Remarks on multi-marginal symmetric Monge-Kantorovich problems

Symmetric Monge-Kantorovich transport problems involving a cost function given by a family of vector fields were used by Ghoussoub-Moameni to establish polar decompositions of such vector fields into $m$-cyclically monotone maps composed with measure preserving $m$-involutions ($m\geq 2$). In this note, we relate these symmetric transport problems to the Brenier solutions of the Monge and Monge-Kantorovich problem, as well as to the Gangbo-\'Swi\c{e}ch solutions of their multi-marginal counterparts, both of which involving quadratic cost functions.


Introduction
The natural question here is the following: Problem (1): For which cost functions c, problem (K) admits a solutionθ that is supported on a "graph", that is a measure of the formθ = (I, T 1 , T 2 , ..., T m−1 ) # µ 0 for a suitable family of point transformations (T 1 , T 2 , ..., T m−1 ).
Whereas the case when m = 2 is already well understood, the problem when m ≥ 3 remains elusive since existence and uniqueness in (M) as well as uniqueness in (K) are still largely open for general cost functions. There is however one important case where this problem has been resolved by Gangbo andŚwiȩch [7]. This is when the cost function is given by Other extensions were also given by Pass [16,17,18,19] and by Carlier-Nazaret [4].
In this note, we are interested in the symmetric versions of the Monge-Kantorovich problem. They are of the following type.
c(x 0 , x 1 , ..., x m−1 )dθ; θ ∈ P sym (Ω m , µ) where c is an appropriate cost function and P sym (Ω m , µ) denotes the set of all probability measures on Ω m , which are invariant under the cyclical permutation σ(x 0 , x 1 , ..., x m−1 ) = (x 1 , x 2 , ..., x m−1 , x 0 ) and whose marginals are all equal to a given measure µ. Note that one can then assume that the cost function c is cyclically symmetric, since one can replace it by its symmetrizatioñ and in this case, one can minimize over the set P(Ω m , µ) of all probability measures on Ω m whose all marginals are equal to µ. Standard results for fairly general cost functions c show that there existsθ ∈ P sym (Ω m , µ), where the supremum above is attained. The natural question here is the following: Problem (2): For which cost functions c, problem (K sym ) admits as a solution a probability measureθ of the formθ = (I, S, S 2 , ..., S m−1 ) # µ, where S is a µ-measure preserving transformation on Ω such that S m = I a.e.
Problem (2) was resolved by Ghoussoub and Moameni for m = 2 in [13] and for m ≥ 3 in [14] in the case where the cost function is of the form where u 1 , ..., u m−1 are bounded vector fields from Ω → R d . Their work was in the context of establishing polar decompositions of vector fields in terms of monotone operators, which we will briefly describe below. The raison-d'être of this paper is however to make a link between the results of Gangbo andŚwiȩch dealing with the quadratic cost (1) but for marginals of the form µ i = σ i # µ for i = 0, ...., m − 1, and the symmetric Monge-Kantorovich problems considered by Ghoussoub and Moameni for the cost (4).

Polar decompositions
Recall that a vector field u : Ω → R d on a domain Ω in R d is said to be monotone on Ω if for all (x, y) in Ω, A result of E. Krauss [15] states that a map u : Ω → R d is monotone if and only if where H is a concave-convex anti-symmetric Hamiltonian on R d × R d . More recently, Galichon-Ghoussoub [6] extended Krauss' result to the case of m-cyclically monotone vector fields, where m is a fixed integer larger than 2. Recall that these are the maps u from Ω to R d that satisfy for any m + 1 points ( For that, Galichon and Ghoussoub consider the class H m (Ω) of all m-cyclically antisymmetric Hamiltonians on Ω m , that is the set where σ is the cyclical permutation σ(x 0 , x 1 , ..., x m−1 ) = (x 1 , x 2 , ..., x m−1 , x 0 ). They then show that if a vector field u is m-cyclically monotone, then there exists a Hamiltonian H ∈ H m (Ω) such that Moreover, H can be assumed to be concave in the first variable, convex in the last (m − 1) variables, though only m-cyclically sub-antisymmetric on Ω m , that is It is worth comparing the above to a classical theorem of Rockafellar [20], which yields that a single-valued map u from Ω to R d is a maximal cyclically monotone operator (i.e., satisfies (7) for every m ≥ 2), if and only if More remarkable is the polar decomposition that Y. Brenier [3] establishes for a general non-degenerate vector field, and which follows from his celebrated mass transport theorem. Recall that a mapping u : Ω → R d is said to be non-degenerate if the inverse image u −1 (N ) of every zero-measure N ⊆ R d has also zero measure. Brenier proved that any non-degenerate vector field u ∈ L ∞ (Ω, R d ) can be decomposed as with ϕ : R d → R being a convex function and S :Ω →Ω a measure preserving transformation. Later, and in the same spirit as Brenier's, Ghoussoub and Moameni established in [13] another decomposition for non-degenerate vector fields, which can be seen as the general version of Krauss' characterization of monotone operators. Assuming the boundary ∂Ω has measure zero, they show that if u ∈ L ∞ (Ω, R d ) is a nondegenerate vector field, then there exists a measure preserving transformation S : Ω → Ω such that S 2 = I (i.e., an involution), and a globally Lipschitz anti-symmetric concave-convex Hamiltonian H : In other words, up to a measure preserving involution, essentially every bounded vector field is monotone, where the latter correspond to when S is the identity map on Ω. More recently, Ghoussoub-Moameni [14] extended this result by showing that any family u 1 , ..., u m−1 of non-degenerate bounded vector field can be represented as where H ∈ H m (Ω) and S is a measure preserving m-involution (i.e., S m = I). Moreover, H could be replaced by a Hamiltonian that is concave in the first variable and convex in the other (m − 1)-variables, though only m-cyclically sub-antisymmetric. The proofs of the representations (11) (when m = 2) and of (12) (when m ≥ 3) rely on symmetric versions of the Monge problem and of its multi-marginal Monge-Kantorovich version as mentioned above. We shall give here another formulation that is closer to the original Monge and multi-marginal Monge-Kantorovich problems corresponding to a quadratic cost.

