We end up by discussing the more complicated question of the existence of an optimal pair $(\beta, \varphi)$ for the dual problem. As a first step we have to relax a bit the notion of dual potentials and assume that $\varphi$ is in $L^\infty(d\rho)$ instead of $C_b$:

$$ \tilde{D}(\rho) := \sup \left\{ \beta + \int_X \varphi(x) d\rho(x) : \beta \le c_0, \ \varphi \in L^\infty(d\rho), \right. \\ \left. \beta + \sum_{j=1}^n \varphi(x_j) \le c_n(x_1, \dots, x_n) \otimes^n \rho \text{-a.e. } \forall n \ge 1 \right\}. \quad (2.23) $$

The following is a rather simple result which provides the existence of the dual pair $(\beta, \varphi)$.

**Theorem 2.2.9 (Existence of a dual potential [DLN22])**. Let $X$ be any open set in $\mathbb{R}^d$ and $\rho$ be any finite measure on $X$. Let $c = (c_n)_{n \ge 0}$ be a pairwise superstable family of lower semi-continuous costs such that $c_0 = c_1 = 0$, $c_2 > 0$ and for $n \ge 2$, $\{c_n < \infty\}$ is an open subset of $X$ on which $c_n$ is continuous. Then we have $D(\rho) = \tilde{D}(\rho) = \text{GCOT}_0^c(\rho)$. If $\text{GCOT}_0^c((1+\varepsilon)\rho) < +\infty$ for some $\varepsilon > 0$. Then there exists $(\beta^*, \varphi^*) \in \mathbb{R} \times L^\infty(X, d\rho)$ which is optimal for $\tilde{D}(\rho)$ in (2.23). For any optimizer $\gamma^* = (\gamma_n^*)_{n \ge 0}$ for $\text{GCOT}_0^c(\rho)$, we have

$$ \beta^* + \sum_{j=1}^{n} \varphi^*(x_j) = c_n(x_1, \dots, x_n) \quad \gamma_n^*-\text{a.e., for all } n \ge 0. \quad (2.24) $$

**Remark 2.2.10.** Notice that the original statement of the previous theorem (see [DLN22][Thm 4.2]) we have considered a generic family of costs so that we have to make some extra assumptions

* $c_1 \in L^\infty(X, d\rho)$;

* Monotonicity $c_{n+1}(x_1, \dots, x_{n+1}) \ge c_n(x_1, \dots, x_n) - A$ for some $A \in \mathbb{R}$, all $n \ge 0$ and all $x_1, \dots, x_{n+1} \in X$.

The $\text{GCOT}_0^c((1+\varepsilon)\rho) < +\infty$ assumption (strongly inspired by [CCL84]) means that, in some sense, $\rho$ must be in the interior of the convex set of densities for which $\text{GCOT}_0^c(\rho) < +\infty$.

**Remark 2.2.11.** We finally remark that in the special case of Coulomb cost (even if [DLN22][Thm 4.3] holds for a more general class of grand canonical costs) we are also able to prove that the optimal potential $\varphi^*$ for (2.23) is Lipschitz with constant depending only on the marginal $\rho$. in the case in which $\text{Supp}(\rho) = \mathbb{R}^d$ then $\varphi^* \in C_b(\mathbb{R}^d)$ and it is also optimal for $D(\rho)$.

### 2.2.2 Entropic Grand Canonical Optimal Transport

As we did for the standard multi-marginal optimal transport problem we discuss the entropic regularization of $\text{GCOT}_0^c(\rho)$ which, from a statistical mechanics point of view, amounts to adding temperature in the system. In the Physics literature, the problem of finding the external potential for which the associated grand-canonical equilibrium classical (Gibbs) state at a temperature $\varepsilon > 0$ has the given density $\rho$, plays an important role in the density functional theory of classical inhomogeneous systems [Eva79]. Notice that the canonical model is just the entropic regularization of the multi-marginal problem $\text{MOT}_0^c(\rho)$, see for instance Section 1.3.

**Remark 2.2.12.** On the mathematical side, our work relies on the fundamental works of Chayes, Chayes and Lieb [CCL84; CC84] in 1984 who, to our knowledge, were the first to prove the existence of the dual potential $v$ for such systems with entropy, both in the $N$-marginal and grand-canonical cases.
In order to have a correct behavior it is necessary to place the right 'Boltzmann n!' factor in the definition of the entropy, an issue which does not occur for the multi-marginal problem at fixed n. The entropy of a grand-canonical probability $\gamma = (\gamma_n)_{n \ge 0}$ is defined by [Rue99; CCL84; CC84]

$$ S(\gamma) := - \sum_{n \ge 0} \int_{X^n} \gamma_n \log (n!\gamma_n), \qquad (2.25) $$

with the condition that each $\gamma_n$ is absolutely continuous with respect to the Lebesgue measure on $X^n$. Otherwise, $S(\gamma)$ is taken equal to $-\infty$. Note the minus sign in the definition (the usual convention in statistical mechanics). Notice the $n!$ which is due to the fact that our system describes independent agents/particles and so we should in principle not work on $X^n$ but, rather, in the simplex of volume $1/n!$ obtained by moving out the permutations. Consider now the following grand canonical relative entropy

$$ \mathrm{Ent}(\gamma|\pi_{\rho}) = \sum_{n \ge 0} \int_{X^n} \gamma_n \log \left( \frac{\gamma_n n! e^{\rho(X)}}{\rho^{\otimes n}} \right), $$

where $\pi_\rho$ is the Poisson grand-canonical probability $\pi_\rho = (\pi_{\rho,n})_{n \ge 0} \in \Pi_{\mathrm{GC}}(\rho)$ given by

$$ \pi_{\rho,0} = e^{-\rho(X)}, \quad \pi_{\rho,n} = e^{-\rho(X)} \frac{\otimes^n \rho}{n!}. $$

For any finite non-negative measure $\rho$ on $X$, the grand-canonical entropic optimal transport problem reads

$$ \mathrm{GCOT}_{\varepsilon}^{c}(\rho) := \inf_{\gamma \in \Pi_{\mathrm{GC}}(\rho)} \{ \langle \gamma, c \rangle + \varepsilon \mathrm{Ent}(\gamma|\pi_{\rho}) \}. \qquad (2.26) $$

We will assume that the cost $c$ is stable, $c_n \ge -A - Bn$, so that $\langle \gamma, c \rangle \ge -A - B\rho(X)$. The infimum is thus finite for one $\varepsilon > 0$ if and only if there exists a $\gamma \in \Pi_{\mathrm{GC}}(\rho)$ such that $\langle \gamma, c \rangle$ and $\mathrm{Ent}(\gamma|\pi_\rho)$ are simultaneously finite. In this case $\mathrm{GCOT}_\varepsilon^c(\rho)$ is actually finite for all $\varepsilon > 0$. Otherwise, we have $\mathrm{GCOT}_\varepsilon^c(\rho) = +\infty$ for all $\varepsilon > 0$. For instance we can simply assume that the cost $\langle \pi_\rho, c \rangle$ is finite for the Poisson state. Note also that $\mathrm{GCOT}_\varepsilon^c(\rho)$ is non-decreasing and concave in $\varepsilon$. The following existence result is a consequence of the known properties of the relative entropy $\mathrm{Ent}$.

**Theorem 2.2.13 (Existence for the positive temperature problem).** Let $X \subset \mathbb{R}^d$ be any Borel set. Let $c = (c_n)_{n \ge 0}$ be a stable family of lower semi-continuous costs. Let $\rho$ be a finite measure such that $\mathrm{GCOT}_\varepsilon^c(\rho) < \infty$ for one (hence all) $\varepsilon > 0$.

(i) $\mathrm{GCOT}_\varepsilon^c(\rho)$ admits a unique minimizer $\gamma^{(\varepsilon)}$ for all $\varepsilon > 0$.

(ii) In the limit $\varepsilon \to 0^+$ we have

$$ \lim_{\varepsilon \to 0^+} \mathrm{GCOT}_{\varepsilon}^{c}(\rho) = \inf_{\substack{\gamma \in \Pi_{\mathrm{GC}}(\rho) \\ \mathrm{Ent}(\gamma|\pi_{\rho}) < \infty}} \langle \gamma, c \rangle. \qquad (2.27) $$

If $c = (c_n)_{n \ge 0}$ is superstable and the right side is equal to $\mathrm{GCOT}_0^c(\rho)$, then $\gamma^{(\varepsilon)}$ converges, up to subsequences, to a minimizer $\gamma^* \in \Pi_{\mathrm{GC}}(\rho)$ for $\mathrm{GCOT}_0^c(\rho)$, in the sense that $\gamma_n^{(\varepsilon)} \to \gamma_n^*$ tightly for all $n \ge 0$.
(iii) If $\langle \pi_{\rho}, \mathbf{c} \rangle < \infty$, then in the limit $\varepsilon \to \infty$ we have

$$ \lim_{\varepsilon \to \infty} \mathrm{GCOT}_{\varepsilon}^{\mathbf{c}}(\rho) = \langle \pi_{\rho}, \mathbf{c} \rangle, $$

$$ \lim_{\varepsilon \to \infty} \varepsilon \mathrm{Ent}(\gamma^{(\varepsilon)}|\pi_{\rho}) = \lim_{\varepsilon \to \infty} \sum_{n \ge 0} \int_{X^n} |\gamma_n^{(\varepsilon)} - \pi_{\rho,n}| = 0. $$

We emphasize that for the existence of $\gamma^{(\varepsilon)}$ the cost $\mathbf{c} = (c_n)_{n \ge 0}$ is only assumed to be *stable* and not *superstable* as we required at $\varepsilon = 0$ in Theorem 2.2.4. This is because the entropy gives us an additional control on the large-$n$ behavior.

We end the section by discussing the dual problem. Consider a measurable potential $\psi: X \to \mathbb{R}$ such that $\int_X e^{\frac{\psi(x)}{\varepsilon}} d\rho(x) < \infty$. With $\Psi_n = \sum_{j=1}^n \psi(x_j)$ we define

$$ D_{\varepsilon,\rho}(\psi) := \inf_{\gamma} \{\langle \gamma, \mathbf{c} - \Psi \rangle + \varepsilon \mathrm{Ent}(\gamma|\pi_{\rho})\} = -\varepsilon \log Z_{\varepsilon,\rho}(\psi) \quad (2.28) $$

with the partition function

$$ Z_{\varepsilon,\rho}(\psi) := e^{-\frac{c_0}{\varepsilon} - \rho(X)} + \sum_{n \ge 1} \frac{e^{-\rho(X)}}{n!} \int_{X^n} \exp \left( \frac{-c_n(x_1, \dots, x_n) + \sum_{j=1}^n \psi(x_j)}{\varepsilon} \right) d^{\otimes n} \rho. \quad (2.29) $$

Recalling that $c_n \ge -A - Bn$ we find

$$ Z_{\varepsilon,\rho}(\psi) \le \exp \left( \frac{A}{\varepsilon} - \rho(X) + e^{\frac{B}{\varepsilon}} \int_X e^{\frac{\psi}{\varepsilon}} d\rho \right) \quad (2.30) $$

and since $Z_{\varepsilon,\rho}(\psi) \ge e^{-c_0/\varepsilon - \rho(X)}$, we obtain

$$ -A + \varepsilon\rho(X) - \varepsilon e^{\frac{B}{\varepsilon}} \int_X e^{\frac{\psi}{\varepsilon}} d\rho \le D_{\varepsilon,\rho}(\psi) \le c_0 + \varepsilon\rho(X). \quad (2.31) $$

The formula $D_{\varepsilon,\rho}(\psi) = -\varepsilon \log Z_{\varepsilon,\rho}(\psi)$ is well known. The corresponding unique minimizer is the Gibbs state $\gamma_{\psi}$ given by

$$ \gamma_{\psi,0} = \frac{e^{-\frac{c_0}{\varepsilon} - \rho(X)}}{Z_{\varepsilon,\rho}(\psi)}, \quad \gamma_{\psi,n} = \frac{e^{\frac{-c_n + \sum_{j=1}^n \psi(x_j)}{\varepsilon} - \rho(X)} \rho^{\otimes n}}{Z_{\varepsilon,\rho}(\psi)n!} \quad (2.32) $$

To prove this claim, just notice that

$$ \langle \gamma, \mathbf{c} \rangle + \varepsilon \mathrm{Ent}(\gamma|\pi_{\rho}) - \langle \gamma^*, \mathbf{c} \rangle - \varepsilon \mathrm{Ent}(\gamma^*|\pi_{\rho}) = \varepsilon \mathrm{Ent}(\gamma|\gamma^*) $$

is positive and vanishes only at $\gamma^*$. Next we discuss the existence of a dual potential. The following is a small adaptation of results proved in [CCL84; CC84].

**Theorem 2.2.14** (Duality at $\varepsilon > 0$ [CCL84; CC84]). Let $X \subset \mathbb{R}^d$ be any Borel set. Let $\mathbf{c} = (c_n)_{n \ge 0}$ be a stable family of lower semi-continuous costs with $c_1 \in L^{\infty}(X, d\rho)$. Let $\rho$ be a finite measure such that $\mathrm{GCOT}_{\varepsilon}^{\mathbf{c}}(\rho) < \infty$ for one (hence all) $\varepsilon > 0$. Then we have

$$ \mathrm{GCOT}_{\varepsilon}^{\mathbf{c}}(\rho) = \sup_{\int_X e^{\psi/\varepsilon} d\rho < \infty} \left\{ \int_X \psi d\rho + D_{\varepsilon,\rho}(\psi) \right\}. \quad (2.33) $$
If there exists $\varepsilon > 0$ such that for one (hence all) $\varepsilon > 0$

$$ \mathrm{GCOT}_{\varepsilon}^{c}((1 + \varepsilon)\rho) < \infty, \qquad (2.34) $$

then there exists a unique potential $\psi$ realizing the supremum in (2.33) and it is such that $\psi \in L^1(X, d\rho)$. The corresponding unique minimizer $\gamma^{(\varepsilon)}$ from Theorem 2.2.13 is equal to the Gibbs state $\gamma_{\psi}$ in (2.32).

## 2.3 Quantum Optimal Transport and Lieb Functional

In this section we focus on a moment constraints approximation of the Lieb Functional. Recall that Lieb functional takes the following form

$$ F_L[\rho] := \inf_{\substack{\Gamma \in \mathcal{S}_1^+(\mathcal{H}_0^m) \\ \rho \Gamma = \rho}} \mathrm{Tr}(H_m \Gamma), \qquad (2.35) $$

where $H_m := -\frac{1}{2}\Delta + V$ is a self-adjoint operator on $\mathcal{H}_0^m$ with domain $D(H_m) = \mathcal{H}_2^m := \bigwedge_{i=1}^m H^2(\mathbb{R}^3)$, $\mathcal{S}_1^+(\mathcal{H}_0^m)$ denotes the set of trace-class self-adjoint non-negative operators on $\mathcal{H}_0^m$. Actually, problem (2.35) is a particular instance of quantum optimal transport problem. We refer the reader to [GMP16; CGP21] for references on earlier works on closely related types of problems. The aim of the joint work with V. Ehrlacher [EN23] is to prove that solutions of approximations of problems (2.35) where the partial trace constraint is relaxed by a finite number of moment constraints enjoy similar sparsity properties than solutions of moment constrained multi-marginal symmetric classical optimal transport problems, such as those which were established in [Alf+21]. More precisely:

* we prove by using a generalization of Tchakaloff's theorem, that the solutions of moment constrained approximations of (2.35) can be written under the form $\Gamma = \sum_{k=1}^K \alpha_k |\Psi_k\rangle\langle\Psi_k|$, where $K \in \mathbb{N}^*$ scales at most linearly with the number of moment constraints, and where for all $1 \le k \le K$, $\alpha_k \in [0, 1]$, $\Psi_k \in \mathcal{H}_1^m$ and $|\Psi_k\rangle\langle\Psi_k|$ is the orthogonal projector of $\mathcal{H}_0^m$ onto the vectorial space spanned by $\Psi_k$ (using bra-ket notation);

* under appropriate assumptions on the set of moment functions we show that the value of the moment constraints approximation functional converges to the value of the exact Lieb functional as the number of moments go to infinity;

* we study the mathematical properties of the associated dual problem;

* we obtain some rates of convergence on the associated approximation of the ground state energy.

Let us finally mention here that particular moment-constrained approximations of the Lieb functional have already been considered in [Gar22] for the construction of Kohn-Sham potentials. The novel results brought by our contribution in comparison to this previous work is (i) the extension of existence and convergence results to more general moment constraints that the one considered in [Gar22]; (ii) the results on the sparsity structure of associated minimizers; (iii) convergence rate of the approximate ground state energy; and (iv) study of the mathematical properties of the associated dual problem.
**Remark 2.3.1** (Classical DFT). Classical DFT is typically used at interfaces between liquid-gas, liquid-liquid (in fluid mixtures), crystal-liquid and crystal-gas phases at bulk coexistence. In this framework the free energy for m particles takes the following form

$$ \mathcal{F}_{\varepsilon}(\rho) = \inf_{\gamma \in \Pi(\rho)} \int_{\mathbb{R}^{dm}} \sum_{i<j} w(x_i - x_j) d\gamma + \varepsilon \int_{\mathbb{R}^{dm}} \gamma \log(\gamma), $$

which is exactly the entropic multi-marginal optimal transport problem for a cost function given by the interaction potential $\sum_{i<j} w(x_i - x_j)$.

