in\mathbb{B}$

such that the diagonal net
$\\{\bm{\gamma}_{\beta}\\}_{\beta\in\mathbb{B}}:=\\{\bm{\gamma}^{h_{1}(\beta)}_{h_{2}(\beta)}\\}_{\beta\in\mathbb{B}}\subset\operatorname{\mathcal{M}_{+,f}}(\mathsf{X})\cap\\{\mu\in\mathcal{M}_{+}(\mathsf{X}):\mu(\mathsf{X})=\alpha(\mathsf{X})\\}$
converges to $\alpha$. We only need to prove that also the integral of $f$
converges:

$\displaystyle\left|\int_{\mathsf{X}}f\,\mathrm{d}\bm{\gamma}_{\beta}-\int_{\mathsf{X}}f\,\mathrm{d}\alpha\right|$
$\displaystyle\leq\left|\int_{\mathsf{X}}f\,\mathrm{d}\bm{\gamma}_{\beta}-\int_{\mathsf{X}}f\,\mathrm{d}\alpha_{h_{1}(\beta)}\right|+\left|\int_{\mathsf{X}}f\,\mathrm{d}\alpha_{h_{1}(\beta)}-\int_{\mathsf{X}}f\,\mathrm{d}\alpha\right|$
$\displaystyle=\left|\int_{\mathsf{X}}f\,\mathrm{d}\bm{\gamma}^{h_{1}(\beta)}_{h_{2}(\beta)}-\int_{\mathsf{X}}f\,\mathrm{d}\alpha_{h_{1}(\beta)}\right|+\left|\int_{\mathsf{X}}f\,\mathrm{d}\alpha_{h_{1}(\beta)}-\int_{\mathsf{X}}f\,\mathrm{d}\alpha\right|$
$\displaystyle\leq\frac{1}{h_{1}(\beta)}+\left|\int_{\mathsf{X}}f\,\mathrm{d}\alpha_{h_{1}(\beta)}-\int_{\mathsf{X}}f\,\mathrm{d}\alpha\right|,$

where we have used (2.6). Now it is enough to observe that
$h_{1}:\mathbb{B}\to\mathbb{N}$ is a final monotone function i.e. it is an
increasing monotone sequence converging to $+\infty$. Passing to
$\lim_{\beta\in\mathbb{B}}$ and using (2.5), we conclude. ∎

###  2.1. The cone construction

It will be natural to state some definitions and results in the context of the
so called geometric cone: we introduce on $\mathsf{X}\times\mathbb{R}_{+}$ the
equivalence relation

$(x,r)\sim(y,s)\quad\overset{\text{def}}{\Leftrightarrow}\quad\left[x=y,r=s\neq
0\quad\vee\quad r=s=0\right]$ (2.7)

and the corresponding geometric cone
$\mathfrak{C}[\mathsf{X}]:=(\mathsf{X}\times\mathbb{R}_{+})/\sim$, whose
points are denoted by gothic letters as $\mathfrak{y}$. We denote by
$\mathfrak{p}$ the quotient map
$\mathfrak{p}:\mathsf{X}\times\mathbb{R}_{+}\to\mathfrak{C}[\mathsf{X}]$
sending a point $(x,r)$ to its equivalence class $[x,r]$. Notice that
$\mathfrak{p}$ is just the identity map except for those points with $r=0$,
which are all sent to the same equivalence class, the so called vertex of the
cone, that we denote with $\mathfrak{o}$. We will consider the natural product
operation on the cone given by

$\lambda[x,r]:=[x,\lambda r]\quad\text{ for every }\lambda,r\geq
0,\,x\in\mathsf{X}.$

In certain cases it is useful to add a further isolated point $o$ (which plays
the role of a source or a sink) to $\mathsf{X}$ and to consider the spaces
$\mathsf{X}_{o}:=\mathsf{X}\sqcup\\{o\\}$ and
$\mathsf{Y}_{o}:=(\mathsf{X}\times\mathbb{R}_{+})\sqcup\\{(o,0)\\}\subset\mathsf{X}_{o}\times\mathbb{R}_{+}$.
We can identify $\mathsf{X}$ (resp. $\mathsf{X}\times\mathbb{R}_{+}$) with a
(closed) subset of $\mathsf{X}_{o}$ (resp. $\mathsf{Y}_{o}$) and
$\mathcal{M}_{+}(\mathsf{X})$ (resp.
$\mathcal{M}_{+}(\mathsf{X}\times\mathbb{R}_{+})$) with the (closed) subset of
$\mathcal{M}_{+}(\mathsf{X}_{o})$ (resp. $\mathcal{M}_{+}(\mathsf{Y}_{o})$)
given by all the measures not charging $\\{o\\}$ (resp. $\\{(o,0)\\}$). Notice
that $\mathfrak{C}[\mathsf{X}]=\mathsf{Y}_{o}/\sim$ and we can trivially
extend $\mathfrak{p}$ to $\mathsf{Y}_{o}$.

On the cone we introduce the projections on $\mathbb{R}_{+}$ and
$\mathsf{X}_{o}$ simply defined as $\mathsf{r}([x,r])=r$ and
$\mathsf{x}([x,r])=x$ if $r>0$ and
$\mathsf{x}([x,r])=\mathsf{x}(\mathfrak{o})=o$ if $r=0$.

We can define a right inverse of
$\mathfrak{p}:\mathsf{Y}_{o}\to\mathfrak{C}[\mathsf{X}]$ as

$\mathfrak{q}([x,r])=(\mathsf{x}([x,r]),\mathsf{r}([x,r])).$

Notice that also that
$\mathfrak{q}(\mathfrak{C}[\mathsf{X}]\setminus\\{{\mathfrak{o}}\\})=\mathsf{X}\times(0,+\infty).$

On $\mathfrak{C}[\mathsf{X}]$ we consider the following topology, weaker then
the quotient one: a local system of neighbourhoods of a point $[x,r]$ is just
the image trough $\mathfrak{p}$ of the local system of neighbourhoods given by
the product topology at $(x,r)\in\mathsf{X}\times\mathbb{R}_{+}$, if $r>0$. A
local system of neighbourhoods at $\mathfrak{0}$ is given by

$\left\\{\left\\{[x,r]\in\mathfrak{C}[\mathsf{X}]:0\leq
r<\varepsilon\right\\}\right\\}_{\varepsilon>0}.$ (2.8)

If the topology of $\mathsf{X}$ is induced by a metric $\mathsf{d}$, then the
topology of $\mathfrak{C}[\mathsf{X}]$ is induced by the metric
$\mathsf{d}_{\mathfrak{C}}:\mathfrak{C}[\mathsf{X}]\times\mathfrak{C}[\mathsf{X}]\to[0,+\infty)$
defined as

$\mathsf{d}_{\mathfrak{C}}([x,r],[y,s]):=\left(r^{2}+s^{2}-2rs\cos(\mathsf{d}(x,y)\wedge\pi)\right)^{\frac{1}{2}},\quad[x,r],[y,s]\in\mathfrak{C}[\mathsf{X}].$
(2.9)

With the above topology, $\mathfrak{C}[\mathsf{X}]$ is completely regular and
it is the right object to consider when one wants to represent elements in
$\Delta_{+}(\mathsf{X})$; in particular we have the following result.

######  Lemma 2.4.

Let $\mathsf{X}$ be a completely regular space. Then $\Delta_{+}(\mathsf{X})$
is homeomorphic to $\mathfrak{C}[\mathsf{X}]$.

###### Proof.

The homeomorphism is given by the map
$\varphi:\mathfrak{C}[\mathsf{X}]\to\Delta_{+}(\mathsf{X})$ defined as

$\varphi([x,r]):=\begin{cases}r\delta_{x}\quad&\text{ if }r>0,\\\
\bm{0}_{\mathsf{X}}\quad&\text{ if }r=0.\end{cases}$

∎

######  Remark 2.5.

Let us point out an obvious but useful fact that we will extensively use in
the following without having to recall it further: if a measure
${\bm{\beta}}\in\mathcal{M}_{+}\big{(}\mathfrak{C}[\mathsf{X}]\big{)}$ does
not charge $\\{{\mathfrak{o}}\\}$ then $\mathsf{x}_{\sharp}{\bm{\beta}}$ does
not charge $\\{o\\}$ and therefore we will identify it with a measure in
$\mathcal{M}_{+}(\mathsf{X})$. In particular, this construction does not
depend on the choice of the point $o$ and the map $\mathsf{x}$ takes value in
$\mathsf{X}$ ${\bm{\beta}}$-a.e.

If $R>0$, we define

$\mathfrak{C}_{R}[\mathsf{X}]:=\left\\{[x,r]\in\mathfrak{C}[\mathsf{X}]:0\leq
r\leq R\right\\}$ (2.10)

and we will often identify measures on $\mathfrak{C}[\mathsf{X}]$ with support
contained in $\mathfrak{C}_{R}[\mathsf{X}]$ with elements of
$\mathcal{M}(\mathfrak{C}_{R}[\mathsf{X}])$. For every $p\geq 1$, we introduce
moreover the set

$\mathcal{\mathfrak{M}}_{+}^{p}(\mathfrak{C}[\mathsf{X}]):=\left\\{\alpha\in\mathcal{M}_{+}(\mathfrak{C}[\mathsf{X}]):\int_{\mathfrak{C}[\mathsf{X}]}\mathsf{r}^{p}\,\mathrm{d}\alpha<+\infty\right\\},$

and the map (see the above Remark 2.5)

$\mathfrak{h}^{p}:\mathcal{\mathfrak{M}}_{+}^{p}(\mathfrak{C}[\mathsf{X}])\to\mathcal{M}_{+}(\mathsf{X}),\quad\mathfrak{h}(\alpha)=\mathsf{x}_{\sharp}(\mathsf{r}^{p}\alpha).$

We stress the fact that the map $\mathfrak{h}^{p}$ does not depend on the
point $o$ occuring in the definition of $\mathsf{x}$.

We introduce now the product cone: given $\mathsf{X}_{1}$ and $\mathsf{X}_{2}$
completely regular spaces, we define
$\mathfrak{C}[\mathsf{X}_{1},\mathsf{X}_{2}]:=\mathfrak{C}[\mathsf{X}_{1}]\times\mathfrak{C}[\mathsf{X}_{2}]$
endowed with the product topology. Points in the product cone are denoted by
bold gothic letters as
$\bm{\mathfrak{y}}=(\mathfrak{y}_{1},\mathfrak{y}_{2})=([x_{1},r_{1}],[x_{2},r_{2}])$.
We denote with $\mathfrak{o}_{i}$ (resp. $o_{i}$) the vertex of
$\mathfrak{C}[\mathsf{X}_{i}]$ (resp. isolated points added to
$\mathsf{X}_{i}$ and forming the disjoint union
$\mathsf{X}_{i,o}:=\mathsf{X}_{i}\sqcup\\{o_{i}\\}$), $i=1,2$ and we set
$\bm{{\mathfrak{o}}}:=({\mathfrak{o}}_{1},{\mathfrak{o}}_{2}),$
$\bm{o}:=(o_{1},o_{2}),$
$\mathfrak{C}_{\bm{{\mathfrak{o}}}}[\mathsf{X}_{1},\mathsf{X}_{2}]:=\mathfrak{C}[\mathsf{X}_{1},\mathsf{X}_{2}]\setminus\\{\bm{{\mathfrak{o}}}\\}.$

On the product cone we can consider the projections on the two components
$\pi^{i}:\mathfrak{C}[\mathsf{X}_{1},\mathsf{X}_{2}]\to\mathfrak{C}[\mathsf{X}_{i}]$
sending $([x_{1},r_{1}],[x_{2},r_{2}])$ to $[x_{i},r_{i}]$ and the projections
on $\mathbb{R}_{+}$ and $\mathsf{X}_{i}$ simply defined as
$\mathsf{r}_{i}:=\mathsf{r}\circ\pi^{i}$ and
$\mathsf{x}_{i}:=\mathsf{x}\circ\pi^{i}$ ($\mathsf{x}_{i}$ maps
$\mathfrak{C}[\mathsf{X}_{1},\mathsf{X}_{2}]$ into $\mathsf{X}_{i,o}$). In
analogy with (2.9), if the topologies of $\mathsf{X}_{1}$ and $\mathsf{X}_{2}$
are induced by distances $\mathsf{d}_{1}$ and $\mathsf{d}_{2}$ respectively,
the topology of the product cone is induced by the distance

$(\mathsf{d}_{1}\otimes_{\mathfrak{C}}\mathsf{d}_{2})((\mathfrak{y}_{1},\mathfrak{y}_{2}),(\mathfrak{w}_{1},\mathfrak{w_{2}})):=\left(\mathsf{d}_{1,\mathfrak{C}}^{2}(\mathfrak{y}_{1},\mathfrak{w}_{1})+\mathsf{d}_{2,\mathfrak{C}}^{2}(\mathfrak{y}_{2},\mathfrak{w}_{2})\right)^{\frac{1}{2}},\quad(\mathfrak{y}_{1},\mathfrak{y}_{2}),(\mathfrak{w}_{1},\mathfrak{w}_{2})\in\mathfrak{C}[\mathsf{X}_{1},\mathsf{X}_{2}].$
(2.11)

As in (2.10), given $R>0$, we define

$\mathfrak{C}_{R}[\mathsf{X}_{1},\mathsf{X}_{2}]:=\mathfrak{C}_{R}[\mathsf{X}_{1}]\times\mathfrak{C}_{R}[\mathsf{X}_{2}]=\left\\{\bm{\mathfrak{y}}\in\mathfrak{C}[\mathsf{X}_{1},\mathsf{X}_{2}]:0\leq\mathsf{r}_{i}(\bm{\mathfrak{y}})\leq
R,\,i=1,2\right\\}$

and we will identify measures on $\mathfrak{C}[\mathsf{X}_{1},\mathsf{X}_{2}]$
with support contained in $\mathfrak{C}_{R}[\mathsf{X}_{1},\mathsf{X}_{2}]$
with elements of
$\mathcal{M}(\mathfrak{C}_{R}[\mathsf{X}_{1},\mathsf{X}_{2}])$. For every
$p\geq 1$, we introduce the set

$\mathcal{\mathfrak{M}}_{+}^{p}(\mathfrak{C}[\mathsf{X}_{1},\mathsf{X}_{2}]):=\left\\{\bm{\alpha}\in\mathcal{M}_{+}(\mathfrak{C}[\mathsf{X}_{1},\mathsf{X}_{2}]):\int_{\mathfrak{C}[\mathsf{X}_{1},\mathsf{X}_{2}]}(\mathsf{r}_{1}^{p}+\mathsf{r}_{2}^{p})\,\mathrm{d}\bm{\alpha}<+\infty\right\\},$

and the maps (see the above Remark 2.5)

$\mathfrak{h}_{i}^{p}:\mathcal{\mathfrak{M}}_{+}^{p}(\mathfrak{C}[\mathsf{X}_{1},\mathsf{X}_{2}])\to\mathcal{M}_{+}(\mathsf{X}_{i}),\quad\mathfrak{h}_{i}^{p}(\bm{\alpha})=(\mathsf{x}_{i})_{\sharp}(\mathsf{r}_{i}^{p}\bm{\alpha})\quad
i=1,2.$

Finally we define, for every
$(\mu_{1},\mu_{2})\in\mathcal{M}_{+}(\mathsf{X}_{1})\times\mathcal{M}_{+}(\mathsf{X}_{2})$
and every $p\geq 1$, the set

$\mathfrak{H}^{p}(\mu_{1},\mu_{2}):=\left\\{\bm{\alpha}\in\mathcal{\mathfrak{M}}_{+}^{p}(\mathfrak{C}[\mathsf{X}_{1},\mathsf{X}_{2}]):\mathfrak{h}_{i}^{p}(\bm{\alpha})=\mu_{i},\,i=1,2\right\\}.$
(2.12)

If $\bm{\alpha}\in\mathfrak{H}^{p}(\mu_{1},\mu_{2})$, we say that $\mu_{1}$
and $\mu_{2}$ are the $p$-homogeneous marginals of $\bm{\alpha}$ and that
$\bm{\alpha}$ is a $p$-homogeneous plan/coupling between $\mu_{1}$ and
$\mu_{2}$. Notice that if $\bm{\alpha}\in\mathfrak{H}^{p}(\mu_{1},\mu_{2})$
then also the restriction of $\bm{\alpha}$ to
$\mathfrak{C}_{\bm{{\mathfrak{o}}}}[\mathsf{X}_{1},\mathsf{X}_{2}]$,
$\bm{\alpha}_{{\bm{{\mathfrak{o}}}}}:=\bm{\alpha}\mathbin{\vrule
height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule
height=0.55974pt,depth=0.0pt,width=5.59721pt}\mathfrak{C}_{\bm{{\mathfrak{o}}}}[\mathsf{X}_{1},\mathsf{X}_{2}]$,
belongs to $\mathfrak{H}^{p}(\mu_{1},\mu_{2})$.

######  Remark 2.6 (Trivial cases).

If $\mu_{i}(\mathsf{X}_{i})=0$ for $i=1,2$ ($\mu_{i}$ are the null measures in
$\mathsf{X}_{i}$), then
$\mathfrak{H}^{p}(\mu_{1},\mu_{2})=\\{\lambda\delta_{{\bm{{\mathfrak{o}}}}}:\lambda\geq
0\\}.$

If $\mu_{1}(\mathsf{X}_{1})>\mu_{2}(\mathsf{X}_{2})=0$ then
$\mathfrak{H}^{p}(\mu_{1},\mu_{2})=\left\\{\alpha\otimes\delta_{{\mathfrak{o}}_{2}}:\alpha\in\mathcal{M}_{+}(\mathfrak{C}[\mathsf{X}_{1}]),\
\mathfrak{h}^{p}(\alpha)=\mu_{1}\right\\}.$ A similar characterization holds
when $0=\mu_{1}(\mathsf{X}_{1})<\mu_{2}(\mathsf{X}_{2}).$

We say that $(\mu_{1},\mu_{2})$ is a non-trivial pair if
$\mu_{i}(\mathsf{X}_{i})>0$ for $i=1,2$.