Mass transport with quadratic cost in the presence of symmetry
Given two probability measures with finite second moment µ 0 , µ 1 on R d , with X := support(µ 0 ) and Y := support(µ 1 ), the Wasserstein distance W 2 (µ 0 , µ 1 ) between them is defined by the formula where S(µ 0 , µ 1 ) is the class of all Borel measurable maps T : X → Y such that T # µ 0 = µ 1 , i.e., those which satisfy the change of variables formula, Whether the infimum describing the Wasserstein distance W 2 (µ 0 , µ 1 ) is achieved by an optimal mapT is a variation on the original mass transport problem of G. Monge, who inquired about finding the optimal way for rearranging µ 0 into µ 1 against the cost function c(x) = |x|. Our cost function here c(x) = 1 2 |x| 2 is quadratic, and the existence, uniqueness and characterization of an optimal map that we give below, was established by Y. Brenier.
Theorem 2.1 (Brenier) Assume µ 0 is absolutely continuous with respect to Lebesgue measure, then there exists a unique optimal mapT in S(µ 0 , µ 1 ), where the infimum in (13) is achieved. Moreover, the map T : X → Y is one-to-one and onto, µ 0 a.e., and is equal to ∇ϕ µ 0 a.e on X, for some convex function ϕ : R d → R. Moreover, the Brenier mapT is the unique map (up to µ 0 a.e. equivalence) of the form ∇ϕ with ϕ convex such that ∇ϕ # µ 0 = µ 1 . Now we consider the above theorem in the presence of symmetry.
Corollary 2.2 Let µ be a probability measure on R d that is absolutely continuous with respect to Lebesgue measure, and letμ be its image by a self-adjoint unitary transformation σ on R d (i.e., σ = σ * and σ 2 = I).
Then, there exists a convex function ϕ : where ϕ * is the Legendre transform of ϕ.
In the rest of this section, we try to connect the above corollary to the following polar decomposition established in [13].

Theorem 2.3 (Ghoussoub-Moameni)
Let Ω be an open bounded set in R d such that ∂Ω has zero Lebesgue measure.
is a non-degenerate vector field, then there exists a measure preserving transformation S :Ω →Ω such that S 2 = I (i.e., an involution), and a globally Lipschitz anti-symmetric concaveconvex Hamiltonian H : The involution S is obtained by solving the following variational problem sup Ω u(x), Sx dx; S is a measure preserving involution on Ω .
2. Moreover, u is a monotone map if and only if there is a representation as (16) with S being the identity.

Lagrangians and Hamiltonians associated to monotone maps
An important example of an involutive transformation is the transpose σ on R d × R d defined by σ(x, y) = (y, x), since the convex functions L on R d × R d satisfying L * (y, x) = L(x, y) for all (x, y) ∈ R d × R d are connected to central notions in nonlinear analysis and PDEs [11]. Duality theory is at the heart of this concept and it is therefore enlightening to describe it in the case where R d is replaced by any reflexive Banach space X. Recall from [11] the notion of a vector field∂L that is derived from a convex lower semi-continuous Lagrangian on phase space L : X × X * → R ∪ {+∞} in the following way: for each x ∈ X, the -possibly empty-subset∂L(x) of X * is defined as Here ∂L is the subdifferential of the convex function L on X × X * , which should not be confused with ∂L. Of particular interest are those vector fields derived from self-dual Lagrangians, i.e., those convex lower semi-continuous Lagrangians L on X × X * that satisfy the following duality property: where here L * is the Legendre transform in both variables, i.e., Such Lagrangians satisfy the following basic property: Moreover, which means that the associated vector field at x ∈ X is simplȳ These so-called selfdual vector fields are natural but far reaching extensions of subdifferentials of convex lower semi-continuous functions. Indeed, the most basic selfdual Lagrangians are of the form where ϕ is a convex and lower semi-continuous function on X, and ϕ * is its Legendre conjugate on X * , in which case∂ More interesting examples of self-dual Lagrangians are of the form where ϕ is as above, and Γ : X → X * is a skew adjoint operator. The corresponding selfdual vector field is then∂ Actually, it turned out that any maximal monotone operator A is a self-dual vector field and vice-versa [11]. That is, there exists a selfdual Lagrangian L such that A =∂L. This fact was proved and reproved by several authors. See for example, R. S. Burachik and B. F. Svaiter [2], B. F. Svaiter [22]. This result means that self-dual Lagrangians can be seen as the potentials of maximal monotone operators, in the same way as the Dirichlet integral is the potential of the Laplacian operator (and more generally as any convex lower semi-continuous energy is a potential for its own subdifferential). Check [11] to see how this characterization leads to variational formulations and resolutions of most equations involving monotone operators.
Proposition 2.1 Let u : Ω → R d be a possibly set-valued map. The following properties are then equivalent: 1. u is a maximal monotone map with domain Ω.