### 2.3.1 Moment approximation of Lieb functional: the primal problem

Recall that $A^m = \{\rho \in L^1(\mathbb{R}^3) : \rho \ge 0, \sqrt{\rho} \in H^1, \rho(\mathbb{R}^3) = m\}$, we fix an electronic density $\rho \in A^m$. We have $\mathcal{F} := L^{3/2}(\mathbb{R}^3) + L^\infty(\mathbb{R}^3) \subset L_\rho^1(\mathbb{R}^3)$ and for any $f \in \mathcal{F}$, we denote by

$$ \|f\|_{\mathcal{F}} := \inf_{\substack{f_{3/2} \in L^{3/2}(\mathbb{R}^3), f_\infty \in L^\infty(\mathbb{R}^3), \\ f_{3/2} + f_\infty = f}} \|f_{3/2}\|_{L^{3/2}(\mathbb{R}^3)} + \|f_\infty\|_{L^\infty(\mathbb{R}^3)}. $$

Let $M \in \mathbb{N}^*$, given a collection of $M$ functions $\Phi := (\varphi_1, \dots, \varphi_M) \in \mathcal{F}^M$, the main idea of the moment-constrained approximation consists in replacing the density constraint in (2.35) with the $M$ scalar moment constraints associated to the functions $\varphi_1, \dots, \varphi_M$, that is

$$ \int_{\mathbb{R}^3} \varphi_n \rho_{\Gamma} = \int_{\mathbb{R}^3} \varphi_n \rho, \quad \forall n = 1, \dots, M. \qquad (2.36) $$

Notice that (2.36) is equivalent to

$$ \int_{\mathbb{R}^3} \varphi \rho_{\Gamma} = \int_{\mathbb{R}^3} \varphi \rho, \quad \forall \varphi \in \operatorname{Span}\{\Phi\}. \qquad (2.37) $$

We denote by $\mathcal{S}_1^+(\mathcal{H}_0^m, \Phi, \rho)$ the set of $\Gamma \in \mathcal{S}_1^+(\mathcal{H}_0^m)$ satisfying constraints (2.36) (or equivalently (2.37)). For any Hilbert space $\mathcal{H}$, we denote by $\Sigma_{\mathcal{H}}$ the Borel $\sigma$-algebra of $\mathcal{H}$. and we have the following version of Tchakaloff's theorem.

**Proposition 2.3.2** (Tchakaloff). Let $\mu$ be a Borelian measure on a Hilbert space $\mathcal{H}$ concentrated on a Borel set $A \in \Sigma_{\mathcal{H}}$. Let $J_0 \in \mathbb{N}^*$ and $\Lambda : \mathcal{H} \to \mathbb{R}^{J_0}$ a Borel measurable map. Assume that the first moments of $\Lambda_{\sharp}\mu$ exists, that is

$$ \int_{\mathbb{R}^{J_0}} \|x\| d\Lambda_{\sharp}\mu(x) = \int_{\mathcal{H}} \|\Lambda(\Psi)\| d\mu(\Psi) < +\infty, $$

where $\|\cdot\|$ denotes the Euclidean norm of $\mathbb{R}^{J_0}$. Then there exists an integer $1 \le K \le J_0$, elements $\Psi_1, \dots, \Psi_K \in A$ and weights $\omega_1, \dots, \omega_K > 0$ such that

$$ \forall j = 1, \dots, J_0, \quad \int_{\mathcal{H}} \Lambda_j(\Psi) d\mu(\Psi) = \sum_{k=1}^{K} \omega_k \Lambda_j(\Psi_k) = \int_{\mathcal{H}} \Lambda_i(\Psi) d\mu_d(\Psi), $$

where $\Lambda_j$ is the j-th component of $\Lambda$, and $\mu_d = \sum_{k=1}^{K} \omega_k \delta_{\Psi_k}$.
## 2.3. QUANTUM OPTIMAL TRANSPORT AND LIEB FUNCTIONAL

The main idea of the proof of the sparsity result announced above is to define a measure associated to an operator $\Gamma \in \mathfrak{S}_1^+(\mathcal{H}_0^m)$. Assume that the operator $\Gamma$ can be written as

$$ \Gamma = \sum_{i=1}^{+\infty} \alpha_i |\Psi_i\rangle\langle\Psi_i| \qquad (2.38) $$

for some sequence $(\Psi_i)_{i \in \mathbb{N}^*}$ of normalized functions of $\mathcal{H}_0^m$ and non-negative real numbers $(\alpha_i)_{i \in \mathbb{N}^*}$ such that $\sum_{i \in \mathbb{N}^*} \alpha_i = m$. Then we can define a Borelian measure $\mu_\Gamma : \Sigma_{\mathcal{H}_0^m} \to \mathbb{R}_+$ associated to the decomposition (2.38) of the operator $\Gamma$ as

$$ \mu_\Gamma = \sum_{i=1}^{+\infty} \alpha_i \delta_{\Psi_i}. $$

Naturally, there is no unique such measure $\mu_\Gamma$ associated with an operator $\Gamma$ since it heavily depends on the decomposition (2.38). We denote by $\mathbf{1}$ the function defined over $\mathbb{R}^3$ which is identically equal to 1. Then our main result concerning existence for the approximate problem reads as follows.

**Theorem 2.3.3 (Existence for the primal, [EN23]).** Let $\rho \in \mathcal{A}^m$, $M \in \mathbb{N}^*$ and $\Phi := (\varphi_1, \dots, \varphi_M) \in \mathcal{F}^M$ such that $\mathbf{1} \in \text{Span}\{\Phi\}$. Let us assume in addition that

(A$\theta$) there exists a non-negative non-decreasing continuous function $\theta: \mathbb{R}_+ \to \mathbb{R}_+$ such that $\theta(r) \xrightarrow[r \to +\infty]{} +\infty$ and $C_\rho := \int_{\mathbb{R}^3} \theta(|x|)\rho(x) dx < +\infty$.

For all $C > 0$, let us introduce the Moment-Constrained Approximation of the Lieb functional (MCAL)

$$ F_{L,\theta}^{\Phi,C}[\rho] := \inf_{\substack{\Gamma \in \mathfrak{S}_1^+(\mathcal{H}_0^m, \Phi, \rho) \\ \text{Tr}(\Theta\Gamma) \le C}} \text{Tr}(H_m \Gamma), \qquad (\text{MCAL}) $$

where $\Theta(x_1, \dots, x_m) := \frac{1}{m} \sum_{i=1}^m \theta(|x_i|)$ for all $x_1, \dots, x_m \in \mathbb{R}^3$. Then, for all $C \ge C_\rho$, $F_{L,\theta}^{\Phi,C}[\rho]$ is finite and a minimum. Moreover, for all $C \ge C_\rho$, there exists a minimizer $\Gamma_{\text{opt},\theta}^{\Phi,C}$ to (MCAL) such that $\Gamma_{\text{opt},\theta}^{\Phi,C} = \sum_{k=1}^K \omega_k |\Psi_k\rangle\langle\Psi_k|$, for some $1 \le K \le M+1$, with $\omega_k \ge 0$ and $\Psi_k \in \mathcal{H}_1^m$ for all $1 \le k \le K$.

Notice that assumption (A$\theta$) is needed in order to obtain tightness and then use a similar argument as in [Lie83]. This is needed because we are considering operators defined on the space $\mathcal{H}_0^m = \bigwedge_{i=1}^m L^2(\mathbb{R}^3)$. Such a technical assumption is not needed in the case when one considers operators acting on functions acting on a finite domain with Dirichlet boundary conditions.

### 2.3.2 Convergence to the exact Lieb functional

Let us denote here by $\mathcal{D}(\mathbb{R}^3)$ the set of $\mathbb{C}^\infty$ real-valued functions defined on $\mathbb{R}^3$ with compact support. More precisely, let $\rho \in \mathcal{A}^m$ such that there exists a function $\theta: \mathbb{R}_+ \to \mathbb{R}_+$ satisfying assumption (A$\theta$). Let $C_\rho := \int_{\mathbb{R}^3} \theta(|x|)\rho(x) dx$ and let $C > C_\rho$.

For all $n \in \mathbb{N}^*$, let $M_n \in \mathbb{N}^*$ and $\Phi^n := (\varphi_k^n)_{1 \le k \le M_n} \subset \mathcal{F}$ be a sequence of functions belonging to $\mathcal{F}$ and which satisfies $\mathbf{1} \in \text{Span}\{\Phi^n\}$ for all $n \in \mathbb{N}^*$ together with the following density conditions:
## 2.3. QUANTUM OPTIMAL TRANSPORT AND LIEB FUNCTIONAL

$(A\Phi)$ for all $f \in \mathcal{D}(\mathbb{R}^3)$,

$$ \inf_{g_n \in \mathrm{Span}\{\Phi^n\}} \|f - g_n\|_{\mathcal{F}} \xrightarrow{n \to +\infty} 0. $$

**Remark 2.3.4.** One example of sequence $(\Phi_n)_{n \in \mathbb{N}^*}$ satisfying $(A\Phi)$ is the following: for all $n \in \mathbb{N}^*$, let $\Omega_n := (-n, n)^3$ and let $\mathcal{T}_n := \{T_1^n, \dots, T_{N_n}\}$ (with $N_n := \#\mathcal{T}_n$) be a regular conforming triangular mesh of $\Omega_n$, the elements of which have a maximal diameter size $h_n$ such that $h_n \le \frac{1}{n}$. Let $M_n := \#\mathcal{T}_n + 1 = N_n + 1$. Denoting by $\varphi_k^n := 1|_{T_k^n}$ for $1 \le m \le M_n - 1$ and by $\varphi_{M_n}^n := 1|_{\Omega_n^c}$ and by $\Phi^n = (\varphi_k^n)_{1 \le k \le M_n}$ for all $n \in \mathbb{N}^*$, one can easily check that the sequence $(\Phi^n)_{n \in \mathbb{N}^*}$ satisfies $(A\Phi)$.

We then have the following convergence result, which may be seen as an extension of [Gar22][Theorem 3.2] to more general set of moment functions, up to the additional tightness assumption $(A\theta)$.

**Theorem 2.3.5 (Convergence, [EN23])**. Let $\rho \in \mathcal{A}^m$ such that there exists a function $\theta: \mathbb{R}_+ \to \mathbb{R}_+$ satisfying assumption $(A\theta)$. Let $C_\rho := \int_{\mathbb{R}^3} \theta(|x|)\rho(x) dx$ and $C \ge C_\rho$. For all $n \in \mathbb{N}^*$, let $M_n \in \mathbb{N}^*$ and $\Phi^n := (\varphi_k^n)_{1 \le k \le M_n} \subset \mathcal{F}$ such that assumption $(A\Phi)$ holds. We assume in addition that there exists $n_0 \in \mathbb{N}^*$ such that $1 \in \mathrm{Span}\{\Phi^n\}$ for all $n \ge n_0$. Then, for all $n \ge n_0$, there exists at least one sparse minimizer to (MCAL) with $\Phi = \Phi^n$ in the sense of Theorem 2.3.3. Besides, it holds that

$$ \lim_{n \to +\infty} F_{L,\theta}^{\Phi^n, C}[\rho] = F_L[\rho]. \qquad (2.39) $$

Moreover, from any sequence $(\Gamma_n)_{n \ge n_0}$ such that $\Gamma_n$ is a minimizer for (MCAL) with $\Phi = \Phi^n$, one can extract a subsequence which strongly converges in $\mathcal{S}_{1,1}(\mathcal{H}_0^m)$ to $\Gamma_\infty$, where $\Gamma_\infty$ is a minimizer of (2.2).

Like for the existence theorem 2.3.3, we can state a similar result with less technical assumptions in the case when we consider operators acting on functions defined on a bounded subdomain $X \subset \mathbb{R}^3$ with Dirichlet boundary conditions.

### 2.3.3 Convergence rate of the ground state energy in the bounded domain case

We restrict ourselves to the case of a bounded subdomain $X \subset \mathbb{R}^3$. Let $M \in \mathbb{N}^*$, $\Phi := (\varphi_k)_{1 \le k \le M} \subset \mathcal{F}(X)$ be a set of moment functions. For all $v \in \mathcal{F}(X)$, let us introduce the ground state energy associated to the potential $v$:

$$ E[v] := \inf_{\Psi \in \mathcal{H}_1^m(X)} \langle \Psi | H_{m,X}^v | \Psi \rangle = \inf_{\Gamma \in \mathcal{S}_1^+(\mathcal{H}_0^m(X))} \mathrm{Tr}(H_{m,X}^v \Gamma), $$

where

$$ H_{m,X}^v := H_{m,X} - \sum_{i=1}^{m} v(x_i). $$

Rewriting the minimization over $\Gamma$ as an external minimization over $\rho \in \mathcal{A}_m(X)$ and then as an internal one over all $\Gamma$ such that $\mathrm{Tr} \, \Gamma = \rho$, it can easily be checked that

$$ E[v] = \inf_{\rho \in \mathcal{A}_m(X)} \left\{ F_{L,X}[\rho] - \int_X v \, d\rho \right\}, \qquad (2.40) $$
## 2.3. QUANTUM OPTIMAL TRANSPORT AND LIEB FUNCTIONAL

where $F_{L,X}$ is the Lieb functional defined on a bounded domain. Let us also define by

$$ E^{\Phi}[v] := \inf_{\rho \in \mathcal{A}_m(X)} \left\{ F_{L,X}^{\Phi}[\rho] - \int_X v \, d\rho \right\}, \qquad (2.41) $$

where $F_{L,X}^{\Phi}$ is the functional (MCAL) defined on a bounded domain so that we can drop off the assumption $(A\theta)$. Similarly, let us point out that, if $v \in \text{Span}\{\Phi\}$, rewriting the minimization over $\Gamma$ as an external minimization over $\rho \in \mathcal{A}_m(X)$ and then as an internal one over all $\Gamma \in \mathcal{S}_1^+(\mathcal{H}_0^m(X), \Phi, \rho)$, it holds that

$$ E[v] = E^{\Phi}[v], \quad \forall v \in \text{Span}\{\Phi\}. $$

We have then the following approximation result.

**Proposition 2.3.6** ([EN23]). Let us assume that $v \in L^{\infty}(X)$ and that $\Phi = (\varphi_k)_{1 \le k \le M} \subset L^{\infty}(X)$. Then, it holds that

$$ |E[v] - E^{\Phi}[v]| \le 2m \min_{w \in \text{Span}\{\Phi\}} \|v - w\|_{L^{\infty}(X)}. \qquad (2.42) $$

Proposition 2.3.6 then enables to quantify the rate of convergence of $|E[v] - E^{\Phi^n}[v]|$ as $n$ goes to infinity for some particular sequences of moment functions $(\Phi^n)_{n \in \mathbb{N}}$ provided that $v$ is regular enough. As an illustration, we analyze here the rate of convergence of a numerical method inspired from the external dual charge approach recently proposed in [Lel22].