######  Remark 2.7.

The class $\mathfrak{H}^{p}(\mu_{1},\mu_{2})$ satisfies a natural invariance
property with respect to inclusion of the ambient spaces $\mathsf{X}_{i}$.
First of all, if $\mathsf{X}_{i}$ are subsets of completely regular spaces
$\tilde{\mathsf{X}}_{i}$ (and the topology of $\mathsf{X}_{i}$ coincides with
the relative topology induced by the inclusion in $\tilde{\mathsf{X}}_{i}$)
there is a natural identification between (pair of) Radon measures
$\mu_{i}\in\mathcal{M}_{+}(\mathsf{X}_{i})$ and (pair of) Radon measures
$\tilde{\mu}_{i}\in\mathcal{M}_{+}(\tilde{\mathsf{X}}_{i})$ concentrated on
$\mathsf{X}_{i}$ (i.e. such that
$\tilde{\mathsf{X}}_{i}\setminus\mathsf{X}_{i}$ is $\mu_{i}$-negligible).
Since $\mathfrak{C}[\mathsf{X}_{1},\mathsf{X}_{2}]$ can be considered as a
subset of $\mathfrak{C}[\tilde{\mathsf{X}}_{1},\tilde{\mathsf{X}}_{2}]$ (with
identification of the corresponding vertexes $\mathfrak{o}_{i}$ and
$\tilde{\mathfrak{o}}_{i}$) every homogeneous coupling
$\bm{\alpha}\in\mathfrak{H}^{p}(\mu_{1},\mu_{2})$ can be considered as a
measure in
$\mathcal{M}_{+}(\mathfrak{C}[\tilde{\mathsf{X}}_{1},\tilde{\mathsf{X}}_{2}])$
which belongs to $\mathfrak{H}^{p}(\tilde{\mu}_{1},\tilde{\mu}_{2})$.
Conversely, if $\tilde{\mu}_{i}\in\mathcal{M}_{+}(\tilde{\mathsf{X}}_{i})$ are
concentrated in $\mathsf{X}_{i}$ then every plan
$\tilde{\bm{\alpha}}\in\mathfrak{H}^{p}(\tilde{\mu}_{1},\tilde{\mu}_{2})$ is
concentrated in $\mathfrak{C}[\mathsf{X}_{1},\mathsf{X}_{2}]$ and its
restriction to $\mathfrak{C}[\mathsf{X}_{1},\mathsf{X}_{2}]$ belongs to
$\mathfrak{H}^{p}(\mu_{1},\mu_{2}).$ In particular, recalling the construction
of the spaces $\mathsf{X}_{i,o}$ at the beginning of Section 2.1, for every
pair of measures $\mu_{i}\in\mathcal{M}_{+}(\mathsf{X}_{i})$ the set
$\mathfrak{H}^{p}(\mu_{1},\mu_{2})$ can also be considered as a subset of
$\mathcal{M}_{+}\big{(}\mathfrak{C}[\mathsf{X}_{1,o},\mathsf{X}_{2,o}]\big{)}$.

The following renormalization result comes from [LMS18].

######  Lemma 2.8.

Let $\mathsf{X}_{i}$ for $i=1,2$ be completely regular spaces and let $p\geq
1$. Given
$\bm{\alpha}\in\mathcal{M}_{+}(\mathfrak{C}[\mathsf{X}_{1},\mathsf{X}_{2}])$
and $\vartheta:\mathfrak{C}[\mathsf{X}_{1},\mathsf{X}_{2}]\to(0,+\infty)$
Borel measurable in
$L^{p}(\mathfrak{C}[\mathsf{X}_{1},\mathsf{X}_{2}],\bm{\alpha})$ we can define

$\displaystyle\operatorname{\mathrm{prd}}_{\vartheta}(\bm{\mathfrak{y}}):=\left(\vartheta(\bm{\mathfrak{y}})^{-1}\mathfrak{y}_{1},\vartheta(\bm{\mathfrak{y}})^{-1}\mathfrak{y}_{2}\right),\quad\bm{\mathfrak{y}}\in\mathfrak{C}[\mathsf{X}_{1},\mathsf{X}_{2}],\quad\quad\quad\operatorname{\mathrm{dil}}_{\vartheta,p}(\bm{\alpha}):=(\operatorname{\mathrm{prd}}_{\vartheta})_{\sharp}(\vartheta^{p}\bm{\alpha}).$

Then we have

$\mathfrak{h}_{i}(\operatorname{\mathrm{dil}}_{\vartheta,p}(\bm{\alpha}))=\mathfrak{h}_{i}^{p}(\bm{\alpha}),\quad
i=1,2.$

In particular, if we define

$\vartheta_{\bm{\alpha},p}(\bm{\mathfrak{y}}):=\frac{1}{r^{*}(\bm{\alpha})}\begin{cases}\mathsf{r}_{1}^{p}(\bm{\mathfrak{y}})+\mathsf{r}_{2}^{p}(\bm{\mathfrak{y}})\quad&\text{
if }\bm{\mathfrak{y}}\neq(\mathfrak{o}_{1},\mathfrak{o}_{2})\\\ 1\quad&\text{
if }\bm{\mathfrak{y}}=(\mathfrak{o}_{1},\mathfrak{o}_{2})\end{cases},$

where $r^{*}(\bm{\alpha})$ is a normalization constant s.t.
$\int_{\mathfrak{C}[\mathsf{X}_{1},\mathsf{X}_{2}]}\vartheta_{\bm{\alpha},p}^{p}\,\mathrm{d}\bm{\alpha}=1$
given by

$r^{*}(\bm{\alpha}):=\int_{\mathfrak{C}[\mathsf{X}_{1},\mathsf{X}_{2}]}(\mathsf{r}_{1}^{p}+\mathsf{r}_{2}^{p})\,\mathrm{d}\bm{\alpha}+\bm{\alpha}(\\{(\mathfrak{o}_{1},\mathfrak{o}_{2})\\}),$

we have that
$\operatorname{\mathrm{dil}}_{\vartheta_{\bm{\alpha},p},p}(\bm{\alpha})\in\mathcal{P}(\mathfrak{C}[\mathsf{X}_{1},\mathsf{X}_{2}])$,
it has the same $p$-homogeneous marginals of $\bm{\alpha}$ and its support is
contained in
$\mathfrak{C}_{r^{*}(\bm{\alpha})}[\mathsf{X}_{1},\mathsf{X}_{2}]$.

The following Lemma is a compactness result for homogeneous marginals (the
corresponding statement for classical marginals is [SS22, Theorem 3.1]).

######  Lemma 2.9.

Let $p\geq 1$, $R>0$ and let $\mathsf{X},\mathsf{X}_{i}$, $i=1,2$ be
completely regular spaces. Then:

  1. (1)

if $\left(\alpha_{\lambda}\right)_{\lambda\in\mathbb{L}}$ is a net in
$\mathcal{P}(\mathfrak{C}_{R}[\mathsf{X}])$ such that
$\mu_{\lambda}:=\mathfrak{h}^{p}(\alpha_{\lambda})\in\mathcal{M}_{+}(\mathsf{X})$,
$\lambda\in\mathbb{L}$, converges to some $\mu\in\mathcal{M}_{+}(\mathsf{X})$,
then there exists a subnet $(\alpha^{\prime}_{\beta})_{\beta\in\mathbb{B}}$ of
$\left(\alpha_{\lambda}\right)_{\lambda\in\mathbb{L}}$ convergent to some
$\alpha\in\mathcal{P}(\mathfrak{C}_{R}[\mathsf{X}])$ with
$\mathfrak{h}^{p}(\alpha)=\mu$;

  2. (2)

if $\left(\bm{\alpha}_{\lambda}\right)_{\lambda\in\mathbb{L}}$ is a net in
$\mathcal{P}(\mathfrak{C}_{R}[\mathsf{X}_{1},\mathsf{X}_{2}])$ such that
$\mu_{i,\lambda}:=\mathfrak{h}_{i}^{p}(\bm{\alpha}_{\lambda})\in\mathcal{M}_{+}(\mathsf{X}_{i})$,
$i=1,2$, $\lambda\in\mathbb{L}$, converge to some $\mu_{i}$ in
$\mathcal{M}(\mathsf{X}_{i})$, then there exists a subnet
$(\bm{\alpha}^{\prime}_{\beta})_{\beta\in\mathbb{B}}$ of
$\left(\bm{\alpha}_{\lambda}\right)_{\lambda\in\mathbb{L}}$ convergent to some
$\bm{\alpha}\in\mathfrak{H}^{p}(\mu_{1},\mu_{2})$.

###### Proof.

Thanks to [SS22, Theorem 3.1], it is enough to prove only the first claim.
Define, for every $\lambda\in\mathbb{L}$,

$\vartheta_{\lambda}:=\mathfrak{q}_{\sharp}(\mathsf{r}^{p}\alpha_{\lambda})\in\mathcal{M}_{+}(\mathsf{X}\times[0,R]).$

Notice that this definition does not depend on the point $\bar{x}$ w.r.t.
$\mathfrak{q}$ is defined. Observe that
$\pi^{[0,R]}_{\sharp}\vartheta_{\lambda}\in\mathcal{M}_{+}([0,R])$ with mass
bounded by $R^{p}$ and
$\pi^{\mathsf{X}}_{\sharp}\vartheta_{\lambda}=\mu_{\lambda}$. Then we can
apply [SS22, Theorem 3.1] to
$\left(\vartheta_{\lambda}\right)_{\lambda\in\mathbb{L}}$ and obtain that, up
to passing to a subnet, there exists
$\vartheta\in\mathcal{M}_{+}(\mathsf{X}\times[0,R])$ s.t.
$\lim_{\lambda\in\mathbb{L}}\vartheta_{\lambda}=\vartheta$. Now we define

$O_{n}:=\left\\{[x,r]\in\mathfrak{C}[\mathsf{X}]:0\leq
r\leq\frac{1}{n}\right\\},\quad n\geq 1$

and, for every $n\geq 1$, the nets of real numbers

$m_{\lambda,n}:=\bm{\alpha}_{\lambda}(O_{n}).$

Observe that $0\leq m_{\lambda,n}\leq 1$ for every $n\geq 1$ and
$\lambda\in\mathbb{L}$ then, up to passing to a subnet (the same for every
$n\in\mathbb{N}$), they converge in $\lambda\in\mathbb{L}$ to some
$m_{n}\in[0,1]$. Define then $m:=\inf_{n\geq 1}m_{n}$. We claim then that

$\lim_{\lambda\in\mathbb{L}}\bm{\alpha}_{\lambda}=\frac{1}{\mathsf{r}^{p}}\mathfrak{p}_{\sharp}\vartheta+m\delta_{\mathfrak{o}}=:\bm{\alpha}.$

Take any $\Omega\subset\mathfrak{C}[\mathsf{X}]$ open; if
$\mathfrak{o}\notin\Omega$, we have

$\displaystyle\liminf_{\lambda\in\mathbb{L}}\alpha_{\lambda}(\Omega)$
$\displaystyle=\liminf_{\lambda\in\mathbb{L}}\int_{\mathsf{X}\times[0,R]}({\raise
1.29167pt\hbox{$\chi$}}_{\Omega}\circ\mathfrak{q})(x,r)\frac{1}{(\mathsf{r}^{p}\circ\mathfrak{q})(x,r)}\,\mathrm{d}\vartheta_{\lambda}(x,r)$
$\displaystyle\geq\int_{\mathsf{X}\times[0,R]}({\raise
1.29167pt\hbox{$\chi$}}_{\Omega}\circ\mathfrak{q})(x,r)\frac{1}{(\mathsf{r}^{p}\circ\mathfrak{q})(x,r)}\,\mathrm{d}\vartheta(x,r)$
$\displaystyle=\bm{\alpha}(\Omega).$

If, on the other hand, $\mathfrak{o}\in\Omega$, we have that
$O_{N}\subset\Omega$ for some $N\geq 1$; calling $\Omega_{n}:=\Omega\setminus
O_{n}$ (which is an open set), we have, for every $n\geq N$, that

$\displaystyle\liminf_{\lambda\in\mathbb{L}}\bm{\alpha}_{\lambda}(\Omega)$
$\displaystyle\geq\liminf_{\lambda\in\mathbb{L}}\bm{\alpha}_{\lambda}(O_{n})+\liminf_{\lambda\in\mathbb{L}}\bm{\alpha}_{\lambda}(\Omega_{n})$
$\displaystyle\geq\bm{\alpha}(\Omega_{n})+m_{n}.$

Now we pass to the limit as $n\to+\infty$ and, using the monotone convergence
theorem and the fact that
$\Omega_{n}\uparrow\Omega\setminus\\{\mathfrak{o}\\}$, we obtain

$\liminf_{\lambda\in\mathbb{L}}\bm{\alpha}_{\lambda}(\Omega)\geq\bm{\alpha}(\Omega\setminus\\{\mathfrak{o}\\})+m=\bm{\alpha}(\Omega),$

and this concludes the proof thanks to Portmanteau theorem (see e.g. [Bog06,
Corollary 8.2.10]). ∎

###  2.2. Functions on the product cone

We describe a few properties of functions defined on the product cone that
will be useful in the sequel. In this subsection $\mathsf{X}_{1}$ and
$\mathsf{X}_{2}$ are completely regular spaces. Recall that the indicator
function $\mathsf{I}_{G}:X\to\mathbb{R}\cup\\{+\infty\\}$ associated with a
subset $G$ of a set $X$ is defined by

$\mathsf{I}_{G}(x):=\begin{cases}0&\text{if }x\in G\\\
+\infty&\text{otherwise}\end{cases},\quad x\in X.$

######  Definition 2.10.

Consider a function
$\mathsf{H}:\mathfrak{C}[\mathsf{X}_{1},\mathsf{X}_{2}]\to[0,+\infty]$ and a
subset $\Gamma\subset\mathfrak{C}[\mathsf{X}_{1},\mathsf{X}_{2}]$. For every
$(x_{1},x_{2})\in\mathsf{X}_{1}\times\mathsf{X}_{2}$ we define

$\displaystyle\mathsf{H}_{x_{1},x_{2}}$
$\displaystyle:{}\,\,\mathbb{R}_{+}^{2}\to[0,+\infty]:\qquad(r_{1},r_{2})\mapsto\mathsf{H}([x_{1},r_{1}],[x_{2},r_{2}]),$
$\displaystyle\Gamma_{x_{1},x_{2}}$
$\displaystyle:={}\Big{\\{}(r_{1},r_{2})\in\mathbb{R}^{2}_{+}:([x_{1},r_{1}],[x_{2},r_{2}])\in\Gamma\Big{\\}}.$

We say that

  * •

$\mathsf{H}$ is radially $p$-homogeneous, $p\in[1,+\infty)$, if
$\mathsf{H}_{x_{1},x_{2}}$ is positively $p$-homogeneous for every
$(x_{1},x_{2})\in\mathsf{X}_{1}\times\mathsf{X}_{2}$ i.e.

$\mathsf{H}_{x_{1},x_{2}}(\lambda r_{1},\lambda
r_{2})=\lambda^{p}\mathsf{H}_{x_{1},x_{2}}(r_{1},r_{2})\quad\text{ for every
}\lambda>0,\,(r_{1},r_{2})\in\mathbb{R}_{+}^{2};$

  * •

$\mathsf{H}$ is radially convex if $\mathsf{H}_{x_{1},x_{2}}$ is convex for
every $(x_{1},x_{2})\in\mathsf{X}_{1}\times\mathsf{X}_{2}$.

We say that $\Gamma\subset\mathfrak{C}[\mathsf{X}_{1},\mathsf{X}_{2}]$ is a
radial cone (resp. radial convex set) if its indicator function is radially
$1$-homogeneous (resp. radially convex). Notice that we do not assume that a
radially $p$-homogenoeus function vanishes at
$({\mathfrak{o}}_{1},{\mathfrak{o}}_{2})$ (or, similarly, that a radial cone
contains $({\mathfrak{o}}_{1},{\mathfrak{o}}_{2})$); however such a property
follows immediately under lower semicontinuity and properness conditions, see
Remark 2.11 below.
We define the (radially) $1$-homogeneous, convex, and closed convex envelopes
$\operatorname{\mathrm{hom}}\left(\mathsf{H}\right),\operatorname{\mathrm{co}}\left(\mathsf{H}\right),\operatorname{\overline{\mathrm{co}}}\left(\mathsf{H}\right):\mathfrak{C}[\mathsf{X}_{1},\mathsf{X}_{2}]\to[0,+\infty]$
of $\mathsf{H}$ as (recall (2.2,2.3,2.4))

$\displaystyle\operatorname{\mathrm{hom}}\left(\mathsf{H}\right)([x_{1},r_{1}],[x_{2},r_{2}])$
$\displaystyle:=\inf_{\lambda\geq 0}\mathsf{H}([x_{1},\lambda
r_{1}],[x_{2},\lambda r_{2}]),$
$\displaystyle\operatorname{\mathrm{co}}\left(\mathsf{H}\right)([x_{1},r_{1}],[x_{2},r_{2}])$
$\displaystyle:=\operatorname{\mathrm{co}}\left(\mathsf{H}_{x_{1},x_{2}}\right)(r_{1},r_{2})\quad\text{
for every
}(x_{1},x_{2})\in\mathsf{X}_{1}\times\mathsf{X}_{2},\,(r_{1},r_{2})\in\mathbb{R}_{+}^{2},$
$\displaystyle\operatorname{\overline{\mathrm{co}}}\left(\mathsf{H}\right)$
$\displaystyle:=\overline{\operatorname{\mathrm{co}}\left(\mathsf{H}\right)}.$

Similarly, in the case of
$\Gamma\subset\mathfrak{C}[\mathsf{X}_{1},\mathsf{X}_{2}]$, we define
$\operatorname{\mathrm{hom}}\left(\Gamma\right),\operatorname{\mathrm{co}}\left(\Gamma\right),$
and $\operatorname{\overline{\mathrm{co}}}\left(\Gamma\right)$ so that
$\mathsf{I}_{\operatorname{\mathrm{hom}}\left(\Gamma\right)}:=\operatorname{\mathrm{hom}}\left(\mathsf{I}_{\Gamma}\right)$,
$\mathsf{I}_{\operatorname{\mathrm{co}}\left(\Gamma\right)}:=\operatorname{\mathrm{co}}\left(\mathsf{I}_{\Gamma}\right)$
and
$\mathsf{I}_{\operatorname{\overline{\mathrm{co}}}\left(\Gamma\right)}:=\operatorname{\overline{\mathrm{co}}}\left(\mathsf{I}_{\Gamma}\right)$
respectively.

######  Remark 2.11.

We added the terms _radial and radially_ just to avoid ambiguities in the case
when $\mathsf{X}_{1},\mathsf{X}_{2}$ are linear spaces and the notions of
$1$-homogeneity and convexity could also refer to the joint behaviour of
$\mathsf{H}$ w.r.t. all the variables $x_{i},r_{i}$. In this paper, we will
always interpret convexity and $1$-homogeneity w.r.t. the radial variables
$r_{i}$.
Notice that $\operatorname{\mathrm{hom}}\left(\Gamma\right)$ (resp.
$\operatorname{\mathrm{co}}\left(\Gamma\right),\operatorname{\overline{\mathrm{co}}}\left(\Gamma\right)$)
is the set whose sections
$\operatorname{\mathrm{hom}}\left(\Gamma\right)_{x_{1},x_{2}}$ are the cone
(resp. convex, closed convex) envelopes of the corresponding sections
$\Gamma_{x_{1},x_{2}}$ of $\Gamma$. In particular, racalling Carathéodory
Theorem in $\mathbb{R}^{2}$, we have

$\displaystyle\operatorname{\mathrm{hom}}\left(\Gamma\right):={}$
$\displaystyle\bigcup_{\lambda>0}\Big{\\{}([x_{1},\lambda
r_{1}],[x_{2},\lambda
r_{2}]):([x_{1},r_{1}],[x_{2},r_{2}])\in\Gamma\Big{\\}},$
$\displaystyle\operatorname{\mathrm{co}}\left(\Gamma\right):={}$
$\displaystyle\bigcup\Big{\\{}\Big{(}\big{[}x_{1},\sum_{i=0}^{2}\alpha_{i}r_{1}^{i}\big{]},\big{[}x_{2},\sum_{i=0}^{2}\alpha_{i}r_{2}^{i}\big{]}\Big{)}:([x_{1},r_{1}^{i}],[x_{2},r_{2}^{i}])\in\Gamma,\
\alpha_{i}\geq 0,\ \sum_{i=0}^{2}\alpha^{i}=1\Big{\\}}.$

We also note that if $\mathsf{H}$ is proper (i.e. not identically $+\infty$),
lower semicontinuous, and radially $1$-homogeneous, then
$\mathsf{H}(\mathfrak{0}_{1},\mathfrak{0}_{2})=0$.

The following result is a simple consequence of the $1$-homogeneity property.

######  Lemma 2.12.

Let $\mathsf{X}_{i}$, $i=1,2$ be completely regular spaces, let
$\mathsf{H}:\mathfrak{C}[\mathsf{X}_{1},\mathsf{X}_{2}]\to[0,+\infty]$ be a
radially $1$-homogeneous Borel function and let
$(\mu_{1},\mu_{2})\in\mathcal{M}_{+}(\mathsf{X}_{1})\times\mathcal{M}_{+}(\mathsf{X}_{2})$.
Then

$\inf\left\\{\int\mathsf{H}\,\mathrm{d}\bm{\alpha}:\bm{\alpha}\in\mathfrak{H}^{1}(\mu_{1},\mu_{2})\right\\}=\inf\left\\{\int\mathsf{H}\,\mathrm{d}\bm{\alpha}:\bm{\alpha}\in\mathfrak{H}^{1}(\mu_{1},\mu_{2})\cap\mathcal{P}(\mathfrak{C}_{R(\mu_{1},\mu_{2})}[\mathsf{X}_{1},\mathsf{X}_{2}])\right\\},$

where

$R(\mu_{1},\mu_{2}):=\mu_{1}(\mathsf{X}_{1})+\mu_{2}(\mathsf{X}_{2}).$ (2.13)

###### Proof.

It is of course enough to prove the $\geq$ inequality. If
$\bm{\alpha}\in\mathfrak{H}^{1}(\mu_{1},\mu_{2})$ we can assume that
$\bm{\alpha}(\\{(\mathfrak{o}_{1},\mathfrak{o}_{2})\\})=0$ (if not, we can
replace $\bm{\alpha}$ with
$\bm{\alpha}_{{\bm{{\mathfrak{o}}}}}=\bm{\alpha}\mathbin{\vrule
height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule
height=0.55974pt,depth=0.0pt,width=5.59721pt}\mathfrak{C}_{\bm{{\mathfrak{o}}}}[\mathsf{X}_{1},\mathsf{X}_{2}]$
which has the same homogeneous marginals and a lower $\mathsf{H}$-cost, since
$\mathsf{H}(\mathfrak{o}_{1},\mathfrak{o}_{2})\geq 0$). By Lemma 2.8, we have
that

$\tilde{\bm{\alpha}}:=\operatorname{\mathrm{dil}}_{\vartheta_{\bm{\alpha},1},1}(\bm{\alpha})\in\mathfrak{H}^{1}(\mu_{1},\mu_{2})\cap\mathcal{P}(\mathfrak{C}_{R(\mu_{1},\mu_{2})}[\mathsf{X}_{1},\mathsf{X}_{2}])$

and

$\int_{\mathfrak{C}[\mathsf{X}_{1},\mathsf{X}_{2}]}\mathsf{H}\,\mathrm{d}\bm{\alpha}=\int_{\mathfrak{C}[\mathsf{X}_{1},\mathsf{X}_{2}]}\mathsf{H}\,\mathrm{d}\tilde{\bm{\alpha}}.$

This concludes the proof. ∎

######  Remark 2.13 ($1$-homogeneity and $q$-homogeneity).

Given $q\in[1,+\infty)$, we define the map
$\mathsf{T}_{q}:\mathfrak{C}[\mathsf{X}_{1},\mathsf{X}_{2}]\to\mathfrak{C}[\mathsf{X}_{1},\mathsf{X}_{2}]$
as

$\mathsf{T}_{q}([x_{1},r_{1}],[x_{2},r_{2}]):=([x_{1},r_{1}^{1/q}],[x_{2},r_{2}^{1/q}]).$
(2.14)

It is easy to check that $(\mathsf{T}_{q})_{\sharp}$ is a bijective
transformation from $\mathfrak{H}^{1}(\mu_{1},\mu_{2})$ to
$\mathfrak{H}^{q}(\mu_{1},\mu_{2})$ for any pair
$(\mu_{1},\mu_{2})\in\mathcal{M}_{+}(\mathsf{X}_{1})\times\mathcal{M}_{+}(\mathsf{X}_{2})$.
Moreover, if
$\mathsf{H}_{q}:\mathfrak{C}[\mathsf{X}_{1},\mathsf{X}_{2}]\to[0,\\+\infty]$
is radially $q$-homogeneous, then
$\mathsf{H}:=\mathsf{H}_{q}\circ\mathsf{T}_{q}$ is radially $1$-homogeneous
and

$\inf\left\\{\int_{\mathfrak{C}[\mathsf{X}_{1},\mathsf{X}_{2}]}\mathsf{H}_{q}\,\mathrm{d}\bm{\alpha}_{q}:\bm{\alpha}_{q}\in\mathfrak{H}^{q}(\mu_{1},\mu_{2})\right\\}=\inf\left\\{\int_{\mathfrak{C}[\mathsf{X}_{1},\mathsf{X}_{2}]}\mathsf{H}\,\mathrm{d}\bm{\alpha}:\bm{\alpha}\in\mathfrak{H}^{1}(\mu_{1},\mu_{2})\right\\}.$
(2.15)

As an immediate consequence of the above formula, many of the results of this
work can also be stated in the case of a $q$-homogeneous cost function
$\mathsf{H}_{q}$, provided that the unbalanced optimal transport functional is
defined by using the $q$-homogeneous marginals $\mathfrak{h}^{q}$ and the
corresponding set $\mathfrak{H}^{q}$ as in (2.15). For this reason, in order
to keep a simpler notation we will limit our analysis to the $1$-homogeneous
case.

###  2.3. Examples of cost functions

In this subsection we present some examples of functions
$\mathsf{H}:\mathfrak{C}[\mathsf{X}_{1},\mathsf{X}_{2}]\to[0,+\infty]$, where
$\mathsf{X}_{1}$ and $\mathsf{X}_{2}$ are completely regular spaces,
satisfying (some of) the hypotheses we will assume throughout the paper. See
also the examples of [Chi+18a, Section5].

Notice that given two radially $1$-homogeneous, convex, lower semicontinuous
and proper functions
$\mathsf{H}_{1},\mathsf{H}_{2}:\mathfrak{C}[\mathsf{X}_{1},\mathsf{X}_{2}]\to[0,+\infty)$,
and two lower semicontinuous and proper functions
$\mathsf{c}_{1},\mathsf{c}_{2}:\mathsf{X}_{1}\times\mathsf{X}_{2}\to[0,+\infty)$
also the function

$\mathsf{H}([x_{1},r_{1}],[x_{2},r_{2}])=\mathsf{c}_{1}(x_{1},x_{2})\mathsf{H}_{1}([x_{1},r_{1}],[x_{2},r_{2}])+\mathsf{c}_{2}(x_{1},x_{2})\mathsf{H}_{2}([x_{1},r_{1}],[x_{2},r_{2}])$

is radially $1$-homogeneous, convex, lower semicontinuous and proper. In this
way, many other examples can be obtained starting from the ones presented
below. Notice moreover that the radial behavior of a $1$-homogeneous convex
function
$\mathsf{H}_{x_{1},x_{2}}:\mathbb{R}_{+}\times\mathbb{R}_{+}\to[0,+\infty]$
can be characterized by the reduced function and its recession slope

$\mathsf{h}(x_{1},x_{2};g)=\mathsf{h}_{x_{1},x_{2}}(g):=\mathsf{H}_{x_{1},x_{2}}(1,g),\quad
h^{\infty}_{x_{1},x_{2}}:=\lim_{g\to+\infty}g^{-1}\mathsf{h}_{x_{1},x_{2}}(g),$
(2.16)

since

$\mathsf{H}_{x_{1},x_{2}}(r_{1},r_{2})=\begin{cases}r_{1}\mathsf{h}_{x_{1},x_{2}}(r_{2}/r_{1})&\text{if
}r_{1}>0,\\\ r_{2}\mathsf{h}^{\infty}_{x_{1},x_{2}}&\text{if }r_{1}=0,\
r_{2}>0,\\\ 0&\text{if }r_{1}=r_{2}=0.\end{cases}$ (2.17)

####  2.3.1. Mass-space product costs

We consider cost functions of the form

$\mathsf{H}([x_{1},r_{1}],[x_{2},r_{2}]):=\mathsf{H}_{+}(r_{1},r_{2})+\mathsf{H}_{-}(r_{1},r_{2})\mathsf{c}(x_{1},x_{2}),\quad([x_{1},r_{1}],[x_{2},r_{2}])\in\mathfrak{C}[\mathsf{X}_{1},\mathsf{X}_{2}],$

where $\mathsf{H}_{+},\mathsf{H}_{-}:\mathbb{R}^{2}_{+}\to\mathbb{R}$ are
convex, $1$-homogeneous and lower semicontinuous and
$\mathsf{c}:\mathsf{X}_{1}\times\mathsf{X}_{2}\to[0,+\infty]$ is a lower
semicontinuous function satisfying

$\mathsf{H}_{+}(r_{1},r_{2})+\mathsf{H}_{-}(r_{1},r_{2})\mathsf{c}(x_{1},x_{2})\geq
0\quad\text{ for every
}(x_{1},x_{2})\in\mathsf{X}_{1}\times\mathsf{X}_{2},\,r_{1},r_{2}\geq 0.$

Possible choices of $\mathsf{H}_{+}$ and $\mathsf{H}_{-}$ (to be then coupled
with a suitable cost $\mathsf{c}$) are given by e.g.

  1. (1)

$\displaystyle
m_{p}(r_{1},r_{2}):=\left(r_{1}^{p}+r_{2}^{p}\right)^{\frac{1}{p}},\quad
p\in[1,+\infty)$,

  2. (2)

$\displaystyle m_{p}(r_{1},r_{2}):=-m_{p}(r_{1},r_{2}),\quad
p\in(-\infty,0)\cup(0,1)$,

  3. (3)

$\displaystyle m_{\infty}(r_{1},r_{2})=r_{1}\vee r_{2},\quad
m_{-\infty}(r_{1},r_{2})=-(r_{1}\wedge r_{2}),\quad m_{0}=-\sqrt{r_{1}r_{2}}$,

  4. (4)

$\displaystyle
n_{\alpha}(r_{1},r_{2}):=\frac{|r_{1}-r_{2}|^{\alpha}}{(r_{1}+r_{2})^{\alpha-1}},\quad\alpha\geq
1$,

  5. (5)

$\displaystyle|r_{1}^{\alpha}-r_{2}^{\alpha}|^{1/\alpha},\quad-|r_{1}^{\alpha}+r_{2}^{\alpha}|^{1/\alpha},\quad
0<\alpha\leq 1$.

In the various examples we are adopting standard conventions when the
expressions are not defined. In particular, for $p<0$ we set
$m_{p}(r_{1},r_{2})=0$ if $r_{1}r_{2}=0$, $n_{\alpha}(0,0)=0$.

####  2.3.2. Homogeneous marginal perspective functional

Following [LMS18, Section 5] we can build $\mathsf{H}$ starting from two
entropy functions $F_{i}:\mathsf{X}_{i}\to[0,+\infty]$, $i=1,2$ and a proper
and lower semicontinuous cost function
$\mathsf{c}:\mathsf{X}_{1}\times\mathsf{X}_{2}\to[0,+\infty]$. Assuming that
each $F_{i}$, $i=1,2$ is convex, lower semicontinuous and finite in at least
one positive point, we can define, for every number $c\in[0,+\infty]$, the
function $\mathsf{H}_{c}:\mathbb{R}_{+}^{2}\to[0,+\infty]$, as the lower
semicontinuous envelope of

$\tilde{\mathsf{H}}_{c}(r_{1},r_{2}):=\begin{cases}\inf_{\theta>0}\left\\{r_{1}F_{1}(\theta/r_{1})+r_{2}F_{2}(\theta/r_{2})+\theta
c\right\\},\quad&\text{ if }c\in[0,+\infty)\\\
F_{1}(0)r_{1}+F_{2}(0)r_{2}\quad&\text{ if }c=\infty,\end{cases}\quad
r_{1},r_{2}\in\mathbb{R}_{+}^{2}.$

The function
$\mathsf{H}:\mathfrak{C}[\mathsf{X}_{1},\mathsf{X}_{2}]\to[0,+\infty]$ is then
defined as

$\mathsf{H}([x_{1},r_{1}],[x_{2},r_{2}]):=\mathsf{H}_{\mathsf{c}(x_{1},x_{2})}(r_{1},r_{2}),\quad([x_{1},r_{1}],[x_{2},r_{2}])\in\mathfrak{C}[\mathsf{X}_{1},\mathsf{X}_{2}].$

Such function $\mathsf{H}$ is radially convex and $1$-homogeneous (see [LMS18,
Lemma 5.3]). Possible choices (see e.g. [LMS16, De ̵20]) for $F_{i}$ are given
by:

  1. (1)

power like entropies: for $p\in\mathbb{R}$ we define

$U_{p}(s):=\begin{cases}\frac{1}{p(p-1)}\left(s^{p}-p(s-1)-1)\right)\quad&\text{
if }p\neq 0,1,\\\ s\log s-s+1,\quad&\text{ if }p=1,\\\ s-1-\log s,\quad&\text{
if }p=0,\\\ \end{cases}\quad\text{ for }s>0,$

with $U_{p}(0)=1/p$ if $p>0$ and $U_{p}(0)=+\infty$ if $p\leq 0$;

  2. (2)

indicator functions: for numbers $0\leq a\leq 1\leq b\leq+\infty$ we define

$I_{[a,b]}(s):=\begin{cases}0\quad&\text{ if }s\in[a,b],\\\
+\infty\quad&\text{ if }s\notin[a,b];\end{cases}$

  3. (3)

${\raise 1.29167pt\hbox{$\chi$}}^{\alpha}$ divergences: for a parameter
$\alpha\geq 1$ we define

${\raise 1.29167pt\hbox{$\chi$}}^{\alpha}(s):=|s-1|^{\alpha},\quad
s\in\mathbb{R}.$

Some of the corresponding expression for $\mathsf{H}$ are for example

  1. (1)

In case of power like entropies with $F_{1}=F_{2}=U_{p}$:

$\mathsf{H}([x_{1},r_{1}],[x_{2},r_{2}])=\begin{dcases}\frac{1}{p}\left[(r_{1}+r_{2})-\frac{r_{1}r_{2}}{(r_{1}^{p-1}+r_{2}^{p-1})^{\frac{1}{p-1}}}(2-(p-1)\mathsf{c}(x_{1},x_{2}))_{+}^{\frac{p}{p-1}}\right],\quad&\text{
if }p\neq 0,1,\\\
(\sqrt{r_{1}}-\sqrt{r_{2}})^{2}+2\sqrt{r_{1}r_{2}}(1-e^{-\mathsf{c}(x_{1},x_{2})/2}),\quad&\text{
if }p=1,\\\
r_{1}\log(r_{1})+r_{2}\log(r_{2})-(r_{1}+r_{2})\log\left(\frac{r_{1}+r_{2}}{2+\mathsf{c}(x_{1},x_{2})}\right),\quad&\text{
if }p=0.\end{dcases}$

In particular, in case $p=1$, for
$\mathsf{X}_{1}=\mathsf{X}_{2}=\mathbb{R}^{d}$ we can chose as cost functions
$\mathsf{c}$

$\mathsf{c}_{\mathsf{GHK}}(x_{1},x_{2}):=|x_{1}-x_{2}|^{2},\quad\mathsf{c}_{\mathsf{HK}}(x_{1},x_{2}):=\begin{cases}-\log(\cos^{2}(|x_{1}-x_{2}|))\quad&\text{
if }|x_{1}-x_{2}|<\pi/2,\\\ +\infty\quad&\text{ else}.\end{cases}$

The resulting cost functions $\mathsf{H}$ are thus given respectively by

$\displaystyle\mathsf{H}_{\mathsf{GHK}}([x_{1},r_{1}],[x_{2},r_{2}])$
$\displaystyle=r_{1}+r_{2}-2\sqrt{r_{1}r_{2}}e^{-|x_{1}-x_{2}|^{2}/2},$ (2.18)
$\displaystyle\mathsf{H}_{\mathsf{HK}}([x_{1},r_{1}],[x_{2},r_{2}])$
$\displaystyle=r_{1}+r_{2}-2\sqrt{r_{1}r_{2}}\cos(|x_{1}-x_{2}|\wedge\pi/2).$
(2.19)

(2.18) and (2.19) are metrics on $\mathfrak{C}[\mathbb{R}^{d}]$, inducing the
same canonical cone topology. In particular, $\mathsf{H}_{\mathsf{HK}}$ is
related to (2.9) via the transformation in Remark 2.13 with $q=2$ (apart from
the specific value of the truncation constant). Both functions are considered
in [LMS18] and they generate the Gaussian Hellinger-Kantorovich and the
Hellinger-Kantorovich metrics on non-negative measures, respectively.