**Corollary 2.3.7** ([EN23]). Let $l \ge 0$ and $X$ be a bounded regular subdomain of $\mathbb{R}^3$. Let $\mu \in H^{l+1}(X)$ be an external density of charge and define $v \in H_0^1(X) \cap H^{l+3}(X)$ as the unique solution to

$$ \begin{cases} -\Delta v = \mu & \text{in } X, \\ v = 0 & \text{on } \partial X. \end{cases} $$

Let $(\mathcal{T}_h)_{h>0}$ be a sequence of triangular regular meshes of $X$ such that

$$ h := \max_{K \in \mathcal{T}_h} \text{diam}(K). $$

Let $k \in \mathbb{N}$ and $P_h^k \subset L^{\infty}(X)$ be the subspace of continuous $\mathbb{P}_k$ finite element functions associated to the mesh $\mathcal{T}_h$. We denote by $V_{h,k}$ the subspace of $H_0^1(X) \cap H^2(X)$ containing all functions $v_{h,k} \in H_0^1(X) \cap H^2(X)$ solution to

$$ \begin{cases} -\Delta v_{h,k} = \mu_{h,k} & \text{in } X, \\ v_{h,k} = 0 & \text{on } \partial X, \end{cases} $$

for some $\mu_{h,k} \in P_h^k$. Let $\Phi_{h,k}$ be a basis of $V_{h,k}$. Then, assuming that $l \le k$, there exists a constant $C > 0$ such that for all $h > 0$,

$$ |E[v] - E^{\Phi_{h,k}}[v]| \le Cmh^{l+1} \|v\|_{H^{l+3}(X)}. $$

**Remark 2.3.8.** Denoting by $M_{h,k}$ the dimension of $V_{h,k}$, it holds that $M_{h,k} = \mathcal{O}(\frac{k}{h^3})$. As a consequence, the above result implies that the rate of convergence of $E^{\Phi_{h,k}}[v]$ to $E[v]$ decays like $\mathcal{O}(\frac{m}{M_{h,k}^{(l+1)/3}})$ where $M_{h,k}$ is the number of moment constraints in the MCAL approximation.
### 2.3.4 The dual problem

We end this section by discussing duality for the case in which we restrict our-self to a bounded domain X as above. We know that the dual variable associated to the density $\rho \in \mathcal{J}_N(X)$ is a one-body interaction potential of the form $W^v(x_1, \dots, x_m) := \sum_{i=1}^m v(x_i)$ for a given $v \in \mathcal{F}(X)$. We then consider the following natural dual problem

$$ \mathcal{D}_{L,X}^{\Phi}[\rho] = \sup_{\substack{v \in \operatorname{Span}\{\Phi\}, \\ \forall \Psi \in \mathcal{H}_1^m(X), \langle \Psi | H_{m,X}^v | \Psi \rangle \ge 0}} \int_X v d\rho, \quad (2.43) $$

where

$$ H_{m,X}^v := H_{m,X} - \sum_{i=1}^{m} v(x_i) = H_{m,X} - W^v. $$

If we take any $v := \sum_{k=1}^M \alpha_k \varphi_k \in \operatorname{Span}\{\Phi\}$ satisfying the above constraints and any $\Gamma \in \mathcal{S}_1^+(\mathcal{H}_0^m(X), \Phi, \rho)$ then we have

$$ \begin{aligned} \operatorname{Tr}(H_{m,X}\Gamma) \ge \operatorname{Tr}(W^v\Gamma) &= \int_X v d\rho_\Gamma = \int_X \left(\sum_{k=1}^M \alpha_k \varphi_k\right) d\rho_\Gamma \\ &\ge \int_X \left(\sum_{k=1}^M \alpha_k \varphi_k\right) d\rho = \int_X v d\rho \end{aligned} $$

which proves that $F_{L,X}^{\Phi}[\rho] \ge \mathcal{D}_{L,X}^{\Phi,C}[\rho]$. We would like to prove that this inequality is actually an equality. Let us introduce the ground state energy associated to the potential $v$:

$$ E[v] = \inf_{\Psi \in \mathcal{H}_1^m(X)} \langle \Psi | H_m^v | \Psi \rangle = \inf_{\Gamma \in \mathcal{S}_1^+(\mathcal{H}_0^m(X))} \operatorname{Tr}(H_m^v \Gamma). $$

We rewrite now the minimization over $\Gamma$ as an external minimization over $\rho \in \mathcal{A}^m(X)$ and then as an internal one over all $\Gamma$ in $\mathcal{S}_1^+(\mathcal{H}_0^m(X), \Phi, \rho)$ (we are considering the ground state for a potential $v \in \operatorname{Span}\{\Phi\}$):

$$ E[v] = \inf_{\rho \in \mathcal{A}^m(X)} \left\{ F_{L,X}^{\Phi}[\rho] - \int_X v d\rho \right\}. $$

Notice that $E$ is nothing but the Legendre-Fenchel transform of $F_{L,X}^{\Phi}[\rho]$. On the other hand, we rewrite (2.43) in the form

$$ \mathcal{D}_{L,X}^{\Phi}[\rho] = \sup_{v \in \operatorname{Span}\{\Phi\}} \left\{ \int_X v d\rho - E[v] \right\}. \quad (2.44) $$

Thus, $\mathcal{D}_{L,X}^{\Phi}[\rho]$ is the Legendre transform of $E$. From lower semi-continuity of $F_{L,X}^{\Phi}$ and Fenchel duality theorem for convex lower semi-continuous functions we conclude the following

**Theorem 2.3.9 (Strong duality, [EN23]).** Under the assumptions of Theorem 2.3.3, we have $F_{L,X}^{\Phi}[\rho] = \mathcal{D}_{L,X}^{\Phi}[\rho]$.

We now have the following result which, taking into account the sparsity result of Theorem 2.3.3, gives a more convenient formulation of $\mathcal{D}_{L,X}^{\Phi}[\rho]$.
**Theorem 2.3.10 (Existence for the dual, [EN23]).** Under the assumptions of Theorem 2.3.3, there exists at least one maximizer to (2.43), and it holds that

$$ 
\begin{aligned}
\mathcal{D}_{L,X}^{\Phi}[\rho] &= \max_{\substack{v \in \operatorname{Span}\{\Phi\}, \\ \forall \Psi \in \mathcal{H}_1^m(X), \quad \langle \Psi | H_{m,X}^v | \Psi \rangle \ge 0}} \int_X v d\rho \\
&= \max_{\substack{v \in \operatorname{Span}\{\Phi\}, \\ \forall \Psi \in \operatorname{Span}\{\Psi_1, \dots, \Psi_K\}, \quad \langle \Psi | H_{m,X}^v | \Psi \rangle \ge 0}} \int_X v d\rho,
\end{aligned}
$$

where

$$ \Gamma_{\text{opt},X}^{\Phi} = \sum_{k=1}^{K} \omega_k |\Psi_k\rangle\langle\Psi_k| $$

for some $1 \le K \le M + 1$, with $\omega_k > 0$ and $\Psi_k \in \mathcal{H}_1^m(X)$ for all $1 \le k \le K$ is a minimizer of $F_{L,X}^{\Phi}$.

The proof of existence result for the dual problem exploits the fact that the problem can be re-written as semi-definite positive program, this is clear by looking at the constraint $\forall \Psi \in \operatorname{Span}\{\Psi_1, \dots, \Psi_K\}, \langle \Psi | H_{m,X}^v | \Psi \rangle \ge 0$, and then one can use classical existence results for this kind of problem.

## 2.4 Perspectives

Let us mention in this section some research perspectives:

* **Numerical methods for repulsive MMOT and Grand canonical OT:** As Explained throughout all this chapter the MMOT problem gives a rigorous lower bound of the electron-electron repulsion energy. However numerical simulations to compute the solution to MMOT cannot be afforded only for a very large number of electrons/marginals [BCN16; FP22; Alf+21] as need in computational chemistry. This implies that algorithms for Coulombic optimal transport are still unsuitable to be plugged in self-consistent field methods of computational chemistry. The idea consists in developing efficient numerical methods based on (1) the particle discretization proposed in the previous chapter (2) or on reduced order model approximation and to adopt them, as sub-routine, in order to compute the ground state of molecules and atoms. Moreover, we plan to generalize these numerical methods to the case of grand canonical optimal transport. Notice that the Grand Canonical OT allows to decompose a molecule in sub-domain (with a small number of electrons/marginals in order to bypass the curse of dimensionality) and compute the electronic configuration in each of them. We are interested in understanding better the computational error when one tries to approximate the total electron-electron repulsion with a sum of grand canonical optimal transport problems computed on each sub-molecules. Furthermore, it would be interesting establishing a way to decompose a molecule as already initiated in [Can+04; Gal+05].

* **Moments approximation of Lieb functional:** This sparsity structure we have proven in [EN23] leads, naturally to propose a numerical scheme in order to approximate the solution of the moments Lieb functional. The main idea is to develop an iterative scheme
which shares some common features with the GenCol algorithm [FP22; FSV22], in the
sense that, at each iteration, the "support" of the minimizer is adapted using the solution
of an associated dual problem.
# Chapter 3

## Unequal dimensional Optimal Transport

This chapter is devoted to the case in which we consider an Optimal Transport problem between two probability measures on space having different dimension. In particular we are going to summarize the results contained in the following papers:

* Section 3.1 L. Nenna and B. Pass. "Variational Problems Involving Unequal Dimensional Optimal Transport". In: *Journal de Mathématiques Pures et Appliquées* 139 (2020), pp. 83–108

* Section 3.2 L. Nenna and B. Pass. "Transport Type Metrics on the Space of Probability Measures Involving Singular Base Measures". In: *Applied Mathematics & Optimization* 87.2 (2023), p. 28

Before detailing the above papers let us briefly recall some basics facts of Optimal Transport involving unequal dimensional measures.

Given a bounded and continuous cost function $c: \bar{X}_1 \times \bar{X}_2 \to \mathbb{R}$ and two measures $\mu_1 \in \mathcal{P}(\bar{X}_1)$ and $\mu_2 \in \mathcal{P}(\bar{X}_2)$ defined on the closure of open bounded domains $X_1 \subseteq \mathbb{R}^{d_1}$ and $X_2 \subseteq \mathbb{R}^{d_2}$, then the optimal transport problem

$$ \mathrm{OT}^c(\mu_0, \mu_1) = \inf_{\gamma \in \Pi(\mu_1, \mu_2)} \int_{X_1 \times X_2} c(x_1, x_2) d\gamma(x_1, x_2) \quad (3.1) $$

always admits at least one solution. We will assume throughout this paper that $c \in C^2(\bar{X}_1 \times \bar{X}_2)$ satisfies the *twist condition*, which asserts that for each $x_1 \in X_1$

$$ x_2 \mapsto D_{x_1} c(x_1, x_2) \text{ is injective on } X_2, \quad (3.2) $$

as well as the *non-degeneracy condition*, asserting that the $d_1 \times d_2$ matrix $D_{xy}^2 c(x_1, x_2)$ of mixed second order partial derivatives has full rank for each $(x_1, x_2) \in \bar{X}_1 \times \bar{X}_2$ (as $d_1 > d_2$, this means $D_{x_1 x_2}^2 c(x_1, x_2)$ has rank $d_2$).

Before discussing the case $d_1 > d_2$, we remind that (3.1) admits a useful dual formulation

$$ \mathrm{OT}^c(\mu_1, \mu_2) := \sup_{(u,v) \in L^1(d\mu_1) \times L^1(d\mu_2)} \int_{X_1} u(x_1) d\mu_1(x_1) + \int_{X_2} v(x_2) d\mu_2(x_2). \quad (3.3) $$
Under mild conditions, there is a unique solution $(u, v)$ to the dual problem, up to the addition $(u, v) \mapsto (u + C, v - C)$ of constants adding to 0, known as the Kantorovich potentials, and these potentials are c-concave; that is, they satisfy

$$u(x_1) = v^c(x_1) := \min_{x_2 \in X_2} [c(x_1, x_2) - v(x_2)], \quad v(x_2) = u^c(x_2) := \min_{x_1 \in X} [c(x_1, x_2) - u(x_1)],$$

This solution is used to define generalized nestedness. The following definition is slightly adapted from [MP20].

**Definition 3.0.1.** When $d_1 > d_2$, we will say that the model $(c, \mu_1, \mu_2)$ satisfies the generalized nestedness condition if for $\mu_2$ almost every $x_2$ the potential $v$ is differentiable and we have

$$\begin{aligned} \partial^c v(x_2) &:= \{x_1 : u(x_1) + v(x_2) = c(x_1, x_2)\} \\ &= X_+(x_2, Dv(x_2)) := \{x_1 : Dv(x_2) = D_y c(x_1, x_2)\}. \end{aligned} \quad (3.4)$$

The containment $\partial^c v(x_2) \subseteq X_+(x_2, Dv(x_2))$ holds automatically throughout the domain of $Dv(x_2)$; it is therefore the opposite containment that distinguishes nested from non-nested models. The origin of the term nestedness lies in the one dimensional target setting, in which case the condition is equivalent to nestedness of certain super-level sets of $x_1 \mapsto \frac{\partial c}{\partial x_2}(x_1, x_2)$ [CMP17]. This is discussed in more detail below (see Proposition 3.0.2).

If both $\mu_2$ and $\mu_1$ are absolutely continuous, the potential $v$ satisfies the Monge-Ampere type equation almost everywhere [MP20]<fnref n="1" />:

$$\bar{\mu}_2(x_2) = \int_{\partial^c v(x_2)} \frac{\det(D_{x_2 x_2}^2 c(x_1, x_2) - D^2 v(x_2))}{\sqrt{|\det(D_{x_2 x_1}^2 c D_{x_1 x_2}^2 c)(x_1, x_2)|}} \bar{\mu}_1(x_1) d\mathcal{H}^{d_1-d_2}(x_1), \quad (3.5)$$

where $\bar{\mu}_1(x_1) := \frac{d\mu_1}{dx_1}(x_1)$ and $\bar{\mu}_2(x_2) := \frac{d\mu_2}{dx_2}(x_2)$ are the densities of $\mu_1$ and $\mu_2$. In general, this is a non-local differential equation for $v(x_2)$, since the domain of integration $\partial^c v(x_2)$ is defined using the values of $v$ and $u = v^c$ throughout $X_2$; however, when the model satisfies the generalized nestedness condition (namely $\partial^c v(x_2) = X_+(x_2, Dv(x_2)))$, it reduces to the local equation [MP20]:

$$\bar{\mu}_2(x_2) = \int_{X_+(x_2, Dv(x_2))} \frac{\det(D_{x_2 x_2}^2 c(x_1, x_2) - D^2 v(x_2))}{\sqrt{|\det(D_{y x}^2 c D_{x y}^2 c)(x_1, x_2)|}} \bar{\mu}_1(x_1) d\mathcal{H}^{d_1-d_2}(x_1). \quad (3.6)$$

**Multi-to one-dimensional optimal transport** We consider now the optimal transport problem in the case in which $d_1 > d_2 = 1$ (for more details we refer the reader to [CMP17]). In this case, generalized nestedness follows from a relatively simple condition, related to the following heuristic attempt to construct solutions to (3.1). Let us define the level and super-level sets of $D_y c$ as follows

$$X_+(x_2, k) := \{x_1 \in X_1 : \frac{\partial c}{\partial x_2}(x_1, x_2) = k\},$$

$$X_+(x_2, k) := \{x_1 \in X_1 : \frac{\partial c}{\partial x_2}(x_1, x_2) \geq k\},$$

<footnote n="1">¹Note that, here and below, our notation differs somewhat from [CMP17] and [MP20], since we have adopted the convention of minimizing, rather than maximizing, in (3.1).</footnote>
as well as the strict variant $X_>(x_2, k) := X_\ge(x_2, k) \setminus X_<(x_2, k)$. In order to build an optimal transport map $T$, we take the unique level set splitting the mass proportionately with $x_2$; that is, defining $k(x_2)$ such that

$$ \mu_1(X_\ge(x_2, k(x_2))) = \mu_2((-\infty, x_2]), \quad (3.7) $$

then we set $x_2 = T(x_1)$ for all $x_1$ which belong to $X_<(x_2, k(x_2))$. Notice that if there exists $x_1 \in X_1$ such that $x_1 \in X_<(x_2, k(x_2)) \cap X_<(x'_2, k(x'_2))$ then the map $T$ is not well-defined. The absence of such a degenerate case is equivalent to the super-level sets being nested; this is the definition of nestedness from [CMP17], which implies the more general Definition 3.0.1 when $d_2 = 1$, as the following result from [MP20] affirms.

**Proposition 3.0.2 (Nestedness for one dimensional targets).** The model $(c, \mu_1, \mu_2)$ satisfies the generalized nestedness condition if

$$ \forall x_2, x'_2 \text{ with } x'_2 > x_2, \mu_2([x_2, x'_2]) > 0 \implies X_\ge(x_2, k(x_2)) \subseteq X_>(x'_2, k(x'_2)) \quad (3.8) $$

We will say $(c, \mu_1, \mu_2)$ is *nested* if (3.8) is satisfied.