  2. (2)

In case of indicator functions with $F_{1}=F_{2}=I_{[a,b]}$:

$\mathsf{H}([x_{1},r_{1}],[x_{2},r_{2}])=\begin{cases}0\quad&\text{ if
}\frac{a}{b}\leq\frac{r_{1}}{r_{2}}\leq\frac{b}{a},\\\ +\infty\quad&\text{
else},\end{cases}$

where $\frac{b}{a}=+\infty$ if $a=0$ and $\frac{a}{b}=0$ if $b=+\infty$.

  3. (3)

In case of the $\chi^{1}$ divergence with $F_{1}=F_{2}=\chi^{1}$:

$\mathsf{H}([x_{1},r_{1}],[x_{2},r_{2}])=|r_{2}-r_{1}|+(\mathsf{c}(x_{1},x_{2})\wedge
2)(r_{1}\wedge r_{2}).$

##  3\. Convexification and duality

This section presents the main convexification and duality results. In this
section $\mathsf{X}_{1}$ and $\mathsf{X}_{2}$ are completely regular spaces.

###  3.1. Unbalanced Optimal Transport as convex relaxation of singular cost
on Dirac masses

######  Definition 3.1 (Homogeneous conical formulation of Unbalanced Optimal
Transport problems).

Let $\mathsf{H}:\mathfrak{C}[\mathsf{X}_{1},\mathsf{X}_{2}]\to[0,+\infty]$ be
a proper (i.e. not identically equal to $+\infty$) Borel function. We define
the singular cost and the Unbalanced Optimal Transport cost
$\mathscr{S}_{\mathsf{H}},\mathscr{U}_{\mathsf{H}}:\mathcal{M}(\mathsf{X}_{1})\times\mathcal{M}(\mathsf{X}_{2})\to[0,+\infty]$
respectively as:

$\displaystyle\mathscr{S}_{\mathsf{H}}(\mu_{1},\mu_{2})$
$\displaystyle:=\begin{dcases}\mathsf{H}([x_{1},r_{1}],[x_{2},r_{2}])\quad&\text{
if
}\mu_{1}=r_{1}\delta_{x_{1}},\,\mu_{2}=r_{2}\delta_{x_{2}},\,(x_{1},x_{2})\in\mathsf{X}_{1}\times\mathsf{X}_{2},\,(r_{1},r_{2})\in\mathbb{R}_{+}^{2},\\\
+\infty\quad&\text{ elsewhere}.\end{dcases}$
$\displaystyle\mathscr{U}_{\mathsf{H}}(\mu_{1},\mu_{2})$
$\displaystyle:=\begin{dcases}\inf\left\\{\int_{\mathfrak{C}[\mathsf{X}_{1},\mathsf{X}_{2}]}\mathsf{H}\,\mathrm{d}\bm{\alpha}:\bm{\alpha}\in\mathfrak{H}^{1}(\mu_{1},\mu_{2})\right\\}\quad&\text{
if
}(\mu_{1},\mu_{2})\in\mathcal{M}_{+}(\mathsf{X}_{1})\times\mathcal{M}_{+}(\mathsf{X}_{2}),\\\
+\infty\quad&\text{ elsewhere}.\end{dcases}$

The aim of this section is to study the relation between
$\mathscr{S}_{\mathsf{H}}$ and $\mathscr{U}_{\mathsf{H}}$; in particular we
are interested in studying the lower semicontinuous and convex envelope of
$\mathscr{S}_{\mathsf{H}}$.

Let us first recall two simple properties in the following remarks.

######  Remark 3.2.

The value of $\mathscr{U}_{\mathsf{H}}(\mu_{1},\mu_{2})$ does not depend on
the ambient spaces $\mathsf{X}_{i}$ (recall the discussion of Remark 2.7): if
$\mathsf{X}_{i}\subset\tilde{\mathsf{X}}_{i}$, $\mathsf{H}$ is the restriction
to $\mathfrak{C}[\mathsf{X}_{1},\mathsf{X}_{2}]$ of a function
$\tilde{\mathsf{H}}$ defined in
$\mathfrak{C}[\tilde{\mathsf{X}}_{1},\tilde{\mathsf{X}}_{2}]$, and
$\tilde{\mu}_{i}\in\mathcal{M}_{+}(\tilde{\mathsf{X}}_{i})$ are the canonical
extensions of $\mu_{i}$, then Remark 2.7 yields

$\mathscr{U}_{\mathsf{H}}(\mu_{1},\mu_{2})=\mathscr{U}_{\tilde{\mathsf{H}}}(\tilde{\mu}_{1},\tilde{\mu}_{2}).$
(3.1)

In particular, we can always “embed” Unbalanced Optimal Transport problems in
$\mathsf{X}_{i,o}$ or, equivalently, suppose that there is at least one
$\mu_{i}$-negligible point $o_{i}$ in each space $\mathsf{X}_{i}$.

######  Remark 3.3.

If $\mathsf{H}:\mathfrak{C}[\mathsf{X}_{1},\mathsf{X}_{2}]\to[0,+\infty]$ is a
proper Borel function, then

$\overline{\mathscr{S}_{\mathsf{H}}}=\mathscr{S}_{\overline{\mathsf{H}}}.$

Indeed, both are equal to $+\infty$ outside the closed set
$\Delta_{+}(\mathsf{X}_{1})\times\Delta_{+}(\mathsf{X}_{2})$ and the equality
on $\Delta_{+}(\mathsf{X}_{1})\times\Delta_{+}(\mathsf{X}_{2})$ follows by
Lemma 2.4.

For this reason and to exploit Lemma 2.12, we will usually assume that

$\mathsf{H}:\mathfrak{C}[\mathsf{X}_{1},\mathsf{X}_{2}]\to[0,+\infty]\text{ is
a proper, radially $1$-homogeneous and l.s.c.\leavevmode\nobreak\ function.}$
(3.2)

In the following result we prove that the Unbalanced Optimal Transport cost
$\mathscr{U}_{\mathsf{H}}$ is a lower semicontinuous convex functional in
$\mathcal{M}_{+}(\mathsf{X}_{1})\times\mathcal{M}_{+}(\mathsf{X}_{2})$ and the
infimum in the Definition of $\mathscr{U}_{\mathsf{H}}$ is attained, when the
problem is feasible.

######  Theorem 3.4 (Existence of solutions to the Unbalanced Optimal
Transport problem).

Let $\mathsf{H}$ be as in (3.2) and let $\mathscr{U}_{\mathsf{H}}$ be as in
Definition 3.1.

  1. (1)

For every
$(\mu_{1},\mu_{2})\in\mathcal{M}_{+}(\mathsf{X}_{1})\times\mathcal{M}_{+}(\mathsf{X}_{2})$
such that $\mathscr{U}_{\mathsf{H}}(\mu_{1},\mu_{2})<+\infty$, there exists an
optimal $1$-homogeneous coupling
$\bm{\alpha}\in\mathfrak{H}^{1}(\mu_{1},\mu_{2})\cap\mathcal{P}(\mathfrak{C}_{R(\mu_{1},\mu_{2})}[\mathsf{X}_{1},\mathsf{X}_{2}])$
not charging ${\bm{{\mathfrak{o}}}}=({\mathfrak{o}}_{1},{\mathfrak{o}}_{2})$
such that

$\mathscr{U}_{\mathsf{H}}(\mu_{1},\mu_{2})=\int_{\mathfrak{C}[\mathsf{X}_{1},\mathsf{X}_{2}]}\mathsf{H}\,\mathrm{d}\bm{\alpha},$

where $R(\mu_{1},\mu_{2})$ is as in (2.13).

  2. (2)

$\mathscr{U}_{\mathsf{H}}$ is a lower semicontinuous convex functional in
$\mathcal{M}_{+}(\mathsf{X}_{1})\times\mathcal{M}_{+}(\mathsf{X}_{2})$ and
satisfies

$\mathscr{U}_{\mathsf{H}}(r_{1}\delta_{x_{1}},r_{2}\delta_{x_{2}})\leq\mathsf{H}([x_{1},r_{1}],[x_{2},r_{2}])$
(3.3)

for every $(x_{1},x_{2})\in\mathsf{X}_{1}\times\mathsf{X}_{2}$ and every
$(r_{1},r_{2})\in\mathbb{R}_{+}^{2}$.

  3. (3)

If, in addition, $\mathsf{H}$ is also radially convex, then (3.3) is an
equality:

$\mathscr{U}_{\mathsf{H}}(r_{1}\delta_{x_{1}},r_{2}\delta_{x_{2}})=\mathsf{H}([x_{1},r_{1}],[x_{2},r_{2}]).$

###### Proof.

(1) Let
$(\mu_{1},\mu_{2})\in\mathcal{M}_{+}(\mathsf{X}_{1})\times\mathcal{M}_{+}(\mathsf{X}_{2})$;
by Lemma 2.12, it holds

$\mathscr{U}_{\mathsf{H}}(\mu_{1},\mu_{2})=\inf\left\\{\int_{\mathfrak{C}[\mathsf{X}_{1},\mathsf{X}_{2}]}\mathsf{H}\,\mathrm{d}\bm{\alpha}:\bm{\alpha}\in\mathfrak{H}^{1}(\mu_{1},\mu_{2})\cap\mathcal{P}(\mathfrak{C}_{R(\mu_{1},\mu_{2})}[\mathsf{X}_{1},\mathsf{X}_{2}])\right\\}.$

Thanks to Lemma 2.9, we have that
$\mathfrak{H}^{1}(\mu_{1},\mu_{2})\cap\mathcal{P}(\mathfrak{C}_{R(\mu_{1},\mu_{2})}[\mathsf{X}_{1},\mathsf{X}_{2}])$
is compact and the lower semicontinuity of $\mathsf{H}$ gives that the
functional

$\bm{\alpha}\mapsto\int_{\mathfrak{C}[\mathsf{X}_{1},\mathsf{X}_{2}]}\mathsf{H}\,\mathrm{d}\bm{\alpha}$
(3.4)

is lower semicontinuous. We can thus conclude that a minimizer exists by the
direct method in Calculus of Variations. Possibly replacing $\bm{\alpha}$ by
$\bm{\alpha}_{{\mathfrak{o}}}=\bm{\alpha}\mathbin{\vrule
height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule
height=0.55974pt,depth=0.0pt,width=5.59721pt}\mathfrak{C}_{\bm{{\mathfrak{o}}}}[\mathsf{X}_{1},\mathsf{X}_{2}]$
we obtain an optimal homogeneous coupling not charging
$({\mathfrak{o}}_{1},{\mathfrak{o}}_{2}).$

(2) The convexity of $\mathscr{U}_{\mathsf{H}}$ follows by the convexity of
the constraints and the linearity of the objective function characterizing
$\mathscr{U}_{\mathsf{H}}$: if
$(\mu_{1}^{1},\mu_{2}^{1}),(\mu_{2}^{1},\mu_{2}^{2})\in\mathcal{M}_{+}(\mathsf{X}_{1})\times\mathcal{M}_{+}(\mathsf{X}_{2})$
and $t\in[0,1]$, we can take, thanks to point (1),
$\bm{\alpha}_{1}\in\mathfrak{H}^{1}(\mu_{1}^{1},\mu_{2}^{1})$ and
$\bm{\alpha}_{2}\in\mathfrak{H}^{1}(\mu_{1}^{2},\mu_{2}^{2})$ such that

$\mathscr{U}_{\mathsf{H}}(\mu_{1}^{1},\mu_{2}^{1})=\int_{\mathfrak{C}[\mathsf{X}_{1},\mathsf{X}_{2}]}\mathsf{H}\,\mathrm{d}\bm{\alpha}_{1},\quad\mathscr{U}_{\mathsf{H}}(\mu_{1}^{2},\mu_{2}^{2})=\int_{\mathfrak{C}[\mathsf{X}_{1},\mathsf{X}_{2}]}\mathsf{H}\,\mathrm{d}\bm{\alpha}_{2}.$

It is then enough to observe that
$\bm{\alpha}:=(1-t)\bm{\alpha}_{1}+t\bm{\alpha}_{2}\in\mathfrak{H}^{1}((1-t)\mu_{1}^{1}+t\mu_{1}^{2},(1-t)\mu_{2}^{1}+t\mu_{2}^{2})$.

The lower semicontinuity of $\mathscr{U}_{\mathsf{H}}$ is a consequence of
Lemma 2.12: if
$\\{(\mu_{1}^{\lambda},\mu_{2}^{\lambda})\\}_{\lambda\in\mathbb{L}}\subset\mathcal{M}_{+}(\mathsf{X}_{1})\times\mathcal{M}_{+}(\mathsf{X}_{2})$
is a net converging to
$(\mu_{1},\mu_{2})\in\mathcal{M}_{+}(\mathsf{X}_{1})\times\mathcal{M}_{+}(\mathsf{X}_{2})$,
we can set $R:=\sup_{\lambda}R(\mu_{1}^{\lambda},\mu_{2}^{\lambda})$ (see
(2.13)) and consider, for every $\lambda\in\mathbb{L}$, some
$\bm{\alpha}_{\lambda}\in\mathfrak{H}^{1}(\mu_{1}^{\lambda},\mu_{2}^{\lambda})\cap\mathcal{P}(\mathfrak{C}_{R}[\mathsf{X}_{1},\mathsf{X}_{2}])$
such that

$\mathscr{U}_{\mathsf{H}}(\mu_{1}^{\lambda},\mu_{2}^{\lambda})=\int_{\mathfrak{C}[\mathsf{X}_{1},\mathsf{X}_{2}]}\mathsf{H}\,\mathrm{d}\bm{\alpha}_{\lambda}.$

We can thus use Lemma 2.9 to extract a convergent subnet of
$\left(\bm{\alpha}_{\lambda}\right)_{\lambda\in\mathbb{L}}$ with limit
$\bm{\alpha}\in\mathfrak{H}^{1}(\mu_{1},\mu_{2})$. Using again the lower
semicontinuity of the functional in (3.4), we can conclude that
$\mathscr{U}_{\mathsf{H}}$ is lower semicontinuous.

(3.3) follows by the fact that

$\bm{\alpha}=\delta_{([x_{1},r_{1}],[x_{2},r_{2}])}$

is an element of $\mathfrak{H}^{1}(r_{1}\delta_{x_{1}},r_{2}\delta_{x_{2}})$.

(3) Let us assume that $\mathsf{H}$ is radially convex and let
$\bm{\alpha}\in\mathfrak{H}^{1}(r_{1}\delta_{x_{1}},r_{2}\delta_{x_{2}})$ be
such that

$\mathscr{U}_{\mathsf{H}}(r_{1}\delta_{x_{1}},r_{2}\delta_{x_{2}})=\int_{\mathfrak{C}[\mathsf{X}_{1},\mathsf{X}_{2}]}\mathsf{H}\,\mathrm{d}\bm{\alpha};$

we observe that $\bm{\alpha}$ is concentrated on

$\big{\\{}\lambda_{1}[x_{1},1],\lambda_{2}[x_{2},1]:\lambda_{1},\lambda_{2}\geq
0\big{\\}}$

with

$\int_{\mathfrak{C}[\mathsf{X}_{1},\mathsf{X}_{2}]}\mathsf{r}_{1}\,\mathrm{d}\bm{\alpha}=r_{1},\quad\int_{\mathfrak{C}[\mathsf{X}_{1},\mathsf{X}_{2}]}\mathsf{r}_{2}\,\mathrm{d}\bm{\alpha}=r_{2}.$

Hence, using Jensen’s inequality and the convexity of
$\mathsf{H}_{x_{1},x_{2}}$, we have

$\displaystyle\mathscr{U}_{\mathsf{H}}(r_{1}\delta_{x_{1}},r_{2}\delta_{x_{2}})$
$\displaystyle=\int_{\mathfrak{C}[\mathsf{X}_{1},\mathsf{X}_{2}]}\mathsf{H}\,\mathrm{d}\bm{\alpha}$
$\displaystyle=\int_{\mathbb{R}_{+}^{2}}\mathsf{H}_{x_{1},x_{2}}\,\mathrm{d}(\mathsf{r}_{1},\mathsf{r}_{2})_{\sharp}\bm{\alpha}$
$\displaystyle\geq\mathsf{H}_{x_{1},x_{2}}\left(\int_{\mathbb{R}_{+}^{2}}(\mathsf{r}_{1},\mathsf{r}_{2})\,\mathrm{d}\bm{\alpha}\right)$
$\displaystyle=\mathsf{H}_{x_{1},x_{2}}(r_{1},r_{2})$
$\displaystyle=\mathsf{H}([x_{1},r_{1}],[x_{2},r_{2}]).\qed$

Thanks to Theorem 3.4, given
$(\mu_{1},\mu_{2})\in\mathcal{M}_{+}(\mathsf{X}_{1})\times\mathcal{M}_{+}(\mathsf{X}_{2})$
such that $\mathscr{U}_{\mathsf{H}}(\mu_{1},\mu_{2})<+\infty$, the set

$\mathfrak{H}^{1}_{o}(\mu_{1},\mu_{2}):=\left\\{\bm{\alpha}\in\mathfrak{H}^{1}(\mu_{1},\mu_{2}):\int_{\mathfrak{C}[\mathsf{X}_{1},\mathsf{X}_{2}]}\mathsf{H}\,\mathrm{d}\bm{\alpha}=\mathscr{U}_{\mathsf{H}}(\mu_{1},\mu_{2})\right\\}$
(3.5)

is not empty.