If the model $(c, \mu_1, \mu_2)$ is nested then [CMP17][Theorem 4] assures that $\gamma_T = (\mathrm{id}, T)_{\#}\mu_1$, where the map $T$ is built as above, is the unique minimizer of (3.1) in $\Pi(\mu_1, \mu_2)$. Moreover, the optimal potential $v(x_2)$ is given by $v(x_2) = \int_{-\infty}^{x_2} k(t)dt$, and so (3.6) becomes

$$ \bar{\mu}_2(x_2) = \int_{X_<(x_2, k(x_2))} \frac{D_{x_2 x_2}^2 c(x_1, x_2) - k'(x_2)}{|D_{x_1 x_2}^2 c(x_1, x_2)|} \bar{\mu}_1(x_1) d\mathcal{H}^{d_1-1}(x_1). \quad (3.9) $$

## 3.1 Some variational problems involving Unequal dimensional Optimal Transport

This Section is devoted to the study of functionals of the form

$$ \mathcal{J}(\mu_1, \mu_1) = \mathrm{OT}^c(\mu_1, \mu_2) + \mathcal{F}(\mu_2) + \mathcal{G}(\mu_1) \quad (3.10) $$

depending on probability measures $\mu_1$ and $\mu_2$. We are interested in characterizing the minimizers of $(\mu_1, \mu_2) \mapsto \mathcal{J}(\mu_1, \mu_2)$, as well as the minimizers of the subproblems obtained when either $\mu_1$ or $\mu_2$ is fixed: $\mu_2 \mapsto \mathcal{J}(\mu_1, \mu_2)$ and $\mu_1 \mapsto \mathcal{J}(\mu_1, \mu_2)$. Problems of these general forms, for various choices of the functionals $\mathcal{F}$ and $\mathcal{G}$, arise in a wide variety of applications, including: gradient flows on Wasserstein space (where the minimization $\mu_2 \mapsto \mathcal{J}(\mu_1, \mu_2)$ represents one step in a discrete gradient flow), displacement interpolation (when $\mathcal{F}$ is the Wasserstein distance to a second probability measure), Cournot-Nash equilibria in game theory, city planning problems, hedonic pricing in economics [BC14a; BC16; BC14b; BS05; AC11; CE10], and have consequently received a fair bit of attention in the literature. Most of the analytical progress so far, however, has been restricted to the case where the dimensions of the spaces $X_1$ and $X_2$ coincide, $d_1 = d_2$. In a wide variety of applications, particularly in economics and game theory, however, these dimensions may differ: in the Cournot-Nash problem, for example, $X_1$ parameterizes a space of agents, (differentiated by $d_1$ characteristics of an agent $x$) while $X_2$ represents a space of strategies. The dimensions of these spaces reflect the number of characteristics used to differentiate among agents, and the number of parameters involved in the choice of strategy, respectively, and need not be the same in general.
## 3.1. SOME VARIATIONAL PROBLEMS INVOLVING UNEQUAL DIMENSIONAL
OPTIMAL TRANSPORT

Minimizers of (3.10) when $d_1 = d_2$ have been studied extensively; for the subproblems where one marginal is fixed, under mild conditions on the functionals $\mathcal{F}$ and $\mathcal{G}$, existence and uniqueness of minimizers has been established, and, depending on the precise forms of $\mathcal{F}$ and $\mathcal{G}$, various regularity results and bounds on solutions exist. Our purpose is to study the minimizers of $\mathcal{J}$ when $d_1 > d_2$. As mentioned above when the target $X_2$ is unidimensional ($d_1 > d_2 = 1$), the *nestedness* condition allows one to construct almost closed form solutions from the cost function $c$ and marginals $\mu_1$ and $\mu_2$. For higher dimensional targets, ($d_1 > d_2 > 1$), an analogous condition ensures that a certain, generally non-local, partial differential equation on the lower dimensional space $X_2$ characterizing the solution is, in fact, local and degenerate elliptic [MP20]. Notice that the nestedness conditions are *joint* conditions on the cost $c$ and marginals $\mu_1$ and $\mu_2$, whereas in the present context, only the cost and **one** of the marginals (**neither** in the case of double minimizations) is prescribed; the other marginal is part of the solution to the problem. In [NP20] we prove here that, under various conditions on $c$, $\mu_1$ and $X_2$, $(c, \mu_1, \mu_2)$ is nested whenever $\mu_2$ minimizes $\mu_2 \mapsto \mathcal{J}(\mu_1, \mu_2)$, for a variety of different choices of the functional $\mathcal{F}$; analogous results for certain specific forms of $\mathcal{G}$ are also established for minimizations on the higher dimensional space, $\mu_1 \mapsto \mathcal{J}(\mu_1, \mu_2)$, and for double minimizations $(\mu_1, \mu_2) \mapsto \mathcal{J}(\mu_1, \mu_2)$. We go on to demonstrate that this a priori guarantee of nestedness makes the problem of characterizing or identifying the minimizers much more tractable; in different contexts, depending on the precise form of $\mathcal{F}$, we establish that solutions can be characterized by (local) differential equations, can be computed numerically by a convergent iterative scheme, or can be derived in almost closed form. The main results we establish in [NP20] are the following:

* **Characterization of nestedness:** Theorem 3.1.1 characterizes nestedness for a given model $(c, \mu_1, \mu_2)$ when the target is one dimensional ($d_2 = 1$). Its consequences include Corollaries 3.1.2 and 3.1.4, which give sufficient conditions for nestedness in terms of either a lower bound on $\mu_2$, which depends on $c$ and $\mu_1$, or an upper bound on $\mu_1$, depending on $c$ and $\mu_2$.

* **Nestedness for the congestion case:** Theorem 3.1.7 ensures nestedness of the model when the lower dimensional measure $\mu_2$ minimizes $\mu_2 \mapsto \mathcal{J}(\mu_1, \mu_2)$, $\mathcal{F}$ is a congestion term and the target is one dimensional;

* **Nestedness for the interaction case:** Theorem 3.1.10 provides sufficient conditions for generalized nestedness when the lower dimensional measure $\mu_2$ minimizes $\mu_2 \mapsto \mathcal{J}(\mu_1, \mu_2)$ (with the target dimension $n$ not necessarily one) and the functional $\mathcal{F}$ is composed of interaction and potential terms;

* **Best reply scheme:** Theorem 3.1.13 generalizes a previous result in [BC14b], providing convergence of the best reply scheme to compute minimizers of $\mu_2 \mapsto \mathcal{J}(\mu_1, \mu_2)$ when $\mathcal{F}$ is made of interaction and potential terms;

* **The high dimensional congestion case** Theorem 3.1.9 proves nestedness when the higher dimensional measure $\mu_1$ minimizes $\mu_1 \mapsto \mathcal{J}(\mu_1, \mu_2)$, $\mathcal{G}$ is a congestion functional and the target is one dimensional ($n = 1$);

* **The double minimization:** Theorem 3.1.16 provides conditions guaranteeing nestedness for minimizers $(\mu_1, \mu_2)$ of the double minimization problem $(\mu_1, \mu_2) \mapsto \mathcal{J}(\mu_1, \mu_2)$, when $\mathcal{F}$ is composed of interaction and potential terms, $\mathcal{G}$ is of congestion type, and the target is one dimensional;
• The hedonic pricing and Wasserstein barycenter: Theorem 3.1.19 considers the case in which the functional $\mathcal{F}(\nu) := \text{OT}^{c_2}(\mu_2, \nu)$ reflects the cost of optimal transport to a second fixed measure, as is the case in the hedonic pricing problem in economics, so that, after relabeling $\mu_1$ and $c$ from (3.10) as $\mu_1$ and $c_1$, the goal is to minimize $\nu \mapsto \text{OT}^{c_1}(\mu_1, \nu) + \text{OT}^{c_2}(\mu_2, \nu)$. When the target $\nu$ is one dimensional, the theorem implies that under a condition we call hedonic nestedness one can find solutions $\nu$ to this problem almost explicitly. Among other consequences, this condition ensures nestedness of $(c_i, \mu_i, \nu)$ for both $i=1$ and $2$.

### 3.1.1 A sufficient condition for nestedness

Fix $x_2 < x'_2$ (where $x_2, x'_2 \in Y$), $k_0 \in D_{x_2}c(X, x_2)$ and set

$$k_{\max}(x_2, x'_2, k_0) = \sup\{k : X_{\ge}(x_2, k_0) \subseteq X_{\ge}(x'_2, k)\}.$$

We then define the minimal mass difference, $D_{\mu_1}^{\min}$, as follows:

$$D_{\mu_1}^{\min}(x_2, x'_2, k_0) = \mu_1(X_{\ge}(x'_2, k_{\max}(x_2, x'_2, k_0)) \setminus X_{\ge}(x_2, k_0)).$$

The minimal mass difference represents the smallest amount of mass that can lie between $x_2$ and $x'_2$, and still have the corresponding level curves $X_-(x_2, k_0)$ and $X_+(x'_2, k_1)$ not intersect. In the following we assume that $d\mu_1(x) = \bar{\mu}_1(x_1)dx_1$ and $d\mu_2(x_2) = \bar{\mu}_2(x_2)dx_2$ are absolutely continuous with respect to the Lebesgue measure.

**Theorem 3.1.1 ([NP20])**. Assume that $\mu_1$ and $\mu_2$ are absolutely continuous with respect to Lebesgue measure. If $D_{\mu_1}^{\min}(x_2, x'_2, k(x_2)) < \mu_2([x_2, x'_2])$ for all $x_2 < x'_2$ where $k$ is defined by (3.7), then $(c, \mu_1, \mu_2)$ is nested. Conversely, if $(c, \mu_1, \mu_2)$ is nested, we must have $D_{\mu_1}^{\min}(x_2, x'_2, k(x_2)) \le \mu_2([x_2, x'_2])$ for all $x'_2 > x_2$.

The proof, essentially, follows from the definition of nestedness (3.8). The following sufficient condition is more convenient when one has bounds on the density $\bar{\mu}_2(x_2)$.

**Corollary 3.1.2 ([NP20])**. If for each $x_2 \in X_2$, we have

$$\sup_{x'_2 \in X_2, x_2 \le y \le x'_2} \left[ \frac{D_{\mu_1}^{\min}(x_2, x'_2, k(x_2))}{x'_2 - x_2} - \bar{\mu}_2(y) \right] < 0,$$

where $\bar{\mu}_2 = \frac{d\mu_2(x_2)}{dx_2}$, then $(c, \mu_1, \mu_2)$ is nested.

As a consequence, if the quantity $\frac{D_{\mu_1}^{\min}(x_2, x'_2, k_0)}{x'_2 - x_2}$ is bounded above for all $x_2 < x'_2$ and $k_0 \in D_y c(X, x_2)$, then a corresponding lower bound on $\bar{\mu}_2$ will ensure nestedness. We illustrate this with an example.

**Example 3.1.3.** Letting $\mu_1$ be uniform measure on the quarter disk so that $d\mu_1(x) = \frac{4}{\pi}dx$, $X_2 = (0, \bar{x}_2)$, with $\bar{x}_2 \le \frac{\pi}{2}$, parametrize a segment of the unit circle, and $c(x_1, x_2) = -(x'_1 \cos(x_2) + x'_1 \sin(x_2))$ the bilinear cost, we note that the level curves $X_-(x_2, k)$ are line segments parallel to the line segment joining $(0, 0)$ with $(\cos(x_2), \sin(x_2))$.

Now, for fixed $x_2 < x'_2$, $k_0$, the lines $X_-(x_2, k_0)$ and $X_+(x'_2, k_{\max}(x_2, x_2, k_0))$ intersect on the $x_2$ axis, if $k_0 < 0$ and the $x_1$ axis if $k_0 > 0$. In either case, the region $X_{\ge}(x'_2, k_{\max}(x_2, x'_2, k_0)) \setminus$
## 3.1. SOME VARIATIONAL PROBLEMS INVOLVING UNEQUAL DIMENSIONAL OPTIMAL TRANSPORT

$X_≥(x_2, k_0)$ is the part of the wedge of angle $x'_2 - x_2$ between the two lines which intersects $X_1$; this wedge is smaller than the corresponding wedge $X_≥(x'_2, 0) \setminus X_≥(x_2, 0)$, for which the intersection point is at the origin. Therefore,

$$D_{\mu_1}^{\min}(x_2, x'_2, k_0) \leq \mu_1(X_\geq(x'_2, 0) \setminus X_\geq(x_2, 0)) = (x'_2 - x_2) \frac{2}{\pi}$$

It therefore follows from Corollary 3.1.2 that the model $(c, \mu_1, \mu_2)$ is nested for any $d\mu_2 = \bar{\mu}_2(x_2)dx_2$ such that

$$\bar{\mu}_2(x_2) > \frac{2}{\pi}. \qquad (3.11)$$

When considering problems where the measure $\mu_2$ on $\mathbb{R}$ is fixed but the high dimensional marginal $\mu_1$ is not, the following reformulation is sometimes convenient; it implies that an appropriate upper bound on $\bar{\mu}_1$ yields nestedness.

For $k_0 \in D_{x_2}c(X_1, x_2)$, set

$$S_{\mu_1}^{\min}(x_2, x'_2, k_0) := X_{\ge}(x'_2, k_{\max}(x_2, x'_2, k_0)) \setminus X_{\ge}(x_2, k_0).$$

We then have:

**Corollary 3.1.4 ([NP20])**. Suppose that for all $x_2 \in X_2$

$$\sup_{x'_2 \in X_2, x_2 \le y \le x'_2} \left[ \|\bar{\mu}_1\|_{L^\infty(S_{\mu_1}^{\min}(x_2, x'_2, k(x_2)))} \frac{D_{vol}^{\min}(x_2, x'_2, k(x_2))}{x'_2 - x_2} - \bar{\mu}_2(x_2) \right] < 0.$$

Then $(c, \mu_1, \mu_2)$ is nested.

We now turn our focus to minimizing functionals of the form (3.10), and the subproblems obtained when one of the measures is fixed:

$$\min_{\mu_2 \in \mathcal{P}(\bar{X}_2)} \mathrm{OT}^c(\mu_1, \mu_2) + \mathcal{F}(\mu_2), \qquad (3.12)$$

and

$$\min_{\mu_1 \in \mathcal{P}(\bar{X}_1)} \mathrm{OT}^c(\mu_1, \mu_2) + \mathcal{G}(\mu_1). \qquad (3.13)$$

### 3.1.2 The congestion case

#### Bounds on the density

Let us consider (3.12) where $\mathcal{F}$ is of congestion, or internal energy, form: $\mathcal{F}(\mu_2) := \int_Y f(\bar{\mu}_2(x_2))dx_2$ with $f: [0, \infty) \to \mathbb{R}$ continuously differentiable on $(0, \infty)$, strictly convex with superlinear growth at infinity, satisfying,

$$\lim_{\bar{\mu}_2 \to 0^+} f'(\bar{\mu}_2) = -\infty.$$

A prototypical example is the entropy, $f(\bar{\mu}_2) = \bar{\mu}_2 \ln(\bar{\mu}_2)$. This is a popular type of functional in a variety of settings; in the Cournot-Nash case, it reflects agents' desires to choose strategies that are not too close to each other. Notice that the assumptions on $\mathcal{F}$ guarantees the existence of a minimizer of (3.12).
## 3.1. SOME VARIATIONAL PROBLEMS INVOLVING UNEQUAL DIMENSIONAL
OPTIMAL TRANSPORT

In the proposition below, we let

$$M_c := \sup_{(x_1,x_2,x'_2) \in (\bar{X}_1 \times \bar{X}_2 \times \bar{X}_2)} \frac{|c(x_1, x_2) - c(x_1, x'_2)|}{|(x_1, x_2) - (x_1, x'_2)|},$$

where $|\cdot|$ denotes the Euclidean norm, be a global Lipschitz constant for $x_2 \mapsto c(x_1, x_2)$ and for a bounded real valued function $v: X_2 \to \mathbb{R}$, denote by $K_v: (0, \infty) \to (-\infty, \infty)$ the inverse of the monotone increasing function $z \mapsto \int_Y (f')^{-1}(z - v(x_2))dx_2: (-\infty, \infty) \to (0, \infty)$.

For simplicity, we assume below that $0 \in \bar{X}_2$. The argument below is inspired by [San15, Section 7.4.1].

**Proposition 3.1.5 ([NP20]).** The minimizing $\mu_2$ in (3.12) is absolutely continuous with respect to Lebesgue, and its density $\bar{\mu}_2$ satisfies

$$(f')^{-1}(K_{-M_c|x_2|}(1) - M_c|x_2|) \le \bar{\mu}_2(x_2) \le (f')^{-1}(K_{M_c|x_2|}(1) + M_c|x_2|)$$

We remark that the bounds on the density $\bar{\mu}_2$ we have established above do not depend on the dimensions of $X_1$ and $X_2$. The lower bound is most relevant in (3.12) (since it is the lower bound on $\bar{\mu}_2$ that implies nestedness in the multi-to one-dimensional optimal transport problem, via Corollary 3.1.2); however, since the dimensions play no role, the result also applies to minimizers $\mu_1$ in problem (3.13), in which case the upper bound can be used to prove nestedness via Corollary 3.1.4 (again with $d_2 = 1$).

Before developing these applications, we illustrate how the result above can be used to find an explicit bound in an example.

**Example 3.1.6.** Recall the quarter disk to arc problem from Example 3.1.3: $\mu_1$ is uniform on the quarter disk, so that $d\mu_1(x_1) = \frac{4}{\pi}dx_1$, $Y = (0, \bar{x}_2)$, and the cost $c(x_1, x_2) = -\langle x_1, (\cos x_2, \sin x_2) \rangle$. We take $\mathcal{F}(\mu_2) = \int_X \bar{\mu}_2 \ln(\bar{\mu}_2)dx_2$ so that up to a non-vital constant $f'(\lambda) = \ln(\lambda)$. We get that $K_v$ is the inverse of $z \mapsto \int_0^{\bar{x}_2} \exp(-v(x_2))dx_2 = \exp(z) \int_0^{\bar{x}_2} \exp(-v(x_2))dx_2$; that is, $K_v(1) = \ln([\int_0^{\bar{x}_2} \exp(-v(x_2))dx_2]^{-1})$. Noting that $M_c = 1$, we have, for any minimizer $\mu_2$ of (3.12),

$$\bar{\mu}_2(x_2) \ge \exp\left(\ln\left([\int_0^{\bar{x}_2} \exp(x_2)dx_2]^{-1}\right) - x_2\right) = \frac{e^{-x_2}}{e^{\bar{x}_2} - 1}. \quad (3.14)$$

### The one dimensional target

We now turn our attention to proving that minimizers in (3.12) are nested. We begin by considering one dimensional targets and congestion (or internal energy) forms for $\mathcal{F}$. We consider (3.12), when the dimension of $X_2$ is $d_2 = 1$, and $\mathcal{F}[\mu_2] = \int_{X_2} f(\bar{\mu}_2)dx_2$, and $f$ satisfies the conditions above.

Combining Corollary 3.1.2 with the lower bound on the target density from Proposition 3.1.5 yields the following.

**Theorem 3.1.7 ([NP20]).** Suppose that $\mu_2$ minimizes (3.12) over $\mathcal{P}(\bar{X}_2)$, where $X_2 = (0, \bar{x}_2)$ and $X_1 \subseteq \mathbb{R}^{d_1}$. Then $(c, \mu_1, \mu_2)$ is nested provided

$$\sup_{x'_2 \in X_2, x_2 \le y \le x'_2} \frac{D_{\mu_1}^{\min}(x_2, x'_2, k(x_2))}{x'_2 - x_2} - (f')^{-1}(K_{-M_c y}(1) - M_c y) < 0$$

for all $x_2 \in X_2$.
## 3.1. SOME VARIATIONAL PROBLEMS INVOLVING UNEQUAL DIMENSIONAL OPTIMAL TRANSPORT

In particular, we note the following consequence for our example matching the quarter circle to an arc with the bilinear cost.

**Corollary 3.1.8 ([NP20]).** Suppose that $\mu_2$ minimizes (3.12) over $\mathcal{P}(\bar{X}_2)$, where $\mu_1$ is uniform measure on the quarter disk, $X_2 = (0, \bar{x}_2)$, $F[\mu_2] = \int_{X_2} \bar{\mu}_2 \ln(\bar{\mu}_2)dx_2$ is the entropy and $c(x_1, x_2) = -\langle x_1, (\cos x_2, \sin x_2) \rangle$. Then the model $(c, \mu_1, \mu_2)$ is nested provided $\bar{x}_2 \le \ln(\frac{1+\sqrt{(1+2\pi)}}{2}) \approx 0.61$.

Once we have determined a priori that the solution must be nested, we can characterize it by a differential equation on the lower dimensional space (ie, an ordinary differential equation). Note that, for a nested model, setting $k(x_2) = v'(x_2)$, one has, by (3.9),

$$ \bar{\mu}_2(x_2) = \int_{X=(x_2, k(x_2))} \frac{D_{x_2 x_2}^2 c(x_1, x_2) - k'(x_2)}{|D_{xy}^2 c(x_1, x_2)|} \bar{\mu}_1(x_1) d\mathcal{H}^{d_1-1} := G(x_2, k(x_2), k'(x_2)) $$

and so differentiating the first order condition $v(x_2) + f'(\bar{\mu}_2(x_2)) = C$, we get the following second order differential equation for $k$:

$$ k(x_2) + f''(G(x_2, k(x_2), k'(x_2))) \frac{d}{dx_2}[G(x_2, k(x_2), k'(x_2))] = 0. \quad (3.15) $$

In addition, we can derive boundary conditions for (3.15): since we know the solution is nested, we have $\lim_{x_2 \to 0^+} \mu_1(X_\ge(x_2, k(x_2))) = \lim_{x_2 \to 0^+} \mu_2(0, x_2) = 0$, so that $k(0) := \lim_{x_2 \to 0^+} k(x_2)$ exists, we have $\mu_1(X_\ge(0, k(0))) = 0$ and $X_\in(0, k(0))$ is tangent to $\partial X$. This, and a similar argument as $x_2$ approaches $\bar{x}_2$ suggests that we impose the boundary conditions:

$$ k(0) = \max_{x_1 \in \bar{X}_1} \frac{\partial c}{\partial x_2}(x_1, 0), \quad k(\bar{x}_2) = \min_{x_1 \in \bar{X}_1} \frac{\partial c}{\partial x_2}(x_1, \bar{x}_2). \quad (3.16) $$

Under the conditions of Corollary 3.1.2, any minimizer of (3.12) gives a solution to the above boundary value problem. Conversely, it is not hard to see that given a solution $k(x_2)$ to (3.15) with boundary conditions (3.16), then $\bar{\mu}_2(x_2) = G(x_2, k(x_2), k'(x_2))$ is a minimizer provided that $v(x_2) = \int_0^{x_2} k(y)dy$ is $c$-concave.

### The high dimensional source

We now consider problems where the high dimensional measure $\mu_1$ is allowed to vary, restricting to a fixed one dimensional target setting, $d_2 = 1$. Consider fixing $\mu_2$ and minimizing $\mu_1 \mapsto \text{OT}^c(\mu_1, \mu_2) + \mathcal{G}(\mu_1)$, where $\mathcal{G}(\mu_1) = \int_{X_1} g(\bar{\mu}_1(x_1))dx_1$ is a congestion type functional, with $g$ satisfying the conditions on $f$ we have established in the above sub-section the domains $X_1 \subseteq \mathbb{R}^{d_1}$ and $X_2 \subseteq \mathbb{R}$ is one dimensional. Combining Corollary 3.1.4 and Proposition 3.1.5, we immediately obtain the following.

**Theorem 3.1.9 ([NP20]).** Assume that for all $x_2 < y < x'_2$, and $x_1 \in X_\ge(x'_2, k_{max}(x_2, x'_2, k(x_2))) \setminus X_\ge(x_2, k(x_2))$ we have

$$ (g')^{-1}(K_{M_c|x_1}(1) + M_c|x_1|) < \frac{\bar{\mu}_2(y)(x'_2 - x_2)}{D_{vol}^{min}(x_2, x'_2, k(x_2))}. $$

where $M_c$ is a Lipschitz constant for $x_2 \mapsto c(x_1, x_2)$. Then the model $(c, \mu_1, \mu_2)$ is nested for any minimizer $\mu_1$.
## 3.1. SOME VARIATIONAL PROBLEMS INVOLVING UNEQUAL DIMENSIONAL OPTIMAL TRANSPORT

Note that the equality $\bar{\mu}_1(x_1) = (g')^{-1}(C - u(x_1))$, and the general fact that the potential $u(x_1)$ is Lipschitz implies that the optimal marginal $\bar{\mu}_1(x_1)$ is Lipschitz as well. This allows one to use Theorem 7.1 in [CMP17] to obtain interior $\mathbb{C}^{2,1}$ estimates on $v = u^c$. It is not clear to us whether this can be bootstrapped to obtain higher regularity.

### 3.1.3 The interaction case and fixed point characterization

We now go back to the minimization over the low dimensional marginal and consider the case where $\mathcal{F}$ consists of interaction and potential terms; that is, suppose that $\mu_2$ minimizes the following functional on $\mathcal{P}(\bar{X}_2)$:

$$ \mu_2 \mapsto \mathrm{OT}^c(\mu_1, \mu_2) + \int_{X_2} V(x_2) \mathrm{d}\mu_2(x_2) + \frac{1}{2} \int_{X_2} \int_{X_2} W(x_2, x'_2) \mathrm{d}\mu_2(x'_2) \mathrm{d}\mu_2(x_2) \quad (3.17) $$

where $\mu_1$ is a given probability measure on a set $X_1 \subseteq \mathbb{R}^{d_1}$, $X_2 \subseteq \mathbb{R}^{d_2}$ with $d_1 > d_2$ and the interaction term $W(x_2, x'_2) = W(x'_2, x_2)$ is symmetric. We will denote by $F_{V,W}[\mu_2]$ the first variation of $\mathcal{F}$; that is,

$$ F_{V,W}[\mu_2](x_2) := V(x_2) + \int_{X_2} W(x_2, x'_2) \mathrm{d}\mu_2(x'_2). $$

In this case, we do not generally expect lower bounds on the density $\bar{\mu}_2$, and so the results from the *Bounds on density* Section tell us little about the structure of solutions. However, under certain conditions, we will be able to use the optimality conditions directly to infer generalized nestedness, as we will see below.

Assume throughout this section convexity of $X_2$ and uniform convexity of $y \mapsto c(x, y) + V(y) + W(z, y)$ throughout $X_1 \times X_2 \times X_2$; that is, for all $x, y, z$, we have

$$ D_{x_2 x_2}^2 c(x_1, x_2) + D^2 V(x_2) + D_{x_2 x_2}^2 W(x'_2, x_2) \geq C > 0. \quad (3.18) $$

Also assume that for each $x_1 \in X_1$, $x'_2 \in X_2$ and $x_2 \in \partial X_2$ that

$$ \langle D_{x_2} c(x_1, x_2) + D V(x_2) + D_{x_2} W(x_2, x'_2), \mathbf{n}_{X_2}(x_2) \rangle \geq 0, \quad (3.19) $$

where $\mathbf{n}_{X_2}$ is the outward unit normal to $X_2$.

**Theorem 3.1.10 ([NP20]).** Under the uniform convexity (3.18) and outward gradient (3.19) assumptions, $(c, \mu_1, \mu_2)$ satisfies the generalized nestedness condition for any minimizer $\mu_2$ of (3.17). Furthermore, if $c$, $V$ and $W$ are $\mathbb{C}^k$ smooth (for any integer $k \geq 2$), then the optimal map between $\mu_1$ and $\mu_2$ is $\mathbb{C}^{k-1}$.

Generalized nestedness of the solution and [MP20] now combine to imply the following result:

**Corollary 3.1.11 ([NP20]).** Assume that $c$ is twisted and non-degenerate, and adopt the assumptions of Theorem 3.1.10. Then the minimizer $\mu_2$ is absolutely continuous and its density satisfies the following integral Monge-Ampere type equation almost everywhere.

$$ \bar{\mu}_2(x_2) = \int_{X_2(x_2, Dv(x_2))} \frac{\det[D_{x_2 x_2}^2 c - D^2 v(x_2)]}{\sqrt{\det[D_{x_2 x_1}^2 c D_{x_1 x_2}^2 c]}} \bar{\mu}_1(x_1) \mathrm{d}\mathcal{H}^{d_1-d_2}(x_1). \quad (3.20) $$
**Remark 3.1.12.** The integro Monge-Ampere operator appearing in (3.20) has regularity con-
trolled by a variety of quantities depending on $c$, $X_1$, $X_2$ and $\mu_1$ (see Theorem 11 in [MP20]).
Since the potential $v(x_2) = -V(x_2) - \int_{X_2} W(x_2, x'_2) d\mu_2(x'_2)$ is as smooth as $V$ and $W$ on the
support of $\mu_2$, (3.6) then yields regularity estimates on $\bar{\mu}_2(x_2)$.

Finally, we note that on the support of $\mu_2$, we can eliminate $v$ from (3.6) to obtain the
following partial differential equation for $\bar{\mu}_2(x_2)$:

$$
\bar{\mu}_2(x_2) = G(x_2, -DF_{V,W}[\mu_2](x_2), -D^2(F_{V,W}[\mu_2])(x_2)) \quad (3.21)
$$

where $F_{V,W}[\mu_2](x_2) = V(x_2) + \int_{X_2} W(x_2, x'_2) d\mu_2(x'_2)$ is linear in $\mu_2$ and

$$
G(x_2, p, Q) = \int_{X=(x_2, p)} \frac{\det[D_{x_2 x_2}^2 c(x_1, x_2) - Q]}{\sqrt{\det[D_{x_2 x_1}^2 c(x_1, x_2) D_{x_1 x_2}^2 c(x_1, x_2)]}}
$$

is the integro Monge-Ampere operator from [MP20].

Two complications, absent in the congestion case, arise here: first, the operator $F_{V,W}$ depends
non-locally on $\mu_2$, and so the PDE (3.21) is non-local, even though the model satisfies the
generalized nestedness condition, which eliminates potential non-locality arising from the integro
Monge-Ampere operator $G$ as in [MP20]. Second, we do not know the support of $\mu_2$ in advance,
only that it is a connected subset of $X_2$; we therefore cannot impose boundary conditions.
These issues are not artefacts of the unequal dimensional setting; they arise in equal dimensional
problems as well. Since they seem to make solving the problem via the PDE approach challenging,
they serve as good motivation for an iteration scheme, adapted from Blanchet-Carlier [BC14b]
and developed below.

**Fixed point characterization**

Noting that by differentiating with respect to $x_2$ the optimality condition for (3.17) we obtain

$$
D_{x_2}c(x_1, x_2) + DF_{V,W}[\mu_2](x_2) = 0, \quad (3.22)
$$

we denote by $B_{\mu_2} : X_1 \to X_2$ the map such that

$$
D_{x_2}c(x_1, B_{\mu_2}(x_1)) + DF_{V,W}[\mu_2](B_{\mu_2}(x_1)) = 0, \quad (3.23)
$$

which is well defined under conditions (3.19) and (3.18). Then, the scheme introduced in [BC14b]
consists in iterating the application defined as

$$
\mathcal{B}(\mu_2) := (B_{\mu_2})_{\sharp} \mu_1. \tag{3.24}
$$

The following Theorem establishes the existence of a unique fixed point $\mu_2^*$ of (3.24) which is a
solution to (3.17).

**Theorem 3.1.13.** (The best reply iteration scheme-unequal dimensional case) [[NP20]] Let $\mu_1 \in \mathcal{P}(X_1)$ and the application $\mathcal{B}: \mathcal{P}(X_2) \to \mathcal{P}(X_2)$ defined in (3.24). Assume that the transport cost $c(x_1, x_2)$ is uniformly convex in $x_2$, that is $D_{x_2 x_2}^2 c \ge \eta$ id with $\eta > 0$, $D_{x_1 x_2}^2 c$ has maximal rank and $F_{V,W}[\mu_2]$ satisfies the following hypothesis

$$
D^2 F_{V,W}[\mu_2] \geq \lambda \operatorname{id} \text{ in } X_2, \lambda > 0; \quad (3.25)
$$

$$
\mathcal{H}^{d_1-d_2} (B_{\mu_2}^{-1} (x_2)) \leq M \quad \forall x_2 \in X_2, M \in \mathbb{R}; \qquad (3.26)
$$

$$
JB_{\mu_2} \geq k > 0 \text{ in } X_1; \qquad (3.27)
$$

$$
\int_{X_2} |DF_{V,W}[\nu_1] - DF_{V,W}[\nu_0]| dx_2 \le CW_1(\nu_1, \nu_0) \quad (3.28)
$$
## 3.1. SOME VARIATIONAL PROBLEMS INVOLVING UNEQUAL DIMENSIONAL OPTIMAL TRANSPORT

where $JB_{\mu_2}$ is the $d_2$-dimensional Jacobian of $B_{\mu_2}$. Moreover, let $\mu_1 \in \mathcal{P}(X_1)$ absolutely continuous with respect to Lebesgue and such that $||\mu_1||_\infty MC < k(\eta + \lambda)$.

Then $\mathcal{B}$ is a contraction of $(\mathcal{P}(X_2), W_1)$, where we denote by $W_1$ the 1-Wasserstein distance (namely the Optimal Transport problem with the Monge cost) and the unique fixed point $\mu_2^*$ is solution to (3.17).

**Remark 3.1.14.** One can get rid of hypothesis eq. (3.27) by noticing that the Jacobian of $B_{\mu_2}$ depends on other quantities: $D^2F_{V,W}[\mu_2](x_2)$, $D_{x_2x_2}^2c(x_1, x_2)$ and $D_{x_1x_2}^2c(x_1, x_2)$.

**Remark 3.1.15 (The equal dimensional case).** When $d_1 = d_2$, the above proposition is an extension of [BC14b, Theorem 5.1] to the case in which a general cost function is involved. We highlight the map $B_{\mu_2}(x_1)$ is actually not explicit or simple to compute. However in the special case in which $c(x_1, x_2) = h(x_1 - x_2)$, with $h$ is strictly convex, the map $B_{\mu_2}(x)$ takes the form

$$B_{\mu_2}(x_1) = (\mathrm{id} + Dh^{-1}(-DF_{V,W}[\mu_2]))^{-1}(x_1).$$

So far we have assumed that the cost is double twisted, but we can avoid this assumption and notice that the co-area formula still holds. In this case we have that

$$\mathcal{B}(\mu_2)(x_2) = \int_{B_{\mu_2}^{-1}(x_2)} \frac{\mu_1(x_1)}{JB_{\mu_2}} d\mathcal{H}^0(x_1),$$

and, since $\mathcal{H}^0$ is simply the counting measure (under the non-degeneracy condition), this implies that (3.26) can be interpreted as a bound on the number of points in the pre-image of $B_{\mu_2}$.

### 3.1.4 The double minimization

Consider now the problem where neither measure is fixed, and where $\mathcal{G}(\mu_1) = \int_{X_1} g(\bar{\mu}_1(x_1))dx_1$ and $F_{V,W}$ have the forms in subsections 3.1.2 and 3.1.3, respectively. That is, consider the minimization problem

$$\inf_{(\mu_1, \mu_2) \in \mathcal{P}(\bar{X}_1) \times \mathcal{P}(\bar{X}_2)} \mathrm{OT}^c(\mu_1, \mu_2) + \mathcal{G}(\mu_1) + \int_{X_2} F_{V,W}[\mu_2] d\mu_2(x_2). \quad (3.29)$$

The results established above can be used to prove the following.