In the following we give a direct proof that the Unbalanced Optimal Transport
cost $\mathscr{U}_{\mathsf{H}}$ is the lower semicontinuous convex envelope of
the singular cost $\mathscr{S}_{\mathsf{H}}$. This result will also be
obtained in Theorem 3.15 as a simple consequence of the dual characterization
of $\mathscr{U}_{\mathsf{H}}$. However, the following proof highlights the
role played by the discrete $1$-homogeneous marginals, which is not evident in
the proof of Theorem 3.15.

######  Theorem 3.5 (Convex l.s.c. envelope).

Let $\mathsf{H}$ be as in (3.2) and let $\mathscr{S}_{\mathsf{H}}$ and
$\mathscr{U}_{\mathsf{H}}$ be as in Definition 3.1. Then

$\operatorname{\overline{\mathrm{co}}}\left(\mathscr{S}_{\mathsf{H}}\right)=\mathscr{U}_{\mathsf{H}}\text{
in }\mathcal{M}(\mathsf{X}_{1})\times\mathcal{M}(\mathsf{X}_{2}).$

###### Proof.

We only need to prove equality on
$\mathcal{M}_{+}(\mathsf{X}_{1})\times\mathcal{M}_{+}(\mathsf{X}_{2})$, being
both functions equal to $+\infty$ outside it. First of all let us compute
$\operatorname{\mathrm{co}}\left(\mathscr{S}_{\mathsf{H}}\right)$ on
$\operatorname{\mathrm{co}}\left(\Delta_{+}(\mathsf{X}_{1})\times\Delta_{+}(\mathsf{X}_{2})\right)=\operatorname{\mathrm{co}}\left(\Delta_{+}(\mathsf{X}_{1})\right)\times\operatorname{\mathrm{co}}\left(\Delta_{+}(\mathsf{X}_{2})\right)$
(outside this set
$\operatorname{\mathrm{co}}\left(\mathscr{S}_{\mathsf{H}}\right)$ is equal to
$+\infty$). We take
$(\mu_{1},\mu_{2})\in\operatorname{\mathrm{co}}\left(\Delta_{+}(\mathsf{X}_{1})\right)\times\operatorname{\mathrm{co}}\left(\Delta_{+}(\mathsf{X}_{2})\right)$
and we observe that any finite set in
$\Delta_{+}(\mathsf{X}_{1})\times\Delta_{+}(\mathsf{X}_{2})$ is always
contained in the cartesian product of two finite sets
$M_{1}=\\{\mu_{1}^{i}=r_{1}^{i}\delta_{x_{1}^{i}}:i=1,\cdots,I\\}$ and
$M_{2}=\\{\mu_{2}^{j}=r_{1}^{j}\delta_{x_{2}^{j}}:j=1,\cdots,J\\}$, so that

$\displaystyle\begin{split}\operatorname{\mathrm{co}}\left(\mathscr{S}_{\mathsf{H}}\right)(\mu_{1},\mu_{2})={}&\inf\Biggl{\\{}\sum_{ij}\gamma_{ij}\mathscr{S}_{\mathsf{H}}(r^{i}_{1}\delta_{x_{1}^{i}},r_{2}^{j}\delta_{x_{2}^{j}}):\Big{(}r_{1}^{i},x_{1}^{i},r_{2}^{j},x_{2}^{j},\gamma_{ij}\Big{)}_{ij}\in
A(\mu_{1},\mu_{2})\Biggr{\\}},\\\
A(\mu_{1},\mu_{2}):={}&\Biggr{\\{}\Big{(}r_{1}^{i},x_{1}^{i},r_{2}^{j},x_{2}^{j},\gamma_{ij}\Big{)}_{ij}:(\mu_{1},\mu_{2})=\sum_{ij}\gamma_{ij}(r^{i}_{1}\delta_{x_{1}^{i}},r_{2}^{j}\delta_{x_{2}^{j}})\,,\,\sum_{ij}\gamma_{ij}=1,\gamma_{ij}\geq
0\Biggr{\\}}.\end{split}$

In particular it holds that

$\mu_{1}=\sum_{ij}\gamma_{ij}r_{1}^{i}\delta_{{x}_{1}^{i}},\quad\mu_{2}=\sum_{ij}\gamma_{ij}r_{2}^{j}\delta_{x_{2}^{j}}$

for some $r_{1}^{i},r_{2}^{j}\in\mathbb{R}_{+}$ and some
${x}_{1}^{i}\in\mathsf{X}_{1}$, ${x}_{2}^{j}\in\mathsf{X}_{2}$ not necessarily
distinct points. Setting

$\bm{\alpha}:=\sum_{i,j}\gamma_{ij}\delta_{([x_{1}^{i},r_{1}^{i}],[x_{2}^{j},r_{2}^{j}])}\in\mathcal{P}_{\mathrm{f}}(\mathfrak{C}[\mathsf{X}_{1},\mathsf{X}_{2}])$
(3.6)

one immediately checks that

$\bm{\alpha}\in\mathfrak{H}^{1}(\mu_{1},\mu_{2}),\quad\sum_{ij}\gamma_{ij}\mathscr{S}_{\mathsf{H}}(r^{i}_{1}\delta_{x_{1}^{i}},r_{2}^{j}\delta_{x_{2}^{j}})=\int\mathsf{H}\,\,\mathrm{d}\bm{\alpha}.$
(3.7)

Conversely, every
$\bm{\alpha}\in\mathcal{P}_{\mathrm{f}}(\mathfrak{C}[\mathsf{X}_{1},\mathsf{X}_{2}])\cap\mathfrak{H}^{1}(\mu_{1},\mu_{2})$
can be written as in (3.6) with coefficients
$(r_{1}^{i},x_{1}^{i},r_{2}^{j},x_{2}^{j},\gamma_{ij})_{ij}$ in
$A(\mu_{1},\mu_{2})$; the integral identity of (3.7) eventually shows that
$\operatorname{\mathrm{co}}\left(\mathscr{S}_{\mathsf{H}}\right)$ can be
written as

$\operatorname{\mathrm{co}}\left(\mathscr{S}_{\mathsf{H}}\right)(\mu_{1},\mu_{2})=\inf\left\\{\int_{\mathfrak{C}[\mathsf{X}_{1},\mathsf{X}_{2}]}\mathsf{H}\,\mathrm{d}\bm{\alpha}:\bm{\alpha}\in\mathfrak{H}^{1}(\mu_{1},\mu_{2})\cap\mathcal{P}_{\mathrm{f}}(\mathfrak{C}[\mathsf{X}_{1},\mathsf{X}_{2}])\right\\}.$

Reasoning as in the proof of Lemma 2.12, we have that

$\displaystyle\operatorname{\mathrm{co}}\left(\mathscr{S}_{\mathsf{H}}\right)(\mu_{1},\mu_{2})$
$\displaystyle=\inf\left\\{\int_{\mathfrak{C}[\mathsf{X}_{1},\mathsf{X}_{2}]}\mathsf{H}\,\mathrm{d}\bm{\alpha}:\bm{\alpha}\in\mathfrak{H}^{1}(\mu_{1},\mu_{2})\cap\mathcal{P}_{\mathrm{f}}(\mathfrak{C}_{*})\right\\},$

where
$\mathfrak{C}_{*}:=\mathfrak{C}_{R(\mu_{1},\mu_{2})}[\mathsf{X}_{1},\mathsf{X}_{2}]$,
with $R(\mu_{1},\mu_{2})$ as in (2.13). Thus

$\mathscr{U}_{\mathsf{H}}(\mu_{1},\mu_{2})\leq\operatorname{\mathrm{co}}\left(\mathscr{S}_{\mathsf{H}}\right)(\mu_{1},\mu_{2})\quad\text{
for every
}\,(\mu_{1},\mu_{2})\in\operatorname{\mathrm{co}}\left(\Delta_{+}(\mathsf{X}_{1})\right)\times\operatorname{\mathrm{co}}\left(\Delta_{+}(\mathsf{X}_{2})\right).$

Moreover $\mathscr{U}_{\mathsf{H}}$ is lower semicontinuous and convex hence,
by definition of
$\operatorname{\overline{\mathrm{co}}}\left(\mathscr{S}_{\mathsf{H}}\right)$,
it must hold

$\operatorname{\overline{\mathrm{co}}}\left(\mathscr{S}_{\mathsf{H}}\right)(\mu_{1},\mu_{2})\geq\mathscr{U}_{\mathsf{H}}(\mu_{1},\mu_{2})\quad\text{
for every
}\,(\mu_{1},\mu_{2})\in\mathcal{M}_{+}(\mathsf{X}_{1})\times\mathcal{M}_{+}(\mathsf{X}_{1}).$

Then, in order to prove equality, we only need to prove the other inequality.
To do so, fixed
$(\mu_{1},\mu_{2})\in\mathcal{M}_{+}(\mathsf{X}_{1})\times\mathcal{M}_{+}(\mathsf{X}_{2})$,
we prove that there exists a net
$\\{(\mu_{1}^{\eta},\mu_{2}^{\eta})\\}_{\eta\in\mathbb{E}}\subset\operatorname{\mathrm{co}}\left(\Delta_{+}(\mathsf{X}_{1})\right)\times\operatorname{\mathrm{co}}\left(\Delta_{+}(\mathsf{X}_{2})\right)$
s.t. $\lim_{\eta}(\mu_{1}^{\eta},\mu_{2}^{\eta})=(\mu_{1},\mu_{2})$ and a net

$\\{\bm{\gamma}_{\eta}\\}_{\eta\in\mathbb{E}}\subset\operatorname{\mathcal{M}_{+,f}}(\mathfrak{C}_{*})\cap\mathcal{P}(\mathfrak{C}_{*})$

s.t. $\bm{\gamma}_{\eta}\in\mathfrak{H}^{1}(\mu_{1}^{\eta},\mu_{2}^{\eta})$
for every $\eta\in\mathbb{E}$ satisfying

$\lim_{\eta\in\mathbb{E}}\int_{\mathfrak{C}[\mathsf{X}_{1},\mathsf{X}_{2}]}\mathsf{H}\,\mathrm{d}\bm{\gamma}_{\eta}=\int_{\mathfrak{C}[\mathsf{X}_{1},\mathsf{X}_{2}]}\mathsf{H}\,\mathrm{d}\bm{\alpha}^{*}$

where
$\bm{\alpha}^{*}\in\mathfrak{H}^{1}_{o}(\mu_{1},\mu_{2})\cap\mathcal{P}(\mathfrak{C}_{*})$
(see (3.5)). To do so, we use Lemma 2.3 with $\mathsf{X}:=\mathfrak{C}_{*}$,
$f:=\mathsf{H}$, $\alpha:=\bm{\alpha}^{*}$ and we find
$\\{\bm{\gamma}_{\eta}\\}_{\eta\in\mathbb{E}}\subset\operatorname{\mathcal{M}_{+,f}}(\mathfrak{C}_{*})\cap\mathcal{P}(\mathfrak{C}_{*})$
s.t.

$\lim_{\eta\in\mathbb{E}}\bm{\gamma}_{\eta}=\bm{\alpha}^{*},\quad\lim_{\eta\in\mathbb{E}}\int_{\mathfrak{C}_{*}}\mathsf{H}\,\mathrm{d}\bm{\gamma}_{\eta}=\int_{\mathfrak{C}_{*}}\mathsf{H}\,\mathrm{d}\bm{\alpha}^{*}.$

Finally we can define

$\mu_{1}^{\eta}:=\mathfrak{h}_{1}^{1}(\bm{\gamma}_{\eta}),\quad\mu_{2}^{\eta}:=\mathfrak{h}_{2}^{1}(\bm{\gamma}_{\eta})\quad\text{
for every }\,\eta\in\mathbb{E}.$

Obviously
$\bm{\gamma}_{\eta}\in\mathfrak{H}^{1}(\mu_{1}^{\eta},\mu_{2}^{\eta})$ and
$\mu_{i}=\lim_{\eta\in\mathbb{E}}\mu_{i}^{\eta}$, indeed if
$\varphi_{i}\in\mathrm{C}_{b}(\mathsf{X}_{i})$, then

$\displaystyle\lim_{\eta\in\mathbb{E}}\int_{\mathsf{X}_{i}}\varphi_{i}\,\mathrm{d}\mu_{i}^{\eta}$
$\displaystyle=\lim_{\eta\in\mathbb{E}}\int_{\mathsf{X}_{i}}\varphi_{i}\,\mathrm{d}\mathfrak{h}^{1}_{i}(\bm{\gamma}_{\eta})=\lim_{\eta\in\mathbb{E}}\int_{\mathfrak{C}[\mathsf{X}_{1},\mathsf{X}_{2}]}(\varphi\circ\mathsf{x}_{i})\mathsf{r}_{i}\,\mathrm{d}\bm{\gamma}_{\eta}$
$\displaystyle=\lim_{\eta\in\mathbb{E}}\int_{\mathfrak{C}_{*}}(\varphi_{i}\circ\mathsf{x}_{i})\mathsf{r}_{i}\,\mathrm{d}\bm{\gamma}_{\eta}=\int_{\mathfrak{C}_{*}}(\varphi_{i}\circ\mathsf{x}_{i})\mathsf{r}_{i}\,\mathrm{d}\bm{\alpha}^{*}$
$\displaystyle=\int_{\mathfrak{C}[\mathsf{X}_{1},\mathsf{X}_{2}]}(\varphi_{i}\circ\mathsf{x}_{i})\mathsf{r}_{i}\,\mathrm{d}\bm{\alpha}^{*}=\int_{\mathsf{X}_{i}}\varphi_{i}\,\mathrm{d}\mathfrak{h}^{1}_{i}(\bm{\alpha}^{*})$
$\displaystyle=\int_{\mathsf{X}_{i}}\varphi_{i}\,\mathrm{d}\mu_{i},$

where we have used that
$(\varphi\circ\mathsf{x}_{i})\mathsf{r}_{i}\in\mathrm{C}_{b}(\mathfrak{C}_{*})$
and the convergence of $\bm{\gamma}_{\eta}$ to $\bm{\alpha}^{*}$ in
$\mathcal{P}(\mathfrak{C}_{*})$. Notice that, in general, it is not true that
$(\varphi\circ\mathsf{x}_{i})\mathsf{r}_{i}\in\mathrm{C}_{b}(\mathfrak{C}[\mathsf{X}_{1},\mathsf{X}_{2}])$.
Finally

$\displaystyle\operatorname{\overline{\mathrm{co}}}\left(\mathscr{S}_{\mathsf{H}}\right)(\mu_{1},\mu_{2})$
$\displaystyle=\inf\biggl{\\{}\liminf_{\lambda}\operatorname{\mathrm{co}}\left(\mathscr{S}_{\mathsf{H}}\right)(\mu_{1}^{\lambda},\mu_{2}^{\lambda}):\\{(\mu_{1}^{\lambda},\mu_{2}^{\lambda})\\}_{\lambda\in\mathbb{L}}\subset\operatorname{\mathrm{co}}\left(\Delta_{+}(\mathsf{X}_{1})\times\Delta_{+}(\mathsf{X}_{2})\right),$
$\displaystyle\quad\quad\quad(\mu_{1},\mu_{2})=\lim_{\lambda}(\mu_{1}^{\lambda},\mu_{2}^{\lambda})\biggr{\\}}$
$\displaystyle\leq\liminf_{\eta}\operatorname{\mathrm{co}}\left(\mathscr{S}_{\mathsf{H}}\right)(\mu_{1}^{\eta},\mu_{2}^{\eta})\leq\liminf_{\eta}\int_{\mathfrak{C}[\mathsf{X}_{1},\mathsf{X}_{2}]}\mathsf{H}\,\mathrm{d}\bm{\gamma}_{\eta}$
$\displaystyle=\int_{\mathfrak{C}[\mathsf{X}_{1},\mathsf{X}_{2}]}\mathsf{H}\,\mathrm{d}\bm{\alpha}^{*}=\mathscr{U}_{\mathsf{H}}(\mu_{1},\mu_{2}).\qed$

Theorem 3.5 immediately yields the following useful property.

######  Theorem 3.6 (Sublinearity of $\mathscr{U}_{\mathsf{H}}$).

The functional $\mathscr{U}_{\mathsf{H}}$ is sublinear (i.e. convex and
positively $1$-homogeneous) in
$\mathcal{M}_{+}(\mathsf{X}_{1})\times\mathcal{M}_{+}(\mathsf{X}_{2})$: for
every
$\mu_{i}^{\prime},\mu_{i}^{\prime\prime}\in\mathcal{M}_{+}(\mathsf{X}_{i})$
and every $\lambda^{\prime},\lambda^{\prime\prime}\in[0,+\infty)$

$\mathscr{U}_{\mathsf{H}}(\lambda^{\prime}\mu_{1}^{\prime}+\lambda^{\prime\prime}\mu_{1}^{\prime\prime},\lambda^{\prime}\mu_{2}^{\prime}+\lambda^{\prime\prime}\mu_{2}^{\prime\prime})\leq\lambda^{\prime}\mathscr{U}_{\mathsf{H}}(\mu_{1}^{\prime},\mu_{2}^{\prime})+\lambda^{\prime\prime}\mathscr{U}_{\mathsf{H}}(\mu_{1}^{\prime\prime},\mu_{2}^{\prime\prime}).$
(3.8)

###  3.2. Disintegration, barycentric projection and decomposition of
homogeneous couplings

Every plan
$\bm{\alpha}\in\mathcal{M}_{+}(\mathfrak{C}[\mathsf{X}_{1},\mathsf{X}_{2}])$
can be expressed as a superposition of a Borel family of probability measures
$(\alpha_{\bm{x}})_{\bm{x}}$ in $\mathbb{R}^{2}_{+}$,
$\bm{x}=(x_{1},x_{2})\in\mathsf{X}_{1,o}\times\mathsf{X}_{2,o}.$ It is
sufficient to consider the map
$\bm{\mathsf{x}}:=(\mathsf{x}_{1},\mathsf{x}_{2}):\mathfrak{C}[\mathsf{X}_{1},\mathsf{X}_{2}]\to\mathsf{X}_{1,o}\times\mathsf{X}_{2,o}$,
where
$\mathsf{x}_{i}:\mathfrak{C}[\mathsf{X}_{1},\mathsf{X}_{2}]\to\mathsf{X}_{i,o}$
are defined in Section 2.1, and the plan
$\bm{\gamma}:=(\mathsf{x}_{1},\mathsf{x}_{2})_{\sharp}\bm{\alpha}\in\mathcal{M}_{+}(\mathsf{X}_{1,o}\times\mathsf{X}_{2,o})$.
The disintegration of $\bm{\alpha}$ w.r.t. $\gamma$ yields the Borel family
$(\alpha_{\bm{x}})_{\bm{x}}\subset\mathcal{P}(\mathbb{R}^{2}_{+})$ satisfying

$\bm{\alpha}=\int_{\mathsf{X}_{1,o}\times\mathsf{X}_{2,o}}\alpha_{x_{1},x_{2}}\,\,\mathrm{d}\bm{\gamma}(x_{1},x_{2}),\quad\bm{\gamma}=\bm{\mathsf{x}}_{\sharp}\bm{\alpha}.$
(3.9)

Notice that if $\bm{\alpha}(\\{{\bm{{\mathfrak{o}}}}\\})=0$ then $\bm{\gamma}$
is concentrated on
$\mathsf{X}_{1,o}\times\mathsf{X}_{2,o}\setminus\\{\bm{o}\\}.$ If moreover
$\bm{\alpha}\in\mathcal{M}_{+}^{1}(\mathfrak{C}[\mathsf{X}_{1},\mathsf{X}_{2}])$
then we can define

$\bm{\varrho}:=(\varrho_{1},\varrho_{2}),\quad\varrho_{i}(x_{1},x_{2}):=\int_{\mathbb{R}^{2}_{+}}r_{i}\,\,\mathrm{d}\alpha_{x_{1},x_{2}}(r_{1},r_{2}),\quad\varrho_{i}\in
L^{1}(\mathsf{X}_{1,o}\times\mathsf{X}_{2,o},\bm{\gamma}),\,\,\varrho_{i}\geq
0.$ (3.10)

######  Definition 3.7 (Barycentric projection and reduced couplings).

The barycentric projection of a nontrivial plan
$\bm{\alpha}\in\mathcal{M}_{+}^{1}(\mathfrak{C}[\mathsf{X}_{1},\mathsf{X}_{2}])$
is the plan
$\bm{\alpha}_{b}\in\mathcal{M}_{+}^{1}(\mathfrak{C}[\mathsf{X}_{1},\mathsf{X}_{2}])$
defined by

$\displaystyle\bm{\alpha}_{b}:={}\mathsf{T}[\bm{\varrho}]_{\sharp}\bm{\gamma},\quad\text{where}\quad\mathsf{T}[\bm{\varrho}](\bm{x}):={}\big{(}[x_{1},\varrho_{1}(\bm{x})],[x_{2},\varrho_{2}(\bm{x})]\big{)},\quad\bm{x}\in\mathsf{X}_{1,o}\times\mathsf{X}_{2,o}.$
(3.11)

When $\bm{\alpha}(\mathfrak{C}[\mathsf{X}_{1},\mathsf{X}_{2}])=0$ (i.e.
$\bm{\alpha}$ is the null measure) we set $\bm{\alpha}_{b}:=\bm{\alpha}$.
We say that $\bm{\alpha}$ is a reduced plan if $\bm{\alpha}=\bm{\alpha}_{b}$
or, equivalently, if $\alpha_{\bm{x}}$ is a Dirac mass
$\delta_{\bm{\varrho}(\bm{x})}$ for $\bm{\gamma}$-a.e.
$\bm{x}\in\mathsf{X}_{1,o}\times\mathsf{X}_{2,o}.$

The interest in the barycentric projection and in reduced couplings is
justified by the following result.

######  Proposition 3.8 (Reduced homogeneous couplings and minimizers).

If $\mu_{i}\in\mathcal{M}_{+}(\mathsf{X}_{i})$,
$\bm{\alpha}\in\mathfrak{H}^{1}(\mu_{1},\mu_{2})$ then also
$\bm{\alpha}_{b}\in\mathfrak{H}^{1}(\mu_{1},\mu_{2}).$ If $\mathsf{H}$
satisfies (3.2) and it is radially convex then

$\int_{\mathfrak{C}[\mathsf{X}_{1},\mathsf{X}_{2}]}\mathsf{H}\,\,\mathrm{d}\bm{\alpha}_{b}\leq\int_{\mathfrak{C}[\mathsf{X}_{1},\mathsf{X}_{2}]}\mathsf{H}\,\,\mathrm{d}\bm{\alpha}.$
(3.12)

In particular, if $\mathscr{U}_{\mathsf{H}}(\mu_{1},\mu_{2})<+\infty$, then
the minimum of the Unbalanced Optimal Transport problem 3.1 is attained at a
reduced coupling.

###### Proof.

We can assume that $\bm{\alpha}$ is nontrivial. Let us first check that
$\bm{\alpha}_{b}$ preserves homogeneous marginals. For every
$\zeta_{i}\in\mathrm{C}_{b}(\mathsf{X}_{i})$, we have

$\displaystyle\int(\zeta_{i}\circ\mathsf{x}_{i})\mathsf{r}_{i}\,\,\mathrm{d}\bm{\alpha}_{b}$
$\displaystyle=\int\zeta_{i}(x_{i})\varrho_{i}(x_{1},x_{2})\,\,\mathrm{d}\bm{\gamma}(x_{1},x_{2})=\int\zeta_{i}(x_{i})\Big{(}\int
r_{i}\,\,\mathrm{d}\alpha_{x_{1},x_{2}}(r_{1},r_{2})\Big{)}\,\,\mathrm{d}\bm{\gamma}(x_{1},x_{2})$
$\displaystyle=\int(\zeta_{i}\circ\mathsf{x}_{i})\mathsf{r}_{i}\,\,\mathrm{d}\bm{\alpha}=\int\zeta_{i}\,\,\mathrm{d}\mu_{i}.$

By radial convexity, using Jensen’s inequality, we have

$\displaystyle\int_{\mathfrak{C}[\mathsf{X}_{1},\mathsf{X}_{2}]}\mathsf{H}\,\mathrm{d}\bm{\alpha}$
$\displaystyle=\int_{\mathsf{X}_{1,o}\times\mathsf{X}_{2,o}}\Big{(}\int_{\mathbb{R}_{+}^{2}}\mathsf{H}\,\mathrm{d}\alpha_{x_{1},x_{2}}(r_{1},r_{2})\Big{)}\,\mathrm{d}\gamma(x_{1},x_{2})$
$\displaystyle\geq\int_{\mathsf{X}_{1,o}\times\mathsf{X}_{2,o}}\mathsf{H}([x_{1},\varrho_{1}(x_{1},x_{2})],[x_{2},\varrho_{2}(x_{1},x_{2})])\,\mathrm{d}\gamma(x_{1},x_{2})=\int_{\mathfrak{C}[\mathsf{X}_{1},\mathsf{X}_{2}]}\mathsf{H}\,\mathrm{d}\bm{\alpha}_{b}.\qed$

We can now show that it is possible to improve the previous representation by
decomposing the measures $\mu_{i}$ in two parts, corresponding to complete or
partial distruction/creation of mass. We introduce the partition
$\mathfrak{C}^{\prime},\mathfrak{C}^{\prime\prime}_{1},\mathfrak{C}^{\prime\prime}_{2}$
of $\mathfrak{C}_{\bm{{\mathfrak{o}}}}[\mathsf{X}_{1},\mathsf{X}_{2}]$, where

$\mathfrak{C}^{\prime}:=(\mathfrak{C}[\mathsf{X}_{1}]\setminus\\{{\mathfrak{o}}_{1}\\})\times(\mathfrak{C}[\mathsf{X}_{2}]\setminus\\{{\mathfrak{o}}_{2}\\}),\quad\mathfrak{C}_{1}^{\prime\prime}:=(\mathfrak{C}[\mathsf{X}_{1}]\setminus\\{{\mathfrak{o}}_{1}\\})\times\\{{\mathfrak{o}}_{2}\\},\quad\mathfrak{C}_{2}^{\prime\prime}:=\\{{\mathfrak{o}}_{1}\\}\times(\mathfrak{C}[\mathsf{X}_{2}]\setminus\\{{\mathfrak{o}}_{2}\\}).$

Every homogeneous plan
$\bm{\alpha}\in\mathcal{M}_{+}^{1}(\mathfrak{C}[\mathsf{X}_{1},\mathsf{X}_{2}])$
not charging ${\bm{{\mathfrak{o}}}}=({\mathfrak{o}}_{1},{\mathfrak{o}}_{2})$
can be decomposed into the corresponding sum
$\bm{\alpha}=\bm{\alpha}^{\prime}+\bm{\alpha}^{\prime\prime}_{1}+\bm{\alpha}^{\prime\prime}_{2}$,
where $\bm{\alpha}^{\prime}=\bm{\alpha}\mathbin{\vrule
height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule
height=0.55974pt,depth=0.0pt,width=5.59721pt}\mathfrak{C}^{\prime}$ and
$\bm{\alpha}_{i}^{\prime\prime}:=\bm{\alpha}\mathbin{\vrule
height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule
height=0.55974pt,depth=0.0pt,width=5.59721pt}\mathfrak{C}_{i}^{\prime\prime}.$
Correspondingly we can write
$\bm{\gamma}=\bm{\gamma}^{\prime}+\bm{\gamma}_{1}^{\prime\prime}+\bm{\gamma}_{2}^{\prime\prime}$

$\bm{\gamma}^{\prime}:=\bm{\mathsf{x}}_{\sharp}\bm{\alpha}^{\prime}\in\mathcal{M}_{+}(\mathsf{X}_{1}\times\mathsf{X}_{2}),\quad\bm{\gamma}_{1}^{\prime\prime}=\bm{\mathsf{x}}_{\sharp}\bm{\alpha}_{1}^{\prime\prime}\in\mathcal{M}_{+}(\mathsf{X}_{1}\times\\{o_{2}\\}),\quad\bm{\gamma}_{2}^{\prime\prime}=\bm{\mathsf{x}}_{\sharp}\bm{\alpha}_{2}^{\prime\prime}\in\mathcal{M}_{+}(\\{o_{1}\\}\times\mathsf{X}_{2}).$
(3.13)

######  Theorem 3.9 (Distinguished optimal couplings).

Let us suppose that $\mathsf{H}$ satisfies (3.2) and it is radially convex and
let $\mu_{i}\in\mathcal{M}_{+}(\mathsf{X}_{i})$ non trivial such that
$\mathscr{U}_{\mathsf{H}}(\mu_{1},\mu_{1})<+\infty.$

There exist Borel partitions $\\{S_{i}^{\prime},S_{i}^{\prime\prime}\\}$ of
$\mathsf{X}_{i}$ and an optimal homogeneous and reduced coupling
$\bm{\alpha}\in\mathfrak{H}^{1}(\mu_{1},\mu_{2})$ not charging
${\bm{{\mathfrak{o}}}}$ such that setting

$\mu_{i}^{\prime}:=\mu_{i}\mathbin{\vrule
height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule
height=0.55974pt,depth=0.0pt,width=5.59721pt}S_{i}^{\prime},\quad\mu_{i}(S_{i}^{\prime})>0,\quad\mu_{i}^{\prime\prime}:=\mu_{i}\mathbin{\vrule
height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule
height=0.55974pt,depth=0.0pt,width=5.59721pt}S_{i}^{\prime\prime},$ (3.14)

and keeping the above notation for
$\varrho_{i},\bm{\alpha}^{\prime},\bm{\alpha}_{i}^{\prime\prime},\bm{\gamma},\bm{\gamma}^{\prime},\bm{\gamma}_{i}^{\prime\prime}$
we have

$\displaystyle\bm{\alpha}_{1}^{\prime\prime}$
$\displaystyle=\mathfrak{p}_{\sharp}(\mu_{1}^{\prime\prime}\otimes\delta_{1})\otimes\delta_{{\mathfrak{o}}_{2}},$
$\displaystyle\bm{\alpha}_{2}^{\prime\prime}$
$\displaystyle=\delta_{{\mathfrak{o}}_{1}}\times\mathfrak{p}_{\sharp}(\mu_{2}^{\prime\prime}\otimes\delta_{1}),$
(3.15) $\displaystyle\bm{\gamma}_{1}^{\prime\prime}$
$\displaystyle=\mu_{1}^{\prime\prime}\otimes\delta_{o_{2}},$
$\displaystyle\bm{\gamma}_{2}^{\prime\prime}$
$\displaystyle=\delta_{o_{1}}\otimes\mu_{2}^{\prime\prime},$ (3.16)
$\mathfrak{h}^{1}_{i}(\bm{\alpha}^{\prime})=\mu_{i}^{\prime},\quad\bm{\gamma}^{\prime}=\bm{\gamma}\mathbin{\vrule
height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule
height=0.55974pt,depth=0.0pt,width=5.59721pt}(S_{1}^{\prime}\times
S_{2}^{\prime}),\quad$ (3.17)
$\displaystyle\bm{\alpha}={}\mathsf{T}[\bm{\varrho}]_{\sharp}\bm{\gamma},\quad\bm{\alpha}^{\prime}={}\mathsf{T}[\bm{\varrho}]_{\sharp}\bm{\gamma}^{\prime},\quad\mathsf{T}[\bm{\varrho}](x_{1},x_{2}):={}\big{(}[x_{1},\varrho_{1}(x_{1},x_{2})],[x_{2},\varrho_{2}(x_{1},x_{2})]\big{)}$
(3.18)

and

$\varrho_{i}>0\text{ on $S_{1}^{\prime}\times
S_{2}^{\prime}$},\quad\varrho_{1}=0\text{ on
$S_{1}^{\prime\prime}\times\mathsf{X}_{2}$},\quad\varrho_{2}=0\text{ on
$\mathsf{X}_{1}\times S_{2}^{\prime\prime}$},$ (3.19)
$\varrho_{1}(x_{1},o_{2})=1\text{ on
$S_{1}^{\prime\prime}$},\quad\varrho_{2}(o_{1},x_{2})=1\text{ on
$S_{2}^{\prime\prime}$}\quad\text{$\bm{\gamma}$-a.e.}.$ (3.20)

###### Proof.

Let
$\tilde{\bm{\alpha}}=\tilde{\bm{\alpha}}^{\prime}+\tilde{\bm{\alpha}}_{1}^{\prime\prime}+\tilde{\bm{\alpha}}_{2}^{\prime\prime}$
be an optimal reduced coupling in $\mathfrak{H}^{1}(\mu_{1},\mu_{2})$ not
charging ${\bm{{\mathfrak{o}}}}$.

In the degenerate case when $\tilde{\bm{\alpha}}^{\prime}$ vanishes, then
setting $\mu_{i}=m_{i}\nu_{i}$, $\nu_{i}\in\mathcal{P}(\mathsf{X}_{i}),$
$m_{i}=\mu_{i}(\mathsf{X}_{i})>0,$ we have

$\mathscr{U}_{\mathsf{H}}(\mu_{1},\mu_{2})=\int_{\mathsf{X}_{1}}\mathsf{H}([x_{1},m_{1}],{\mathfrak{o}}_{2})\,\,\mathrm{d}\nu_{1}+\int_{\mathsf{X}_{2}}\mathsf{H}({\mathfrak{o}}_{1},[x_{2},m_{2}])\,\,\mathrm{d}\nu_{2}\geq\int_{\mathsf{X}_{1}\times\mathsf{X}_{2}}\mathsf{H}([x_{1},m_{1}],[x_{2},m_{2}])\,\,\mathrm{d}(\nu_{1}\otimes\nu_{2})$

so that the reduced coupling
$\bm{\alpha}=([x_{1},m_{1}],[x_{2},m_{2}])_{\sharp}(\nu_{1}\otimes\nu_{2})$
satisfies all the assumptions with $S_{i}^{\prime}=\mathsf{X}_{i}.$

We can therefore suppose that $\bm{\alpha}^{\prime}$ is nontrivial and we set

$\tilde{\mu}_{i}^{\prime}:=\mathfrak{h}^{1}_{i}(\tilde{\bm{\alpha}}^{\prime}),\quad\tilde{\mu}_{i}^{\prime\prime}:=\mathfrak{h}^{1}_{i}(\tilde{\bm{\alpha}}_{i}^{\prime\prime}),\quad\tilde{\bm{\gamma}}:=\bm{\mathsf{x}}_{\sharp}\tilde{\bm{\alpha}},\quad\tilde{\bm{\gamma}}^{\prime}:=\bm{\mathsf{x}}_{\sharp}\tilde{\bm{\alpha}}^{\prime}:=\tilde{\bm{\gamma}}\mathbin{\vrule
height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule
height=0.55974pt,depth=0.0pt,width=5.59721pt}(\mathsf{X}_{1}\times\mathsf{X}_{2}),$

and we can write
$\tilde{\bm{\alpha}}=\mathsf{T}[\tilde{\bm{\varrho}}]_{\sharp}\tilde{\bm{\gamma}}$
as in (3.11).