**Theorem 3.1.16 ([NP20])**. Adopt the assumptions on $X_1, X_2, c, V$ and $W$ from the previous section, and assume that $g$ satisfies the conditions in subsection 3.1.2. Then, whenever $(\mu_1, \mu_2)$ minimizes (3.29),

1. $(c, \mu_1, \mu_2)$ is nested.

2. $\mu_1$ is absolutely continuous with an everywhere positive density.

3. The optimal map $T$ is two degrees less smooth than $c, V$ and $W$, while the Kantorovich potential $u(x_1)$ and density $\bar{\mu}_1(x_1)$ are one degree less smooth than $c, V$ and $W$.

4. $\mu_2$ is absolutely continuous.
**3.1.5 The hedonic pricing problem**

In this section, we study the hedonic pricing problem found in [Eke05] and [CMN10]; economically, this problem involves matching distributions $\mu_1$ and $\mu_2$ of buyers and sellers on spaces $X_1 \subseteq \mathbb{R}^{d_1}$ and $X_2 \subseteq \mathbb{R}^{d_2}$ (both assumed bounded and open), with $m_1, m_2 \ge 1$, according to their preferences for goods in a space $Y$ (which we will assume is one dimensional). Mathematically, this amounts to taking $\mathcal{F}(\nu)$ to be the optimal transport distance to another fixed measure in (3.12) [Eke05][CMN10]. We therefore seek to minimize:

$$ \min_{\nu \in \mathcal{P}(\bar{Y})} \mathrm{OT}^{c_1}(\mu_1, \nu) + \mathrm{OT}^{c_2}(\mu_2, \nu), \qquad (3.30) $$

where the $\mu_i \in \mathcal{P}(X_i)$ are absolutely continuous probability measures on the $X_i$ and $Y \subseteq \mathbb{R}$. Each $\mathrm{OT}^{c_i}$ represents the optimal transport distance (3.1) between $\mu_i$ and $\nu$ with respect to a $\mathbb{C}^2$, non-degenerate cost function $c_i(x_i, y)$. We attempt to construct a solution by adapting the construction for the straight optimal transport problem in [CMP17] as follows:

Fix $y$. For each $M \in [0, 1]$, choose the unique $k_i = k_i(y, M)$ such that $\mu_i(X_{\ge}^i(y, k_i)) = M$, where

$$ X_{\ge}^i(y, k_i) := \{x_i \in X_i : D_y c_i(x_i, y) \ge k_i\}; $$

we adopt similar notation for the level sets $X_{\le}^i(y, k_i)$. Now consider the function $M \mapsto k_1(y, M) + k_2(y, M)$. The map is continuous and strictly decreasing.

**Lemma 3.1.17 ([NP20]).** Assume $y$ is in the interior of $Y$ and $y \in \mathrm{argmin}(c_1(x_1, y) + c_2(x_2, y))$ for some $(x_1, x_2) \in X_1 \times X_2$. Then the mapping $M \mapsto k_1(y, M) + k_2(y, M)$ has a unique 0.

We can then state the following definition of hedonic nestedness.

**Definition 3.1.18 (Hedonic nestedness).** Denote the zero from the preceding Lemma by $M(y)$. We say the problem (3.30) is hedonically nested if

$$ X_{\ge}^i(y, k_i(y, M(y))) \subseteq X_{\ge}^i(\bar{y}, k_i(\bar{y}, M(\bar{y}))) \qquad (3.31) $$

for $i = 1, 2$, whenever $y, \bar{y} \in Y$ with $y < \bar{y}$.

Note that this is equivalent to $M$ being the cumulative distribution function of a probability measure $\nu$ and $(c_i, \mu_i, \nu)$ being nested for $i = 1$ and $2$.

**Theorem 3.1.19 ([NP20]).** The problem is hedonically nested if and only if $M(y)$ is the cumulative distribution function of some probability measure $\nu$ and $(c_i, \mu_i, \nu)$ is nested for $i = 1, 2$. In this case, $\nu$ is optimal in (3.30).

**Remark 3.1.20 (Wasserstein barycenter).** Note that a similar construction will hold for the matching for teams problem from [CE10], where one minimizes $\nu \mapsto \sum_{i=1}^N \mathrm{OT}^{c_i}(\mu_i, \nu)$ over probability measures on $Y \subseteq \mathbb{R}$. Notice that in the case in which $c_i(x, y) = \lambda_i |x - y|^2$ with $\lambda_i \ge 0$ and $\sum_i \lambda_i = 1$, this problem can be read as an unequal dimensional version of the Wasserstein barycenters problem introduced in [AC11].

We end this section by noting that the nestedness of either one of the $(c_i, \mu_i, \nu)$ (implied, for instance, by hedonic nesting) implies that the solution $\nu$ vanishes at the boundary; in economic terms, this means that neither the lowest nor highest quality goods are exchanged in equilibrium.
**Corollary 3.1.21 ([NP20])**. Suppose $Y = (y, \bar{y})$ is an interval. Assume $(c_i, \mu_i, \nu)$ is nested for the optimal $\nu$, for either $i = 1$ or 2, and that the density $\bar{\mu}_i$ is bounded. Set $\underline{k} = \max_{x \in \bar{X}} D_y c_i(x, \underline{y})$ and $\bar{k} = \min_{x \in \bar{X}} D_y c_i(x, \bar{y})$.

If $\lim_{k \to \underline{k}^-} \mathcal{H}^{d_i-1}(X_{\underline{k}}^i(y, k)) = 0$, then the optimal density is zero at $\underline{y}$, $\bar{\nu}(\underline{y}) = 0$.

Similarly, if $\lim_{k \to \bar{k}^+} \mathcal{H}^{d_i-1}(X_{\bar{k}}^i(\bar{y}, k)) = 0$, then the optimal density is zero at $\bar{y}$, $\bar{\nu}(\bar{y}) = 0$.

The condition $\lim_{k \to \bar{k}^+} \mathcal{H}^{d_i-1}(X_{\bar{k}}^i(y, k)) = 0$ heuristically means that the first level set of $x \mapsto D_y c_i$ that intersects $\bar{X}_i$ does so in a lower dimensional way. Since this level curve is tangent to $\partial X_i$, this is generically true. Note that when $c_i(x, y) = \langle x, \alpha(y) \rangle$ for some curve $\alpha: \mathbb{R} \to \mathbb{R}^{d_i}$, the level sets are hyperplanes and so strict convexity of $X_i$, or in fact the slightly weaker condition that $\partial X_i$ has no $d_i - 1$ dimensional facets introduced in [FKM11], suffices.

## 3.2 A class of metrics involving singular measures

We now turn our attention to the results obtained in [NP23b] where we exploit the unequal dimensional optimal transport in order to define a new class of metrics involving singular measures. We will also see that these metrics help us to obtain new information about the variational problems we have studied above. Recall that given two probability measures $\mu_0$ and $\mu_1$ (we change a little bit the labelling of the measures with respect to the previous section for sake of simplicity in terms of notations) on a convex, bounded domain $X \subseteq \mathbb{R}^d$, the **Wasserstein distance** between them is defined as the infimal value in the Monge-Kantorovich optimal transport problem; that is,

$$ \mathrm{OT}^2(\mu_0, \mu_1) := \inf_{\pi \in \Pi(\mu_0, \mu_1)} \int_{X \times X} |x_0 - x_1|^2 d\pi(x_0, x_1) \quad (3.32) $$

Among the many important properties of the Wasserstein distance (reviewed in [San15; Vil09; Vil03] for example) is the fact that it is a metric on the set $\mathcal{P}(X)$ of probability measures on $X$. In turn, the geodesics induced by this metric, known as *displacement interpolants* and introduced in [McC97], play key roles in many problems, both theoretical and applied. Variants of displacement interpolants, known as *generalized geodesics* (introduced in [AGS04], see Definition 3.2.1 below), are a natural and important tool in the analysis of problems involving a fixed base measure $\nu \in \mathcal{P}(X)$ as the ones treated in the previous section.

**Definition 3.2.1 (Generalized geodesics).** A generalized geodesic with base measure $\nu$ from $\mu_0$ to $\mu_1$ is a curve $\mu_t$ in $\mathcal{P}(X)$ of the form $\mu_t = ((1-t)e_0 + te_1)\#\gamma$ for some $\gamma \in \mathcal{P}(X \times X \times X)$ where $\gamma_{yx_i} \in \Pi_{\mathrm{opt}}(\nu, \mu_i)$ for $i = 0, 1$ where $\Pi_{\mathrm{opt}}(\nu, \mu_i)$ is the set of optimal couplings between $\nu$ and $\mu_i$ with respect to optimal transport for the quadratic cost function.

It is natural in certain variational problems to consider *singular* base measures; for instance in game theory problems derived from spatial economics, $\nu$ may represent a population of players, parametrized by their location $y \in \mathbb{R}^2$ [BC16]; it is often the case that the population is essentially concentrated along a one dimensional subset, such as a major highway or railroad. Our goal here is to develop an appropriate framework for these: we study a metric on an appropriate subset of $\mathcal{P}(X)$ for each choice of base measure $\nu$, which we call the $\nu$-based Wasserstein metric (see Definition 3.2.2); essentially, this metric arises from minimizing the average squared distance among all couplings of $\mu_0$ and $\mu_1$ corresponding to generalized geodesics. Geodesics with respect to the $\nu$-based Wasserstein metric will always be generalized geodesics with respect to $\nu$, but, for different structures of $\nu$, very different geometry is induced. We pay special attention to the cases
## 3.2. A CLASS OF METRICS INVOLVING SINGULAR MEASURES

when $\nu$ concentrates on a lower dimensional submanifold of $\mathbb{R}^d$, in which case the problem has a natural interplay with the unequal dimensional optimal transport problem explored in [CMP17; MP20; NP20]. In the particular case when $\nu$ concentrates on a line segment, we show that our metric coincides with the layerwise-Wasserstein metric (see (3.34)) introduced in [KPS20] to analyze anisotropic data such as plants' root shapes. We establish the following results:

* We obtain three equivalent characterizations of the $\nu$-based Wasserstein metric, roughly speaking:

1. as an optimal transport problem restricted to couplings which are correlated along $\nu$ (we take this as the definition);

2. by optimally coupling conditional probabilities of $\mu_0$ and $\mu_1$ after disintegrating with respect to optimal transport to $\nu$;

3. as limits of multi-marginal optimal transport between $\mu_0$, $\mu_1$ and $\nu$.

* We establish uniqueness of the corresponding geodesics (although uniqueness of generalized geodesics for regular base measures was established in [AGS04], they are generally not unique when the base measure is singular).

* We study geodesic convexity of several functionals which play a key role in optimal transport research; many of these functionals were originally introduced by McCann [McC97], who established their displacement convexity. Using the standard terminology convexity along $W_\nu$ geodesics of potential energies, interaction energies and the Wasserstein distance to $\nu$ follow immediately from known results, whereas convexity along $W_\nu$ geodesics of the internal energy, under certain conditions, requires a new proof (convexity of the internal energy is known to hold along some, but not all, generalized geodesics [AGS04]). We note that this applies in particular to the layerwise-Wasserstein distance, yielding a far reaching improvement to [KPS20][Corollary 4.2]. We also show that when $\nu$ concentrates on a lower dimensional submanifold, the set of measures $\mu$ for which the model ($|x-y|^2$, $\mu, \nu$) satisfies a strengthening of the generalized nested condition (see Definition 3.0.1) is geodesically convex, see [NP23b][Proposition 33].

* We also introduce a class of metrics, said $\mu$-based Wasserstein metric, relevant in the case when the measure $\mu$ on $X \subset \mathbb{R}^d$ is fixed, and one would like to interpolate between measures on a fixed, lower dimensional submanifold $Y$. In particular we compare and interpolate between Kantorovich potentials on the $Y$ side in order to compare and interpolate between measures on $Y$.

* We, finally, show how the ideas introduced [NP23b] can be applied to prove convergence of computational methods to find equilibria in some game theoretic models. In particular we identify conditions under which equilibria are fixed points of a contractive mapping, implying uniqueness of the equilibrium (although this is easily deduced by other methods as well) and, perhaps more importantly, that it can be computed by iterating the mapping. This iteration had already been introduced as a method of computation by Blanchet-Carlier when $d=1$, but without a proof of convergence, and in higher dimensions, with a proof of convergence but for simpler interaction terms [BC14b]. We prove that the relevant mapping is a contraction with respect to a variant of the $\mu$-based Wasserstein metric.
### 3.2.1 The $\nu$-based Wasserstein metric

We now define our metric with base point $\nu$ as follows.

**Definition 3.2.2** ($\nu$-based Wasserstein metric). Let $\nu \in \mathcal{P}(X)$. For $\mu_0, \mu_1 \in \mathcal{P}(X)$, we define the $\nu$-based Wasserstein metric as

$$ W_{\nu}(\mu_1, \mu_0) := \inf_{\gamma \in \Gamma} \int_{X \times X \times X} |x_0 - x_1|^2 d\gamma(y, x_0, x_1), \qquad (3.33) $$

where $\Gamma := \{\gamma \in \mathcal{P}(X \times X \times X) | \gamma_{yx_i} \in \Pi_{\text{opt}}(\mu_i, \nu), i = 0, 1\}$.

**Remark 3.2.3.** This definition is closely related to the concept of linear optimal transport, introduced in [Wan+13]. The difference is that in linear optimal transport, a fixed optimal transport $\pi_{\mu_i} \in \Pi_{\text{opt}}(\nu, \mu_i)$ is selected for each $\mu_i$ (see equation (3) in [Wan+13]), whereas in the definition of $W_{\nu}$ one minimizes over the entire set $\Pi_{\text{opt}}(\nu, \mu_i)$. For $\mu_i \in \mathcal{P}_{\nu}^u(X)$, the two concepts clearly coincide, and it is on this set that $W_{\nu}$ yields a metric (see Lemma 3.2.4 below). Outside of this set, $W_{\nu}$ still yields a semi-metric, whereas linear optimal transport might be better described as defining a metric on the selected $\pi_{\mu_i} \in \Pi_{\text{opt}}(\nu, \mu_i)$, since it is dependent on these choices.

**Lemma 3.2.4** ([NP23b]). $W_{\nu}$ is a semi-metric on $\mathcal{P}(X)$. It is a metric on $\mathcal{P}_{\nu}^u(X)$.

The proof of the following Lemma is very similar to the proof that the classical Wasserstein metric is in fact a metric (see for example, [San15][Proposition 5.1]).

**Example 3.2.5.** We recall the main example used to introduce generalized geodesics in Section 9.2 of [AGS04]. Let $\nu$ be absolutely continuous with respect to $d$-dimensional Lebesgue measure on $X$. Then by Brenier's theorem [Bre91] there exist unique optimal couplings of the form $(\text{id}, \tilde{T}_i)_{\#\nu}$ in $\Pi_{\text{opt}}(\nu, \mu_i)$, and therefore the only measure $\gamma$ with $\gamma_{yx_i} \in \Pi_{\text{opt}}(\nu, \mu_i)$ for $i = 0, 1$ is $\gamma = (\text{Id}, \tilde{T}_0, \tilde{T}_1)_{\#\nu}$. We then have

$$ W_{\nu}^{2}(\mu_{0}, \mu_{1}) = \int_{X} |\tilde{T}_{0}(y) - \tilde{T}_{1}(y)|^{2} d\nu(y) $$

so that the metric space $(\mathcal{P}(X), W_{\nu})$ is isometric to a subset of the Hilbert space $L^2(\nu)$. Geodesics for this metric take the form $t \mapsto (tT_1+(1-t)T_0)_{\#\nu}$; these are the standard generalized geodesics found in, for example, Definition 7.31 of [San15].

**Example 3.2.6.** At the other extreme, suppose $\nu = \delta_y$ is a Dirac mass. Then for any coupling $\pi \in \Pi(\mu_0, \mu_1)$, the measure $\gamma = \delta_y \otimes \pi$ has $\gamma_{yx_i} = \delta \otimes \mu_i \in \Pi_{\text{opt}}(\nu, \mu_i)$. Since

$$ \int_{X \times X \times X} |x_0 - x_1|^2 d\gamma(y, x_0, x_1) = \int_{X \times X} |x_0 - x_1|^2 d\pi(x_0, x_1) $$

we have

$$ W_{\nu}^{2}(\mu_{0}, \mu_{1}) = \inf_{\pi \in \Pi(\mu_{0}, \mu_{1})} \int_{X \times X} |x_{0} - x_{1}|^{2} d\pi(x_{0}, x_{1}) $$

which is exactly the standard quadratic Wasserstein metric.

We are especially interested in the cases in between these extremes, when $\nu$ is singular with respect to Lebesgue measure but not a Dirac mass. One of the simplest such cases is the following example.
**Example 3.2.7** ([NP23b] | Proposition 16). Suppose that $\nu$ concentrates on a line segment and is absolutely continuous with respect to one dimensional Hausdorff measure. It then turns out that the $\nu$-based Wasserstein metric coincides with the layerwise-Wasserstein metric introduced in [KPS20] and defined as follows

$$d_{\text{LW}}^2(\mu_0, \mu_1) := \text{OT}^2(\mu_0^V, \mu_1^V) + \int_0^1 \text{OT}^2(\tilde{\mu}_0^l, \tilde{\mu}_1^l)dl \quad (3.34)$$

where the $\mu_i^V = (x = (x^1, x^2, ..., x^d) \mapsto x^1)_\# \mu_i$ are the vertical marginals of the $\mu_i$, $\tilde{\mu}_i$ are rescaled versions of the $\mu_i$, defined by $\tilde{\mu}_i = (F_{\mu_i}, \text{id})_\# \mu_i$ where $F_{\mu_i}$ is the cumulative distribution function of $\mu_i^V$, with the identity mapping $\text{id}$ applied on the last $d-1$ coordinates (so that $(F_{\mu_i}, \text{id})(x^1, x^2, ..., x^d) = (F_{\mu_i}(x^1), x^2, ..., x^d)$) and $\tilde{\mu}_i = \mu_i^l \otimes dl$ is disintegrated with respect to its uniform vertical marginal.

As we show below, $W_\nu$ is also related to the following multi-marginal optimal transport problem. Fix $\varepsilon > 0$ and set

$$\text{MOT}^{\varepsilon c}(\nu, \mu_0, \mu_1) := \inf_{\gamma \in \Pi(\nu, \mu_0, \mu_1)} \int_{X \times X \times X} [\varepsilon|x_0 - x_1|^2 + |x_0 - y|^2 + |x_1 - y|^2] d\gamma(y, x_0, x_1). \quad (3.35)$$

The following result establishes two different characterizations of the $\nu$-based Wasserstein metric.

**Theorem 3.2.8** ([NP23b]). The following holds

1.

$$W_\nu^2(\mu_1, \mu_0) = \inf_{\pi_i \in \Pi_{\text{opt}}(\mu_i, \nu), i=0,1} \int_X \text{OT}^2(\mu_0^y, \mu_1^y) d\nu(y),$$

where $\mu_i^y$ is the conditional probability given $y$ of the optimal coupling $\pi_i = \nu(y) \otimes \mu_i^y(x) \in \Pi_{\text{opt}}(\nu, \mu_i)$ between $\nu$ and $\mu_i$.

2. Furthermore, any weak limit point $\bar{\gamma}$ as $\varepsilon \to 0$ of minimizers $\gamma_\varepsilon$ of the multi-marginal problem (3.35) is an optimal coupling between $\mu_0$ and $\mu_1$ for the problem (3.33) defining $W_\nu$.

we can also establish the following $\Gamma$-convergence result relating $MM_\nu^\varepsilon$ and $W_\nu$.

**Proposition 3.2.9** ([NP23b]). The functional on $\mathcal{P}(X) \times \mathcal{P}(X)$ defined by

$$(\mu_0, \mu_1) \mapsto \mathcal{F}_\nu^\varepsilon(\mu_0, \mu_1) := \frac{1}{\varepsilon}\text{MOT}^{\varepsilon c}(\nu, \mu_0, \mu_1) - \frac{1}{\varepsilon}\text{OT}^2(\nu, \mu_0) - \frac{1}{\varepsilon}\text{OT}^2(\nu, \mu_0)$$

$\Gamma$-converges to $(\mu_0, \mu_1) \mapsto W_\nu^2(\mu_0, \mu_1)$ as $\varepsilon \to 0$ with respect to the product on $\mathcal{P}(X) \times \mathcal{P}(X)$ of the weak topology on $\mathcal{P}(X)$ with itself

We end this section on the properties of the $\nu$-based Wasserstein metric by the following corollary which characterizes the optimal solution in the case $d=2$ through the Knothe-Rosenblatt rearrangement.

**Corollary 3.2.10** ([NP23b]). Let $m=2$. Suppose $\nu$ is concentrated on the line segment $\{(t, 0) : t \in \mathbb{R}\}$ and is absolutely continuous with respect to 1-dimensional Hausdorff measure. Then the optimal rearrangement $\gamma$ in (3.33) satisfies $\gamma_{x_0x_1} = (\text{id}, G)_\# \mu_0$, where G is the Knothe-Rosenblatt rearrangement.
### 3.2.2 Structure of geodesics for the $\nu$-based Wasserstein metric

We note that geodesics for the metric $W_\nu$ are generalized geodesics with base $\nu$, according to the general definition in [AGS04]. When $\nu$ is sufficiently regular, generalized geodesics are uniquely determined by $\mu_0$ and $\mu_1$. In this case, $\Pi_{\text{opt}}(\nu, \mu_i)$ consists of a single measure concentrated on the graph of a function $\tilde{T}_i: X \mapsto X$ pushing $\nu$ forward to $\mu_i$; $\gamma = (\text{id}, \tilde{T}_0, \tilde{T}_1)_{\#\nu}$ is then the unique measure satisfying $\gamma_{yx_i} \in \Pi_{\text{opt}}(\nu, \mu_i)$, and therefore the unique coupling minimizing the integral in the definition of $W_\nu(\mu_0, \mu_1)$. If $\nu$ is singular, this reasoning does not apply; existence of minimizers in (3.33) follows immediately from standard continuity-compactness results, but they may be non-unique in general. Consider now the case when $\nu$ concentrates on some lower dimensional submanifold and take as cost function $c(x, y) = -\langle x, f(y) \rangle$ where $f \in \mathbb{C}^2(Y)$ is injective and non-degenerate, that is $D^2f(y)$ has full rank everywhere.

In this setting, if the $\mu_i$ are absolutely continuous with respect to Lebesgue, we obtain a uniqueness result as a consequence of Theorem 3.2.8 and Lemma 3.2.13. We also characterize geodesics with respect to our metric is this setting.

**Theorem 3.2.11 ([NP23b])**. Assume that both $\mu_i$ are absolutely continuous with respect to Lebesgue measure and that $\nu$ is supported on a smooth $n$-dimensional submanifold of $X$, and is absolutely continuous with respect to $n$-dimensional Hausdorff measure. Then the optimal coupling $\gamma$ in (3.33) is unique and its two-fold marginal $\gamma_{x_0x_1}$ is induced by a mapping $G: X \to X$ pushing $\mu_0$ forward to $\mu_1$. Letting $T_i$ denote the optimal maps from $\mu_i$ to $\nu$, the restriction of $G$ to $T_i^{-1}(y)$ is the optimal mapping pushing $\mu_0^y$ forward to $\mu_1^y$ for a.e. $y$, where, as in Theorem 3.2.8, $\mu_i(x) = \mu_i^y(x) \otimes \nu(y)$. This mapping takes the form $\nabla\varphi^y: X_1(y, Dv_0(y)) \to X_1(y, Dv_1(y))$, where $\varphi^y: X_1(y, Dv_0(y)) \to \mathbb{R}$ is a convex function defined on the $(d-n)$-dimensional affine subset $X_1(y, Dv_0(y))$.

We now turn to the study of minimizing $W_\nu$ geodesics.

**Proposition 3.2.12 ([NP23b])**. Suppose that $\pi_0, \pi_1 \in \Pi_{\text{opt}}(\nu)$. Then $\pi_t = ((1-t)e_0 + te_1, e_2)_{\#\gamma}$ is a minimizing geodesic for the $\nu$-based Wasserstein metric on couplings defined by

$$W_\nu(\mu_0, \mu_1) = \inf_{\pi_i \in \Pi_{\text{opt}}(\nu, \mu_i), i=0,1} \tilde{W}_\nu(\pi_0, \pi_1),$$

where

$$\tilde{W}_{\nu}(\pi_0, \pi_1) = \sqrt{\inf_{\gamma \in \mathcal{P}(X \times X \times X)|\gamma_{yx_i} = \pi_i, i=0,1} \int_{X \times X \times X} |x_0 - x_1|^2 d\gamma(y, x_0, x_1)} \quad (3.36)$$

$\gamma$ is optimal in (3.36), where $e_0(x_0, x_1, y) = x_0$, $e_1(x_0, x_1, y) = x_1$, and $e_2(x_0, x_1, y) = y$. Suppose that $\mu_0, \mu_1 \in \mathcal{P}_\nu^u$. Then the generalized geodesic $\mu_t = ((1-t)e_0 + te_1)_{\#\gamma}$ is a minimizing geodesic for the $\nu$ based Wasserstein metric for each optimal $\gamma$ in (3.33), provided that $\mu_t \in \mathcal{P}_\nu^u$ for all $t \in [0, 1]$.

We end this section on the structure of geodesic by giving a uniqueness result. Indeed, under much stronger conditions on the marginals $\mu_0$ and $\mu_1$ and the reference measure $\nu$, we are able to use Theorem 3.2.11 to show that the geodesic for the $W_\nu$ metric between $\mu_0$ and $\mu_1$ is unique.

**Lemma 3.2.13 (Structure of geodesics, [NP23b])**. Let $\mu_0, \mu_1 \in \mathcal{P}(X)$ and $\nu \in \mathcal{P}(Y)$ be absolutely continuous with respect to the $d$-dimensional and $n$-dimensional Lebesgue measure respectively. Then the geodesic $\mu_t$ between $\mu_0$ and $\mu_1$ is uniquely defined. The Kantorovich potential for optimal transport between $\mu_t$ and $\nu$ is $v_t = tv_1 + (1-t)v_0$, where $v_i: Y \to \mathbb{R}$ is the Kantorovich potential
between $\mu_i$ and $\nu$, for $i = 0, 1$. For almost every $y$, the conditional probability $\mu_t^y$, concentrated on $X_1(y, Dv_t(y))$, is the unique displacement interpolant between $\mu_0^y$ and $\mu_1^y$. It is absolutely continuous with respect to $(d - n)$-dimensional Hausdorff measure; denoting its density by $\bar{\mu}_t^y$, and the density of $\mu_t$ with respect to $\mathcal{L}^d$ by $\bar{\mu}_t$, we have,

$$ \bar{\mu}_t^y(x_t) = \frac{\bar{\mu}_t(x_t)}{JT_t^y(x_t)\bar{\nu}(y)} $$

where

$$ JT_t^y(x_t) := \frac{\sqrt{\det(Df^T Df(y))}}{\det[-D^2v_t(y) - x_t \cdot D^2f(y)]} $$

is the Jacobian of the optimal map $T_t$ from $\mu_t$ to $\nu$, evaluated at $x_t \in T_t^{-1}(y) \subseteq X_1(y, Dv_t(y))$ and $f$ is defined at the beginning of this section. The Brenier map between $\mu_0^y$ and $\mu_t^y$ is then given by $\nabla\varphi_t^y = (1-t)I + t\nabla\varphi^y$, where $\nabla\varphi^y: T_0^{-1}(y) \to T_1^{-1}(y)$ is the Brenier map between $\mu_0^y$ and $\mu_1^y$.

### 3.2.3 Geodesic convexity of functionals

We now turn our attention to certain convexity properties of $W_\nu$ geodesics and we start by studying the convexity of various functionals along the geodesics. We consider the functionals

$$ \mu \mapsto \mathrm{OT}^2(\mu, \nu) \qquad (3.37) $$

$$ \mu \mapsto \int_X V(x) d\mu(x) \qquad (3.38) $$

$$ \mu \mapsto \int_{X^2} W(x-z) d\mu(x) d\mu(z) \qquad (3.39) $$

and

$$ \mu \mapsto \begin{cases} \int U(\bar{\mu}(x))dx & \text{if } d\mu = \bar{\mu}(x)dx \text{ is a.c. wrt Lebesgue measure on } X \\ +\infty & \text{otherwise.} \end{cases} \qquad (3.40) $$

We are able show that these functionals are geodesically convex with respect to the metric $W_\nu$. Explicitly, geodesic convexity means that $\mathcal{F}(\mu_t)$ is a convex function of $t \in [0, 1]$, where $\mathcal{F}: \mathcal{P}(X) \to \mathbb{R}$ is any of (3.39), (3.40), (3.38) or (3.37). We will restrict our attention to $W_\nu$ geodesics $\mu_t$ of the form in Proposition 3.2.12; that is, $\mu_t = (te_0 + (1-t)e_1)_{\#}\gamma$, where $\gamma$ is optimal in (3.33), under the assumption that each $\mu_t \in \mathcal{P}_\nu^u(X)$, since in more general situations the existence of a minimizing $W_\nu$ geodesic in the metric space $\mathcal{P}_\nu^u$ joining $\mu_0$ and $\mu_1$ is not clear. We are particularly interested in the case when the endpoints $\mu_0$ and $\mu_1$ and reference measure $\nu$ satisfy the assumptions in Theorem 3.2.11, under which the existence of a unique geodesic follows from Lemma 3.2.13. Since the geodesics we consider for $W_\nu$ are always generalized geodesics, the following result follows immediately from Lemma 9.2.1 and Propositions 9.3.2 and 9.3.5 [AGS04].

**Proposition 3.2.14** ([NP23b]). The following functionals are convex along all minimizing $W_\nu$ geodesics of the form in Proposition 3.2.12, under the corresponding conditions:

1. (3.37) is geodesically 1-convex for any $\nu$.

2. (3.38) is geodesically convex if $V$ is convex. It is geodesically strictly convex if $V$ is strictly convex.
3. (3.39) is geodesically convex if W is convex. It is geodesically strictly convex along the subset of measures with the same barycenter if W is strictly convex.

The geodesic convexity of the internal energy is somewhat more involved; in particular unlike the other three forms, it does not hold for all generalized geodesics. Our proof, under additional restrictions on $\nu$, uses the structure of the transport map $G$ in a crucial way.

**Lemma 3.2.15 ([NP23b])**. Assume that the reference measure $\nu$ is absolutely continuous with respect to $n$-dimensional Hausdorff measure on a smooth $n$-dimensional submanifold $Y$ and that $\mu_0$ and $\mu_1$ are absolutely continuous with respect to Lebesgue measure on $X$. The internal energy at the interpolant is given by

$$ \int_X U(\bar{\mu}_t(x))dx = \int_Y \frac{1}{\bar{\nu}(y)} \int_{T_0^{-1}(y)} U\left(\frac{\bar{\mu}_0^y(x_0)\bar{\nu}(y)JT_t^y(\nabla\varphi_t^y(x_0))}{\det(D^2\varphi_t^y(x_0))}\right) \frac{1}{JT_t^y(\nabla\varphi_t^y(x_0))} \det(D^2\varphi_t^y(x_0))d\mathcal{H}^{d-n}(x_0)dy \quad (3.41) $$

From the previous Lemma it follows the geodesic convexity of the internal energy.

**Corollary 3.2.16 ([NP23b])**. Under the assumption in the preceding Lemma, the internal energy (3.40) is geodesically convex assuming $U(0) = 0$ and $r \mapsto r^m U(r^{-m})$ is convex and non-increasing.

Note that the condition that $r \mapsto r^m U(r^{-m})$ is convex and non-increasing is quite standard and goes back to the seminal paper by McCann [McC97].

### 3.2.4 The $\mu$-based Wasserstein metric

We consider now the case in which the base measure is actually the one in higher dimension. We can define a class of metrics on a lower dimensional space. More precisely, these metrics use the Kantorovich potentials $v(y)$ on the variable $y$ arising from optimal transport to a fixed reference measure $\mu(x)$ to compare measures on the $y$ variable. Notice that on the contrary the $W_\nu$ uses in part the Kantorovich potential $v(y)$ for optimal transport from a reference measure $\nu(y)$ on $Y$ to measures on $X$ in order to compare those free measures. Fix reference measures $\mu \in \mathcal{P}(X)$ and $\sigma \in \mathcal{P}(Y)$ on bounded domains $X \subseteq \mathbb{R}^d$ and $Y \subseteq \mathbb{R}^n$, with $n \le d$, absolutely continuous with respect to $m$ and $n$ dimensional Lebesgue measure, respectively. Let $c: \bar{X} \times \bar{Y} \to \mathbb{R}$ be a cost function which is $\mathbb{C}^1$ up to the boundary and satisfies the twist condition.

We let $\mathcal{P}_{ac,\sigma}(Y)$ be the set of probability measures on $Y$ which are absolutely continuous with respect to $\sigma$. For measures $\nu_0, \nu_1 \in \mathcal{P}_{ac,\sigma}(Y)$, we define

$$ \mathcal{W}_{\mu,\sigma,c,p}(\nu_0, \nu_1) := ||D\nu_0 - D\nu_1||_{L^p(\sigma)} \quad (3.42) $$

for some $p \in [1, \infty]$ where $v_i$ is the $c$-concave Kantorovich potential, that is, $c$-conjugate solutions to the dual problem of (3.1) corresponding to the optimal transport problem between $\mu$ and $\nu_i$ with cost function $c$.