The optimal cost $\mathscr{U}_{\mathsf{H}}(\mu_{1},\mu_{2})$ can be decomposed
as

$\int\mathsf{H}\,\,\mathrm{d}\tilde{\bm{\alpha}}=\int_{\mathsf{X}_{1}\times\mathsf{X}_{2}}\mathsf{H}([x_{1},\tilde{\varrho}_{1}],[x_{2},\tilde{\varrho}_{2}])\,\,\mathrm{d}\tilde{\bm{\gamma}}^{\prime}+\int_{\mathsf{X}_{1}}\mathsf{H}([x_{1},1],{\mathfrak{o}}_{2})\,\,\mathrm{d}\mu_{1}^{\prime\prime}+\int_{\mathsf{X}_{2}}\mathsf{H}({\mathfrak{o}}_{2},[x_{2},1])\,\,\mathrm{d}\mu_{2}^{\prime\prime}.$
(3.21)

Since
$\mathfrak{h}^{1}_{1}(\tilde{\bm{\alpha}}_{2}^{\prime\prime})=\bm{0}_{\mathsf{X}_{1}}$
and
$\mathfrak{h}^{1}_{2}(\tilde{\bm{\alpha}}_{1}^{\prime\prime})=\bm{0}_{\mathsf{X}_{2}}$
we have $\mu_{i}=\tilde{\mu}_{i}^{\prime}+\tilde{\mu}_{i}^{\prime\prime}$: we
call $\theta_{i}^{\prime},\theta_{i}^{\prime\prime}$ Borel representatives of
the Lebesgue densities of $\mu_{i}^{\prime},\mu_{i}^{\prime\prime}$ w.r.t.
$\mu_{i}$ and we call

$S_{i}^{\prime\prime}:=\Big{\\{}x_{i}\in\mathsf{X}_{i}:\theta_{i}^{\prime\prime}(x_{i})=1\Big{\\}},\quad
S_{i}^{\prime}:=\mathsf{X}_{i}\setminus S_{i}^{\prime\prime}.$

Since $\theta_{i}^{\prime}+\theta_{i}^{\prime\prime}=1$ $\mu_{i}$-a.e. in
$\mathsf{X}_{i}$, we clearly have $\theta_{i}^{\prime}=0$ $\mu_{i}$-a.e. in
$S_{i}^{\prime\prime}$ and $\theta_{i}^{\prime}>0$ $\mu_{i}$-a.e. in
$S_{i}^{\prime}$. We also have
$\mu_{1}^{\prime}(S_{1}^{\prime\prime})=\mathfrak{h}^{1}_{1}(\tilde{\bm{\alpha}}^{\prime})(S_{1}^{\prime\prime})=0$
and since
$\mathfrak{h}^{1}_{1}(\tilde{\bm{\alpha}}^{\prime})=(\mathsf{x}_{1})_{\sharp}(\tilde{\varrho}_{1}\tilde{\bm{\gamma}}^{\prime})$
we deduce that $\tilde{\varrho}_{1}=0$ $\tilde{\bm{\gamma}}^{\prime}$-a.e. in
$S_{1}^{\prime\prime}\times\mathsf{X}_{2}.$ With a similar argument, we deduce
that $\tilde{\varrho}_{2}=0$ $\tilde{\bm{\gamma}}^{\prime}$-a.e. in
$\mathsf{X}_{1}\times S_{2}^{\prime\prime}.$

We define $\mu_{i}^{\prime},\mu_{i}^{\prime\prime}$ according to (3.14).
Notice that
$\mu_{i}^{\prime}=(\theta_{i}^{\prime})^{-1}\tilde{\mu}_{i}^{\prime}\ll\tilde{\mu}_{i}^{\prime}$,
whereas $\mu_{i}^{\prime\prime}=\tilde{\mu}_{i}^{\prime\prime}\mathbin{\vrule
height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule
height=0.55974pt,depth=0.0pt,width=5.59721pt}S_{i}^{\prime\prime}.$ The
quotients $q_{i}(x_{i}):=\theta_{i}^{\prime\prime}/\theta_{i}^{\prime}$ are
well defined $\mu_{i}$-a.e. in $S_{i}^{\prime}$ and we have

$\mu_{i}^{\prime}=(1+q_{i})\tilde{\mu}_{i}^{\prime},\quad\tilde{\mu}_{i}^{\prime\prime}=\mu_{i}^{\prime\prime}+\theta_{i}^{\prime\prime}\mu_{i}^{\prime}=\mu_{i}^{\prime\prime}+q_{i}\tilde{\mu}_{i}^{\prime}.$

We set

$\varrho_{1}:=\begin{cases}\tilde{\varrho}_{1}(1+q_{1})&\text{in
}S_{1}^{\prime}\times\mathsf{X}_{2},\\\ 0&\text{in
}S_{1}^{\prime\prime}\times\mathsf{X}_{2},\end{cases}\qquad\varrho_{2}:=\begin{cases}\tilde{\varrho}_{2}(1+q_{2})&\text{in
}\mathsf{X}_{1}\times S_{2}^{\prime},\\\ 0&\text{in }\mathsf{X}_{1}\times
S_{2}^{\prime\prime},\end{cases}$

and we define $\bm{\alpha}_{i}^{\prime\prime}$ as in (3.15). Finally

$\bm{\gamma}^{\prime}:=\tilde{\bm{\gamma}}^{\prime}\mathbin{\vrule
height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule
height=0.55974pt,depth=0.0pt,width=5.59721pt}(S_{1}^{\prime}\times
S_{2}^{\prime}),\quad\bm{\alpha}^{\prime}:=\mathsf{T}[\bm{\varrho}]_{\sharp}\tilde{\bm{\gamma}}^{\prime}=\mathsf{T}[\bm{\varrho}]_{\sharp}\bm{\gamma}^{\prime},\quad\bm{\alpha}:=\bm{\alpha}^{\prime}+\bm{\alpha}_{1}^{\prime\prime}+\bm{\alpha}_{2}^{\prime\prime}.$

Let us check (3.17); since $\bm{\gamma}=\bm{\mathsf{x}}_{\sharp}\bm{\alpha}$,
the equality $\bm{\gamma}^{\prime}=\bm{\gamma}\mathbin{\vrule
height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule
height=0.55974pt,depth=0.0pt,width=5.59721pt}(S_{1}\times S_{2}^{\prime})$ is
immediate, so that we have to prove that
$\mathfrak{h}^{1}_{i}(\bm{\alpha}^{\prime})=\mu_{i}^{\prime}$. We just
consider the case $i=1$ since the calculations in the case $i=2$ are
completely analogous. For every test function
$\zeta\in\mathrm{C}_{b}(\mathsf{X}_{1})$ we get

$\displaystyle\int(\zeta\circ\mathsf{x}_{1})\mathsf{r}_{1}\,\,\mathrm{d}\bm{\alpha}^{\prime}$
$\displaystyle=\int_{\mathsf{X}_{1}\times\mathsf{X}_{2}}\zeta(x_{1})\varrho_{1}(x_{1},x_{2})\,\,\mathrm{d}\tilde{\bm{\gamma}}^{\prime}(x_{1},x_{2})$
$\displaystyle=\int_{S_{1}^{\prime}\times\mathsf{X}_{2}}\zeta(x_{1})\tilde{\varrho}_{1}(x_{1},x_{2})\Big{(}1+q_{1}(x_{1})\Big{)}\,\,\mathrm{d}\tilde{\bm{\gamma}}^{\prime}(x_{1},x_{2})$
$\displaystyle=\int_{S_{1}^{\prime}}\zeta(x_{1})(1+q_{1}(x_{1}))\,\,\mathrm{d}\tilde{\mu}_{1}(x_{1})=\int_{\mathsf{X}_{1}}\zeta\,\,\mathrm{d}\mu_{1}^{\prime}.$

Let us now compute the $\mathsf{H}$-cost of $\bm{\alpha}$:

$\displaystyle\int\mathsf{H}\,\,\mathrm{d}\bm{\alpha}^{\prime}$
$\displaystyle=\int\mathsf{H}([x_{1},\tilde{\varrho}_{1}+q_{1}\tilde{\varrho}_{1}],[x_{2},\tilde{\varrho}_{2}+q_{2}\tilde{\varrho}_{2}])\,\,\mathrm{d}\tilde{\bm{\gamma}}^{\prime}$
$\displaystyle\leq\int\mathsf{H}([x_{1},\tilde{\varrho}_{1}],[x_{2},\tilde{\varrho}_{2}])\,\,\mathrm{d}\tilde{\bm{\gamma}}^{\prime}+\int
q_{1}\tilde{\varrho}_{1}\mathsf{H}([x_{1},1],{\mathfrak{o}}_{2})\,\,\mathrm{d}\tilde{\bm{\gamma}}^{\prime}+\int
q_{2}\tilde{\varrho}_{2}\mathsf{H}({\mathfrak{o}}_{1},[x_{2},1])\,\,\mathrm{d}\tilde{\bm{\gamma}}^{\prime}$
$\displaystyle=\int\mathsf{H}\,\,\mathrm{d}\tilde{\bm{\alpha}}^{\prime}+\int
q_{1}\mathsf{H}([x_{1},1],{\mathfrak{o}}_{2})\,\,\mathrm{d}\tilde{\mu}_{1}^{\prime}+\int
q_{2}\mathsf{H}({\mathfrak{o}}_{1},[x_{2},1])\,\,\mathrm{d}\tilde{\mu}_{2}^{\prime},$
$\displaystyle\int\mathsf{H}\,\,\mathrm{d}\bm{\alpha}$
$\displaystyle=\int\mathsf{H}\,\,\mathrm{d}\bm{\alpha}^{\prime}+\int\mathsf{H}\,\,\mathrm{d}\bm{\alpha}_{1}^{\prime\prime}+\int\mathsf{H}\,\,\mathrm{d}\bm{\alpha}_{2}^{\prime\prime}$
$\displaystyle=\int\mathsf{H}\,\,\mathrm{d}\bm{\alpha}^{\prime}+\int\mathsf{H}([x_{1},1],{\mathfrak{o}}_{2})\,\,\mathrm{d}\mu_{1}^{\prime\prime}+\int\mathsf{H}({\mathfrak{o}}_{1},[x_{2},1])\,\,\mathrm{d}\mu_{2}^{\prime\prime}$
$\displaystyle\leq\int\mathsf{H}\,\,\mathrm{d}\tilde{\bm{\alpha}}^{\prime}+\int\mathsf{H}([x_{1},1],{\mathfrak{o}}_{2})\,\,\mathrm{d}(q_{1}\tilde{\mu}_{1}^{\prime}+\mu_{1}^{\prime\prime})\int\mathsf{H}({\mathfrak{o}}_{1},[x_{2},1])\,\,\mathrm{d}(q_{2}\tilde{\mu}_{2}^{\prime}+\mu_{2}^{\prime\prime})$
$\displaystyle=\int\mathsf{H}\,\,\mathrm{d}\tilde{\bm{\alpha}}^{\prime}+\int\mathsf{H}([x_{1},1],{\mathfrak{o}}_{2})\,\,\mathrm{d}\tilde{\mu}_{1}^{\prime\prime}+\int\mathsf{H}({\mathfrak{o}}_{1},[x_{2},1])\,\,\mathrm{d}\tilde{\mu}_{2}^{\prime\prime}=\int\mathsf{H}\,\,\mathrm{d}\tilde{\bm{\alpha}}=\mathscr{U}_{\mathsf{H}}(\mu_{1},\mu_{2}).$

∎

As an application of the previous properties, we compare the formulation 3.1
of Unbalanced Optimal Transport via homogeneous couplings with the formulation
based on the notion of semi-couplings introduced by [Chi+18a] in a compact
Euclidean setting. We obtain a considerable extension of the semi-coupling
approach to a general, possibly non-compact, setting.

Given $\mu_{i}\in\mathcal{M}_{+}(\mathsf{X}_{i})$, $i=1,2$, we can consider
the set

$\hat{\Gamma}(\mu_{1},\mu_{2}):=\Big{\\{}(\gamma_{1},\gamma_{2})\in(\mathcal{M}_{+}(\mathsf{X}_{1}\times\mathsf{X}_{2}))^{2}:\pi^{i}_{\sharp}(\gamma_{i})=\mu_{i},\
i=1,2\Big{\\}}.$ (3.22)

Every radially $1$-homogeneous cost function $\mathsf{H}$ as in (3.2) induces
a functional on $\hat{\Gamma}(\mu_{1},\mu_{2})$ given by

$J_{\mathsf{H}}(\gamma_{1},\gamma_{2}):=\int_{\mathsf{X}_{1}\times\mathsf{X}_{2}}\mathsf{H}\Big{(}\Big{[}x_{1},\frac{\,\mathrm{d}\gamma_{1}}{\,\mathrm{d}\gamma}\Big{]},\Big{[}x_{2},\frac{\,\mathrm{d}\gamma_{2}}{\,\mathrm{d}\gamma}\Big{]}\Big{)}\,\,\mathrm{d}\gamma,\quad\gamma\in\mathcal{M}_{+}(\mathsf{X}_{1}\times\mathsf{X}_{2}),\
\gamma_{i}\ll\gamma.$ (3.23)

Since $\mathsf{H}$ is radially $1$-homogeneous, (3.23) does not depend on the
choice of the dominating measure $\gamma$ and one can consider the problem of
minimizing $J_{\mathsf{H}}$ in $\hat{\Gamma}(\mu_{1},\mu_{2}):$

$\widehat{\mathscr{U}}_{\mathsf{H}}(\mu_{1},\mu_{2}):=\inf\Big{\\{}J_{\mathsf{H}}(\gamma_{1},\gamma_{2}):(\gamma_{1},\gamma_{2})\in\hat{\Gamma}(\mu_{1},\mu_{2})\Big{\\}}.$
(3.24)

When $\widehat{\mathscr{U}_{\mathsf{H}}}(\mu_{1},\mu_{2})<\infty$,
$\mathsf{H}$ satisfies (3.2) and it is also radially convex, and
$\mathsf{X}_{1},\mathsf{X}_{2}$ are compact, it is not difficult to prove the
existence of an optimal pair
$(\gamma_{1},\gamma_{2})\in\hat{\Gamma}(\mu_{1},\mu_{2})$ attaining the
infimum in (3.24) (see [Chi+18a, Prop. 3.4]). The direct approach when
$\mathsf{X}_{i}$ are not compact is less clear, but existence of an optimal
pair can be obtained as a consequence of the argument we are going to detail
below in Theorem 3.11.

######  Remark 3.10.

The formulation (3.24) also allows for more general (l.s.c., radially
$1$-homogeneous and convex) functions $\hat{\mathsf{H}}$ defined in
$(\mathsf{X}_{1}\times[0,+\infty))\times(\mathsf{X}_{2}\times[0,+\infty))$
which are not compatible with the equivalence relation (2.7) inducing the
topological cone $\mathfrak{C}[\mathsf{X}_{1},\mathsf{X}_{2}]$. This
definition, however, may not satisfy the invariance property stated in Remark
3.2: consider, e.g., the case when $\mathsf{X}_{1}=\mathsf{X}_{2}=[0,a]$,
$a>0$, $\mu_{1}=\delta_{0},$ $\mu_{2}=\bm{0}$, and
$\hat{\mathsf{H}}((x_{1},r_{1}),(x_{2},0))=r_{1}\mathrm{e}^{-|x_{1}-x_{2}|}$.
We clearly have
$\widehat{\mathscr{U}_{\hat{\mathsf{H}}}}(\mu_{1},\mu_{2})=\mathrm{e}^{-a}$.
In order to have an intrinsic formulation, we will assume that $\mathsf{H}$ is
compatible with the cone structure.

######  Theorem 3.11.

If $\mathsf{H}$ satisfies (3.2) then for every
$\mu_{i}\in\mathcal{M}_{+}(\mathsf{X}_{i})$ non trivial we have

$\mathscr{U}_{\mathsf{H}}(\mu_{1},\mu_{2})\leq\widehat{\mathscr{U}}_{\mathsf{H}}(\mu_{1},\mu_{2}).$
(3.25)

If, in addition to (3.2), $\mathsf{H}$ is also radially convex, then

$\mathscr{U}_{\mathsf{H}}(\mu_{1},\mu_{2})=\widehat{\mathscr{U}}_{\mathsf{H}}(\mu_{1},\mu_{2}).$
(3.26)

and the infimum in (3.24) is attained.

###### Proof.

We first observe that every triple
$(\gamma,\gamma_{1},\gamma_{2})\in\mathcal{M}_{+}(\mathsf{X}_{1}\times\mathsf{X}_{2})\times\hat{\Gamma}(\mu_{1},\mu_{2})$
with $\gamma_{i}\ll\gamma$ induces a homogeneous coupling
$\bm{\alpha}\in\mathfrak{H}^{1}(\mu_{1},\mu_{2})$ via the formula

$\bm{\alpha}:=G_{\sharp}\gamma,\quad
G(x_{1},x_{2}):=\Big{(}\big{[}x_{1},\varrho_{1}(x_{1},x_{2})\big{]},\big{[}x_{2},\varrho_{2}(x_{1},x_{2})\big{]}\Big{)},\quad\varrho_{i}:=\frac{\,\mathrm{d}\gamma_{i}}{\,\mathrm{d}\gamma},$
(3.27)

such that

$\int_{\mathfrak{C}[\mathsf{X}_{1},\mathsf{X}_{2}]}\mathsf{H}\,\,\mathrm{d}\bm{\alpha}=J_{\mathsf{H}}(\gamma_{1},\gamma_{2}).$
(3.28)

Therefore we immediately get (3.25).