**Proposition 3.2.17 ([NP23b])**. $\mathcal{W}_{\mu,\sigma,c,p}$ is a metric on $\mathcal{P}_{ac,\sigma}(Y)$.
To help explain the motivation behind this metric, it is useful to first consider the $m = n$ case with $c(x, y) = |x - y|^2$. The metric $W_{\mu,\sigma,c,2}$ is then similar to the $\nu$-based Wasserstein metric. Recall from Example 3.2.5 that in this case $W_{\nu}(\mu_0, \mu_1)$ between two measures in $\mathcal{P}(X)$ coincides with the $L^2$ metric on the optimal maps from $\nu$ to the $\mu_i$; on the other hand, $W_{\mu,\sigma,c,2}(\nu_0, \nu_1)$ for two measures in $\mathcal{P}(Y)$ coincides with the $L^2$ metric on the optimal maps from the $\nu_i$ to $\mu$. That is, $W_{\mu,\sigma,c,2}(\nu_0, \nu_1)$ compares gradients of potential on the opposite, rather than same, side of the problem from the reference measure. The case where $d > n$ is an attempt to generalize this idea in the same way that the $\nu$-based Wasserstein metric generalizes the $L^2$ metric in Example 3.2.5. The metric $W^*_{\mu,\sigma,c,p}$ arises naturally in certain variational problems among measures on $Y$ involving optimal transport to the fixed measure $\mu$ on $X$ like the one treated in Section 3.1.2, we will expand this viewpoint in the section below.

### 3.2.5 Fixed point characterization of solutions to variational problems

We consider a fixed measure $\mu$ on the bounded domain $X \subseteq \mathbb{R}^m$, absolutely continuous with respect to Lebesgue measure with density $\bar{\mu}$, bounded above and below, and a fixed domain $Y := (\underline{y}, \bar{y}) \subset \mathbb{R}$. We are interested in the minimization problem

$$ \inf_{\nu \in \mathcal{P}(Y)} \mathrm{OT}^c(\mu, \nu) + \int \bar{\nu}(y) \log(\bar{\nu}(y)) dy. \quad (3.43) $$

We recall that this problem belongs to a class arising in certain game theory problems, introduced in a series of papers by Blanchet-Carlier [BCN17; BC16; BC14b], in which the fixed measure $\mu$ represents a distribution of players and $Y$ a space of strategies; minimizing measures $\nu$ in (3.10) represent equilibrium distributions of strategies. In general, the dimensions of $d$ and $n$ of the spaces of players and strategies represent the number of characteristics used to distinguish between them. The case $d > n$ (including the case $n = 1$ treated here) are of particular interest, since it is often the case that **players are highly homogeneous**, whereas **the strategies available to them are less varied**. When $m = 1$ (so that both players and strategy spaces are one dimensional), Blanchet-Carlier introduced a characterization of minimizers as fixed points of a certain mapping. They then iterated that mapping as a way to compute solutions [BC14b]; however, they did not prove convergence of this scheme. Here we introduce a similar fixed point characterization for $d \ge 1$. Furthermore, we prove for any $m$, under certain assumptions, that the mapping is a contraction with respect to a strategically chosen metric, implying convergence of the iterative scheme; even for $d = 1$, this is a new result. The key to the proof is choosing a suitable metric; our choice is a slight variant of the metric $W_{\mu,\mathcal{L},c,1}$ introduced in Section 3.2.4. Choose $\underline{d} = \min_{x \in \bar{X}, y \in \bar{Y}} \frac{\partial c}{\partial \bar{y}}(x, y)$, $\bar{d} = \max_{x \in \bar{X}, y \in \bar{Y}} \frac{\partial c}{\partial \bar{y}}(x, y)$. Let $k \in L^1((\underline{y}, \bar{y}); [\underline{d}, \bar{d}])$, the set of $L^1$ functions on $(\underline{y}, \bar{y})$ taking values in $[\underline{d}, \bar{d}]$, and set $v_k(y) := \int_{\underline{y}}^{\bar{y}} k(s) ds$. By using optimality condition of (3.43) we can define the following mapping $L^1((\underline{y}, \bar{y}); [\underline{d}, \bar{d}]) \to \mathcal{P}^{ac}((\underline{y}, \bar{y}))$,

$$ k \mapsto \bar{\nu}_k(y) := \frac{e^{-v_k(y)}}{\int_{\underline{y}}^{\bar{y}} e^{-v_k(s)} ds}. \quad (3.44) $$

This is a probability density on $(\underline{y}, \bar{y})$ (we denote the corresponding measure by $\nu_k(y)$); moreover note that for a given $\nu \in \mathcal{P}^{ac}((\underline{y}, \bar{y}))$ the mass splitting property

$$ \mu(X_{\ge}^c(y, k)) = \nu(\underline{y}, y), $$
selects a unique $k = k(y) \in [\underline{d}, \bar{d}]$, for each $y \in (\underline{y}, \bar{y})$. $k(y)$ is then bounded and so clearly in $L^1((\underline{y}, \bar{y}); [\underline{d}, \bar{d}])$. We can then think of this process as defining a mapping $\mathcal{P}^{ac}((\underline{y}, \bar{y})) \to L^1((\underline{y}, \bar{y}); [\underline{d}, \bar{d}])$, $\nu \mapsto k(\cdot)$. Composing this mapping with (3.44) then yields a $F: L^1((\underline{y}, \bar{y}); [\underline{d}, \bar{d}]) \to L^1((\underline{y}, \bar{y}); [\underline{d}, \bar{d}])$. Explicitly, defining $\tilde{k} := F[k]$, we have

$$ \nu_k(\underline{y}, y) := \mu(X_{\ge}^{c}(\underline{y}, \tilde{k}(y))). \quad (3.45) $$

This approach let us define the following algorithm 2 to compute solution to (3.43). We can then

**Algorithm 2** Algorithm to compute the optimal $\nu$

**Require:** $k^{(0)}$

1: **while** $||k^{(l+1)} - k^{(l)}|| < \text{tol}$ **do**

2:

$$ \bar{\nu}^{(l)}(y) := \frac{e^{-\nu^{(l)}(y)}}{\int_{\underline{y}}^{\bar{y}} e^{-\nu^{(l)}(s)} ds}. $$

3: Compute $k^{(l)}$ by using the mass splitting $\mu(X_{\ge}^{c}(y, k)) = \nu^{(l)}((\underline{y}, y))$.

4: **end while**

characterize the minimizer of (3.43) as a fixed point of this mapping $F$.

**Lemma 3.2.18** ([NP23b]). Let $\nu$ be a minimizer of (3.10) such that $(c, \mu, \nu)$ is nested. Then $k(y) = v'(y)$ is a fixed point of $F$, where $v$ is the corresponding Kantorovich potential.

We can prove that, under certain conditions, $F$ is a contraction. The Banach fixed point theorem will then imply that $F$ has a unique fixed point. Under conditions ensuring nestedness of the minimizer, Lemma 3.2.18, will then ensure that that fixed point is the minimizer of (3.43). We need the following conditions on $X$, $c$ and $\mu$ to ensure $F$ is a contraction:

* $A := \max|D_x c_y(x, y)| < \infty$,

* $B := \min \bar{\mu}(x) > 0$,

* There is a $C > 0$ such that, for all $y \in Y$, $x \in X$, using $c_y(x, y)$ as a shorthand for $\frac{\partial c}{\partial y}(x, y)$,

$$ \mathcal{H}^{d-1}(X_1^c(y, c_y(x, y))) \geq C \min \{\mu(X_{\le}^c(y, c_y(x, y))), \mu(X_{\le}^c(y, c_y(x, y)))\}. \quad (3.46) $$

Let us discuss briefly the third condition above. For any $x, y$, setting $p = c_y(x, y)$, the level set $X_1^c(y, p)$ is an $d-1$ dimensional submanifold, dividing the region $X$ into the sub and super level sets $X_{\ge}^c(y, p)$ and $X_{\le}^c(y, p)$. The condition implies that if $X_1^c(y, p)$ is small, then at least one of $\mu(X_{\ge}^c(y, p))$ or $\mu(X_{\le}^c(y, p))$ must also be (quantitatively) small. In addition, we set

$$ H(y) = \max \left\{ \frac{|e^{(\bar{d}-2\underline{d})(y-\underline{y})} - 1|}{|1 - e^{-\bar{d}(y-\underline{y})}|}, \frac{|e^{(\bar{d}-2\underline{d})(\bar{y}-\underline{y})} - e^{(\bar{d}-2\underline{d})(y-\underline{y})}|}{|(e^{-\bar{d}(y-\underline{y})} - e^{-\bar{d}(\bar{y}-\underline{y})})|} \right\}. $$
**Theorem 3.2.19 ([NP23b])**. Assume the three conditions above, and that

$$ \frac{2A\bar{d}^3(1 - e^{-d(\bar{y}-y)})^2}{BC\bar{d}^2[1 - e^{-d(\bar{y}-y)}]^2|\bar{d} - 2\bar{d}|} \int_{\bar{y}}^{\bar{y}} H(y)dy < 1. $$

Then F is a contraction.

**Remark 3.2.20.** The condition appearing in the statement of the theorem looks complicated; however, all quantities except for B and C can be computed using the cost function alone (B involves the reference marginal $\mu$ while C involves both the cost and $\mu$). Since the factor in front of the integral and the function $H(y)$ are bounded, the limit as $\bar{y} - \bar{y} \to 0$ of the left hand side above is 0; therefore, the condition will always hold for sufficiently small intervals. We illustrate these points in a simple example below, where we compute each of these quantities explicitly.

**Remark 3.2.21 (Relationship to $\mu$-based metrics).** We show that F is a contraction with respect to the $L^1$ metric. Noting that the mapping $\nu \mapsto k$, where $k$ is defined by the mass splitting condition $\nu(y, y) = \mu(X_{\ge}(y, k(y)))$ is a bijection on the set of non-atomic measures $\nu \in \mathcal{P}(Y)$ (and in particular on the set $\mathcal{P}^{ac}(Y)$ of measures which are absolutely continuous with respect to Lebesgue measure $\mathcal{L}$), we can consider the $L^1$ metric to be a metric on $\mathcal{P}^{ac}(Y)$. If $(c, \mu, \nu)$ is nested, then $k(y) = v'(y)$ where $v$ is the Kantorovich dual potential. In this case, this metric is exactly the metric $W_{\mu, \mathcal{L}, c, 1}$ from Section 3.2.4.

Now combining Lemma 3.2.18, condition 3.1.2 ensuring nestedness of the solution, and the Banach Fixed Point Theorem, we get the convergence of the iterative algorithm to compute the solution we discussed above.

**Corollary 3.2.22 ([NP23b])**. Under the assumptions in Theorem 3.2.19 and 3.1.2, for any $k \in L^1((y, \bar{y}); [\bar{d}, \bar{d}])$, the sequence $k^{(l)}$ generated by algorithm 2 converges to a function $k_{fixed}$ whose anti-derivative is the Kantorovich potential corresponding to the unique minimizer $\nu$ of (3.43).

Let us conclude this section by showing some numerical results obtained by using the algorithm 2. In all the simulations we have taken the uniform density on the arc $(0, \frac{\pi}{6})$ as initial density $\nu^{(0)}$, $\mu$ is always the uniform on quarter disk and $f(\nu)$ is the entropy as above. The cost is given by $c(x, y) = |x_1 - \cos(y)|^2 + |x_2 - \sin(y)|^2$ we know that the model is nested provided $\bar{y} \le 0.61$ (which is exactly the case we consider for the numerical tests). Moreover, we have also considered an additional potential term (the convergence proof can be straightforward adapted to this simple variant), which will favour a concentration in a certain area of $(0, \frac{\pi}{6})$, so that the functional $\mathcal{F}$ has the form

$$ \mathcal{F}(\nu) = \int_{Y} f(\nu)dy + \int_{Y} V(y)d\nu(y), $$

where $V(y) = 10|y - 0.1|^2$. In Figure 3.1 we show the final density we have obtained (on the left) and the intersection between the level-set $X_{\ge}(y, k^*(y))$ for the final solution and the support of the fix measure $\mu$. Notice that, as expected, the level set $X_{\ge}(y, k^*(y))$ do not cross each other meaning that the model is nested.
<figure><img src="image_6.png" /><figcaption>Figure 3.1: (Left) Final density $v^*$. (Right) Intersection between the level-set $X_≥(y, k^*(y))$ for the final solution and the support of the fix measure $\mu$. These figures were firstly presented in a joint work with B. Pass [NP22].</figcaption></figure>

3.3 Perspectives

The research perspectives I would like to address in the future can be developed along the two
main axis of this chapter (1) the unequal dimensional optimal transport and (2) the class of
metrics involving singular measures.

* Concerning the former an interesting variational problem where optimal transport arises naturally is the principal agent's problem (see for instance [FKM11; Car01] as well as the seminal work by Rochet and Choné [RC98]) where the population of agents $\mu$ is given and the monopolist must choose the best strategies in order to maximise the utility. However, as in the Cournot-Nash case explained above, there is no need for the space of strategies to have the same dimension as the ones of the agents, actually it is natural to assume that it is one dimensional. It turns out then that an unequal dimensional optimal transport problems arises. As in the joint work with B. Pass it is natural to study if the model (which can suffers a lack of convexity) is nested in order to make the solutions more tractable and numerically computable.

* As for the class of metrics I would like to further develop the possible applications: as we have highlighted above the *v*-based metric is useful to compare/interpolate anisotropic measures (as it happens for the plants’ root [KPS20]) since the geodesic preserves this property of the measures. So it is natural to define a barycenter of anisotropic measures and study its property as well as the possible numerical methods to compute it. Finally, notice that this metric is in some sense a non-linear (and more difficult to compute numerically) version of the sliced Wasserstein metric. As for sliced Wasserstein ([Bon13]) one can study its relation with the standard Wasserstein metric and, up to make it numerically tractable, study the associated gradient flows: does the generated sequence converges to the solution of a specific PDE (in the same flavour as done in [JKO98])? Can be applied to understand better some machine learning problems as for the sliced counterpart [Bon+22] ?
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**Titre** : Sur quelques généralisations du problème du transport optimal

**Mots clés** : Transport optimal multi-matinales, TFD, Grand canonique, Fonctionnelle de Lieb, métriques, transport optimal

**Résumé** : Ce manuscrit est consacré à l'étude de quelques généralisations du problème de transport optimal, ainsi qu'à des applications dans les domaines de la finance mathématique, de la théorie des jeux et de la physique quantique. Plus précisément, nous commençons par étudier le problème du transport optimal multi-marginales (TOM) en affinant la théorie existante pour le cas unidimensionnel. Nous fournissons une caractérisation explicite des mesures de risque spectrales. Nous étudions également la régularisation entropique de MMOT et en particulier, nous dérivons des taux de convergence du coût entropique vers le coût multi-marginal lorsque le paramètre de régularisation tend vers zéro. De plus, dans le cadre du transport entropique, nous proposons une caractérisation du problème discret en utilisant une équation différentielle.

En ce qui concerne l'application du transport optimal à la théorie de la fonctionnelle de densité (TFD), nous proposons une généralisation du TOMM à l'ensemble canonique grand, c'est-à-dire que nous autorisons le nombre de marginales à varier. De plus, nous nous concentrons également sur une approximation parcimonieuse de la fonctionnelle de Lieb via des contraintes de moments. Enfin, nous étudions le problème du transport optimal entre dimensions différentes et certains problèmes variationnels issus de la théorie des jeux et de l'économie, tels que les équilibres de Cournot-Nash ou le problème de hedonic pricing. La dernière partie du manuscrit est consacrée au développement de la théorie de certaines métriques impliquant des mesures singulières.

**Title** : On some generalisations of Optimal Transport problem

**Keywords** : Multi-Marginal Optimal Transport, DFT, Grand Canonical, Lieb functional, metrics, Optimal Transport

**Abstract** : This manuscript is devoted to the study of some generalisation of Optimal Transport problem as well as some applications arising in Mathematical Finance, Game Theory and Quantum Physics. More specifically, we first consider the Multi-Marginal Optimal Transport (MMOT) problem by a careful refinement of the existing theory for the one dimensional case, we provide an explicit characterisation of spectral risk measures. We also study the entropic regularization of MMOT and in particular derive rate of convergence of the entropic cost to the multi-marginal cost as the regularization parameter vanishes. Moreover, in the framework of entropic transport we provide an ODE characterisation of the discrete problem.

Concerning the application of Optimal Transport to Density Functional Theory (DFT) we propose a generalisation of MMOT to the grand canonical ensemble, that is we allow the number of marginals to vary. Furthermore, we also focus on a sparse approximation of the Lieb functional via moments constraints. Finally, we study unequal dimensional Optimal Transport and some related variational problems arising in Game Theory and Economics as, for instance, Cournot-Nash equilibria or the hedonic pricing problem. The last part of the manuscript is then devoted to develop the theory of some metrics involving singular measures