Let us now assume that $\mathsf{H}$ is also radially convex and let
$\bm{\alpha}$ be a distinguished optimal coupling as in Theorem 3.9. Keeping
the same notation of that Theorem let us set $\mu_{i}(S_{i}^{\prime})=m_{i}>0$
and let us set $\nu_{i}:=m_{i}^{-1}\mu_{i}^{\prime}.$ We define
$\gamma:=\bm{\gamma}^{\prime}+\mu_{1}^{\prime\prime}\otimes\nu_{2}+\nu_{1}\otimes\mu_{2}^{\prime\prime}$,
$\gamma_{1}:=\varrho_{1}\gamma^{\prime}+\mu_{1}^{\prime\prime}\otimes\nu_{2}$,
$\gamma_{2}:=\varrho_{2}\gamma^{\prime}+\nu_{1}\otimes\mu_{2}^{\prime\prime}$,
obtaining
$\pi^{i}_{\sharp}\gamma_{i}=\mu_{i}^{\prime}+\mu_{i}^{\prime\prime}=\mu_{i}$
so that $(\gamma_{1},\gamma_{2})\in\hat{\Gamma}(\mu_{1},\mu_{2}).$ Moreover

$\displaystyle J_{\mathsf{H}}(\gamma_{1},\gamma_{2})$
$\displaystyle=\int\mathsf{H}([x_{1},\varrho_{1}],[x_{2},\varrho_{2}])\,\,\mathrm{d}\bm{\gamma}^{\prime}$
$\displaystyle\qquad+\int\mathsf{H}([x_{1},1],[x_{2},0])\,\,\mathrm{d}(\mu_{1}^{\prime\prime}\otimes\nu_{2})+\int\mathsf{H}([x_{1},0],[x_{2},1])\,\,\mathrm{d}(\nu_{1}\otimes\mu_{2}^{\prime\prime})$
$\displaystyle=\int\mathsf{H}\,\,\mathrm{d}\bm{\alpha}=\mathscr{U}_{\mathsf{H}}(\mu_{1},\mu_{2}).\qed$

###  3.3. The dual problem

In the next results we study the dual definition of the Unbalanced Optimal
Transport cost $\mathscr{U}_{\mathsf{H}}$, for which we need the following
definitions.

######  Definition 3.12.

Let $\mathsf{H}:\mathfrak{C}[\mathsf{X}_{1},\mathsf{X}_{2}]\to[0,+\infty]$ be
a function; we define the set of continuous functions

$\Phi_{\mathsf{H}}:=\left\\{(\varphi_{1},\varphi_{2})\in\mathrm{C}_{b}(\mathsf{X}_{1})\times\mathrm{C}_{b}(\mathsf{X}_{2})\
:\begin{array}[]{l}\varphi_{1}(x_{1})r_{1}+\varphi_{2}(x_{2})r_{2}\leq\mathsf{H}([x_{1},r_{1}],[x_{2},r_{2}])\\\
\text{for every
}(x_{1},x_{2})\in\mathsf{X}_{1}\times\mathsf{X}_{2},\,(r_{1},r_{2})\in\mathbb{R}_{+}^{2}\end{array}\right\\}$
(3.29)

and, for every
$(\mu_{1},\mu_{2})\in\mathcal{M}_{+}(\mathsf{X}_{1})\times\mathcal{M}_{+}(\mathsf{X}_{2})$,
the functional
$\mathcal{D}(\cdot;\mu_{1},\mu_{2}):\mathrm{C}_{b}(\mathsf{X}_{1})\times\mathrm{C}_{b}(\mathsf{X}_{2})\to\mathbb{R}$
given by

$\mathcal{D}(\varphi_{1},\varphi_{2};\mu_{1},\mu_{2}):=\int_{\mathsf{X}_{1}}\varphi_{1}\,\mathrm{d}\mu_{1}+\int_{\mathsf{X}_{2}}\varphi_{2}\,\mathrm{d}\mu_{2},\quad(\varphi_{1},\varphi_{2})\in\mathrm{C}_{b}(\mathsf{X}_{1})\times\mathrm{C}_{b}(\mathsf{X}_{2}).$

Before stating the main duality result, let us briefly recall the Fenchel-
Moreau Theorem in the framework of a pair of vector spaces $E,F$ placed in
duality by a nondegenerate bilinear map $\langle\cdot,\cdot,\rangle$, see e.g.
[ET87]. We endow $E$ with the weak topology $\sigma(E,F)$, the coarsest
topology for which all the functions $e\mapsto\langle e,f\rangle$, $f\in F$,
are continuous.

######  Definition 3.13.

Let $\mathscr{F}:E\to(-\infty,+\infty]$ be not identically $+\infty$ and
satisfying

$\mathscr{F}(e)\geq\langle e,f\rangle-c\quad\text{for some $f\in F$,
$c\in\mathbb{R}$ and every }e\in E.$ (3.30)

The polar (or conjugate) function of $\mathscr{F}$ is the function
$\mathscr{F}^{*}:F\to(-\infty,+\infty]$ defined by

$\mathscr{F}^{*}(f):=\sup_{e\in E}\langle
e,f\rangle-\mathscr{F}(e)\quad\text{for every }f\in F.$

######  Theorem 3.14 (Fenchel-Moreau).

Let $E$ and $F$ be vector spaces placed in duality by a nondegenerate bilinear
map $\langle\cdot,\cdot,\rangle$ and let $\mathscr{F}:E\to(-\infty,+\infty]$
be satisfying (3.30) and not identically $+\infty$. Then the lower
semicontinuous (w.r.t. the topology $\sigma(E,F)$) and convex envelope of
$\mathscr{F}$ is given by the dual formula

$\operatorname{\overline{\mathrm{co}}}\left(\mathscr{F}\right)=\mathscr{F}^{**}(e):=\sup_{f\in
F}\langle e,f\rangle-\mathscr{F}^{*}(f)\quad\text{for every }e\in E.$

In particular,

$\text{if $\mathscr{F}$ is convex and lower semicontinuous
then}\quad\mathscr{F}=\mathscr{F}^{**}.$

The following is the main duality result and it gives also an independent
proof of Theorem 3.5.

######  Theorem 3.15 (Duality).

Let $\mathsf{H}$ be as in (3.2) and let $\mathscr{S}_{\mathsf{H}}$ and
$\mathscr{U}_{\mathsf{H}}$ be as in Definition 3.1. Then

$\mathscr{U}_{\mathsf{H}}(\mu_{1},\mu_{2})=\operatorname{\overline{\mathrm{co}}}\left(\mathscr{S}_{\mathsf{H}}\right)(\mu_{1},\mu_{2})=\sup\left\\{\mathcal{D}(\varphi_{1},\varphi_{2};\mu_{1},\mu_{2}):(\varphi_{1},\varphi_{2})\in\Phi_{\mathsf{H}}\right\\}$
(3.31)

for every
$(\mu_{1},\mu_{2})\in\mathcal{M}_{+}(\mathsf{X}_{1})\times\mathcal{M}_{+}(\mathsf{X}_{2})$.

###### Proof.

Set $E:=\mathcal{M}(\mathsf{X}_{1})\times\mathcal{M}(\mathsf{X}_{2})$ and
$F:=\mathrm{C}_{b}(\mathsf{X}_{1})\times\mathrm{C}_{b}(\mathsf{X}_{2})$ with
the bilinear form

$\langle\cdot,\cdot\rangle:E\times
F\to\mathbb{R},\quad\quad((\mu_{1},\mu_{2}),(\varphi_{1},\varphi_{2}))\mapsto\mathcal{D}(\varphi_{1},\varphi_{2};\mu_{1},\mu_{2}).$

This is a well defined nondegenerate bilinear form. We endow then $E$ with the
topology $\sigma(E,F)$ which coincides exactly with the product weak topology.
Consider then the function
$\mathscr{S}_{\mathsf{H}}:E\mapsto(-\infty,+\infty]$ defined as in Definition
3.1. Then we have

$\displaystyle\mathscr{S}_{\mathsf{H}}^{*}(\varphi_{1},\varphi_{2})$
$\displaystyle=\sup_{(\mu_{1},\mu_{2})\in
E}\left\\{\langle(\mu_{1},\mu_{2}),(\varphi_{1},\varphi_{2})\rangle-\mathscr{S}_{\mathsf{H}}(\mu_{1},\mu_{2})\right\\}$
$\displaystyle=\sup_{(\mu_{1},\mu_{2})\in\Delta_{+}(\mathsf{X}_{1})\times\Delta_{+}(\mathsf{X}_{2})}\left\\{\langle(\mu_{1},\mu_{2}),(\varphi_{1},\varphi_{2})\rangle-\mathscr{S}_{\mathsf{H}}(\mu_{1},\mu_{2})\right\\}$
$\displaystyle=\sup_{x_{1},r_{1},x_{2},r_{2}}\left\\{\varphi_{1}(x_{1})r_{1}+\varphi_{2}(x_{2})r_{2}-\mathsf{H}([x_{1},r_{1}],[x_{2},r_{2}])\right\\}$
$\displaystyle=\begin{cases}0\quad&\text{ if
}(\varphi_{1},\varphi_{2})\in\Phi_{\mathsf{H}}\\\ +\infty\quad&\text{
elsewhere}\end{cases}.$

Hence by Theorem 3.14 we have

$\operatorname{\overline{\mathrm{co}}}\left(\mathscr{S}_{\mathsf{H}}\right)(\mu_{1},\mu_{2})=\mathscr{S}_{\mathsf{H}}^{**}(\mu_{1},\mu_{2})=\sup\left\\{\mathcal{D}(\varphi_{1},\varphi_{2};\mu_{1},\mu_{2})\mid(\varphi_{1},\varphi_{2})\in\Phi_{\mathsf{H}}\right\\}.$
(3.32)

By Theorem 3.4 we know that $\mathscr{U}_{\mathsf{H}}$ is convex and lower
semicontinuous and stays below $\mathscr{S}_{\mathsf{H}}$ so that we clearly
have
$\mathscr{U}_{\mathsf{H}}\leq\operatorname{\overline{\mathrm{co}}}\left(\mathscr{S}_{\mathsf{H}}\right)$.
The other inequality is immediate: take any
$\bm{\alpha}\in\mathfrak{H}^{1}(\mu_{1},\mu_{2})$ and any
$(\varphi_{1},\varphi_{2})\in\Phi_{\mathsf{H}}$; then

$\int_{\mathsf{X}_{1}}\varphi_{1}\,\mathrm{d}\mu_{1}+\int_{\mathsf{X}_{2}}\varphi_{2}\,\mathrm{d}\mu_{2}=\int_{\mathfrak{C}[\mathsf{X}_{1},\mathsf{X}_{2}]}\left((\varphi_{1}\circ\mathsf{x}_{1})\mathsf{r}_{1}+(\varphi_{2}\circ\mathsf{x}_{2})\mathsf{r}_{2}\right)\,\mathrm{d}\bm{\alpha}\leq\int_{\mathfrak{C}[\mathsf{X}_{1},\mathsf{X}_{2}]}\mathsf{H}\,\mathrm{d}\bm{\alpha}.$

Passing to the supremum in $\Phi_{\mathsf{H}}$ and to the infimum in
$\mathfrak{H}^{1}(\mu_{1},\mu_{2})$ and using (3.32), we conclude that
$\mathscr{U}_{\mathsf{H}}\geq\operatorname{\overline{\mathrm{co}}}\left(\mathscr{S}_{\mathsf{H}}\right)$.
∎

######  Remark 3.16.

If $\mathsf{H}$ has the form

$\mathsf{H}([x_{1},r_{1}],[x_{2},r_{2}]):=\begin{cases}r_{1}\mathsf{c}(x_{1},x_{2})\quad&\text{
if }r_{1}=r_{2}\geq 0\\\ +\infty\quad&\text{ elsewhere }\end{cases},$

for some $\mathsf{c}:\mathsf{X}_{1}\times\mathsf{X}_{2}\to[0,+\infty]$ lower
semicontinuous function, we have that
$\mathscr{U}_{\mathsf{H}}\raisebox{-1.29167pt}{$|$}_{\mathcal{P}(\mathsf{X}_{1})\times\mathcal{P}(\mathsf{X}_{2})}$
is exactly the classical Optimal Transport cost induced by $\mathsf{c}$. This
was indeed exploited in [SS22].

In Definition 2.10 we have introduced the closed and convex envelope of
$\mathsf{H}$ which is obtained first convexifying $\mathsf{H}$ “slice by
slice” and then closing its graph globally in
$\mathfrak{C}[\mathsf{X}_{1},\mathsf{X}_{2}]$. One could also compute first
the closed convex envelope “slice by slice” and then glue together the
resulting functions (see the definition of
$\widehat{\operatorname{\mathrm{co}}}\left(\mathsf{H}\right)$ below). In the
following result we show, making use of Theorem 3.15, that the two procedures
give raise to same object.

We also show that, even if $\mathsf{H}$ is not convex, taking the closed and
convex envelope of the singular cost $\mathscr{S}_{\mathsf{H}}$ is sufficient
to also recover the Unbalanced Optimal Transport cost induced by the closed
and convex envelope of $\mathsf{H}$.

######  Corollary 3.17.

Let $\mathsf{H}$ be as in (3.2) and let
$\operatorname{\overline{\mathrm{co}}}\left(\mathsf{H}\right)$ be as in
Definition 2.10. Let us define
$\widehat{\operatorname{\mathrm{co}}}\left(\mathsf{H}\right):\mathfrak{C}[\mathsf{X}_{1},\mathsf{X}_{2}]\to[0,+\infty]$
as

$\widehat{\operatorname{\mathrm{co}}}\left(\mathsf{H}\right)([x_{1},r_{1}],[x_{2},r_{2}]):=\operatorname{\overline{\mathrm{co}}}\left(\mathsf{H}_{x_{1},x_{2}}\right)(r_{1},r_{2})\quad\text{
for every
}(x_{1},x_{2})\in\mathsf{X}_{1}\times\mathsf{X}_{2},\,(r_{1},r_{2})\in\mathbb{R}_{+}^{2}.$

Then

$\operatorname{\overline{\mathrm{co}}}\left(\mathsf{H}\right)([x_{1},r_{1}],[x_{2},r_{2}])=\widehat{\operatorname{\mathrm{co}}}\left(\mathsf{H}\right)([x_{1},r_{1}],[x_{2},r_{2}])=\sup\left\\{\varphi(x_{1})r_{1}+\varphi_{2}(x_{2})r_{2}:(\varphi_{1},\varphi_{2})\in\Phi_{\mathsf{H}}\right\\}$

for every $(x_{1},x_{2})\in\mathsf{X}_{1}\times\mathsf{X}_{2}$ and every
$(r_{1},r_{2})\in\mathbb{R}_{+}^{2}$. Moreover

$\operatorname{\overline{\mathrm{co}}}\left(\mathscr{S}_{\mathsf{H}}\right)=\operatorname{\overline{\mathrm{co}}}\left(\mathscr{S}_{\operatorname{\overline{\mathrm{co}}}\left(\mathsf{H}\right)}\right)=\mathscr{U}_{\mathsf{H}}=\mathscr{U}_{\operatorname{\overline{\mathrm{co}}}\left(\mathsf{H}\right)}\text{
in }\mathcal{M}(\mathsf{X}_{1})\times\mathcal{M}(\mathsf{X}_{2}).$ (3.33)

###### Proof.

We denote by
$\mathsf{U}_{\mathsf{H}}:\mathfrak{C}[\mathsf{X}_{1},\mathsf{X}_{2}]\to[0,+\infty]$
the restriction of $\mathscr{U}_{\mathsf{H}}$ to
$\Delta_{+}(\mathsf{X}_{1})\times\Delta_{+}(\mathsf{X}_{2})\cong\mathfrak{C}[\mathsf{X}_{1},\mathsf{X}_{2}]$
(see Lemma 2.4).
It is clear that
$\operatorname{\overline{\mathrm{co}}}\left(\mathsf{H}\right)\leq\widehat{\operatorname{\mathrm{co}}}\left(\mathsf{H}\right)\leq\mathsf{H}$
so that

$\mathscr{U}_{\operatorname{\overline{\mathrm{co}}}\left(\mathsf{H}\right)}\leq\mathscr{U}_{\widehat{\operatorname{\mathrm{co}}}\left(\mathsf{H}\right)}\leq\mathscr{U}_{\mathsf{H}}$

and

$\operatorname{\overline{\mathrm{co}}}\left(\mathsf{H}\right)=\mathsf{U}_{\operatorname{\overline{\mathrm{co}}}\left(\mathsf{H}\right)}\leq\widehat{\operatorname{\mathrm{co}}}\left(\mathsf{H}\right)=\mathsf{U}_{\widehat{\operatorname{\mathrm{co}}}\left(\mathsf{H}\right)}\leq\mathsf{U}_{\mathsf{H}}\leq\mathsf{H},$

where we used Theorem 3.4 and the radial convexity of
$\operatorname{\overline{\mathrm{co}}}\left(\mathsf{H}\right)$ and
$\widehat{\operatorname{\mathrm{co}}}\left(\mathsf{H}\right)$. Moreover, since
$\mathsf{U}_{\mathsf{H}}$ is, lower semicontinuous, convex and stays below
$\mathsf{H}$, we have that
$\mathsf{U}_{\mathsf{H}}\leq\operatorname{\overline{\mathrm{co}}}\left(\mathsf{H}\right)$.
This gives that

$\operatorname{\overline{\mathrm{co}}}\left(\mathsf{H}\right)=\mathsf{U}_{\operatorname{\overline{\mathrm{co}}}\left(\mathsf{H}\right)}=\widehat{\operatorname{\mathrm{co}}}\left(\mathsf{H}\right)=\mathsf{U}_{\widehat{\operatorname{\mathrm{co}}}\left(\mathsf{H}\right)}=\mathsf{U}_{\mathsf{H}}$

and in particular, using Theorem 3.15, that

$\operatorname{\overline{\mathrm{co}}}\left(\mathsf{H}\right)([x_{1},r_{1}],[x_{2},r_{2}])=\widehat{\operatorname{\mathrm{co}}}\left(\mathsf{H}\right)([x_{1},r_{1}],[x_{2},r_{2}])=\sup\left\\{\varphi(x_{1})r_{1}+\varphi_{2}(x_{2})r_{2}:(\varphi_{1},\varphi_{2})\in\Phi_{\mathsf{H}}\right\\}.$

The fact that
$\operatorname{\overline{\mathrm{co}}}\left(\mathsf{H}\right)=\mathsf{U}_{\mathsf{H}}$
gives that