There exists a convex self-dual Lagrangian on
3. There exists a concave-convex anti-symmetric Hamiltonian H on R d × R d such that Sketch of proof: Assuming u is maximal monotone with domain Ω, we consider its associated Fitzpatrick function, that is It is known and easy to see that and that Now consider the following Lagrangian on R d × R d , which interpolates between N and N * , where the infimum is taken over all couples (x 1 , p 1 ) and (x 2 , p 2 ) such that It can be shown (see [11], p. 92) that L is a self-dual Lagrangian on R d × R d , in such a way that which means that if (x, p) ∈ Ω × R d , then L(p, x) = x, p if and only if (x, p) ∈ Graph(u), that is when p ∈ u(x).
To show that 2) implies 3), it suffices to take the Legendre transform of L with respect to the second variable, i.e., K L (x, y) = sup{ y, p − L(x, p); p ∈ R d }.
It is clearly concave-convex. The selfduality of L yields that K L is (at least) sub-antisymmetric, i.e., in Ω as in the theorem of Krauss mentioned above. Finally, assuming 3) that is u(x) ∈ ∂ 2 H(x, x), where H is convex in the second variable, we have for any x, y ∈ Ω, any p ∈ ∂ 2 H L (x, x) and q ∈ ∂ 2 H L (y, y) H(x, y) ≥ H(x, x) + p, y − x and H(y, x) ≥ H(y, y) + q, x − y .
Since H is anti-symmetric (hence H(x, x) = 0), this implies -by adding the two inequalities-that which means that u is monotone. Now we note that Monge transport problems provide a natural way to construct selfdual Lagrangians, hence general monotone operators. Corollary 2.4 Let µ be a probability measure on the product space Ω := Ω 1 × Ω 2 ⊂ R d × R d , and letμ be the probability measure onΩ := Ω 2 × Ω 1 obtained as the image of µ by the transformation σ(x, y) = (y, x). Then, there exists a self-dual Lagrangian L : has a solution θ that is supported on the self-dual Lagrangian manifold Proof: This is a direct application of the above corollary. The map ∇L then solves the Monge problem The self-duality of L follows from applying the above corollary to the transformation σ(x, y) = (y, x).

Involutions as a byproduct of mass transport between graphs
Suppose now that µ = (I ⊗ u) # dx, where dx is normalized Lebesgue measure on a bounded domain Ω in R d , and u : Ω → R d is a vector field in such a way that µ is supported by the graph G := {(x, u(x)); x ∈ Ω} and letμ = σ # µ. Now any map T pushing µ ontoμ can be parameterized by an application S : Ω → Ω via the formula: and the Monge problem between µ andμ can then be formulated as Assume now that -just like in the non-degenerate case-there exists a self-dual Lagrangian L such that ∇L # µ =μ. This means that there exists S : Ω → Ω such that ∇L(x, u(x)) = (u(Sx), Sx) for a.e. x ∈ Ω.
It follows that S is a measure preserving involution on Ω. In other words, problem (33) is equivalent to the problem sup Ω u(x), Sx dx; S measure preserving involution on Ω , which was used by Ghoussoub-Moameni to establish the polar decomposition (11). Actually, if one now considers the Legendre transform H of L with respect to the second variable, then as noted above, H is sub-anti-symmetric, can be assumed to be anti-symmetric, and satisfies, This then yields the polar decomposition u(x) = ∇ 2 H(x, Sx) a.e., which can be then seen as a (self-dual) mass transport problem between the measure µ supported by the graph of u and its transpose. Unfortunately, the measure µ is too degenerate to fall under the framework where we have uniqueness in Brenier's theorem, hence the need to give a direct proof of the result as in [13]. In order to link the polar decomposition with the symmetric Monge-Kantorovich problem, we note that measurable functions S : Ω → Ω, whose graphs {(x, Sx); x ∈ Ω} are the support of measures in P µ sym (Ω × Ω) can be characterized in the following way. Lemma 2.5 Let S : Ω → Ω be a measurable map, then the following are equivalent: 1. The image of µ by the map x → (x, Sx) belongs to P µ sym (Ω × Ω).
Proof. It is clear that 1) implies 3), while 2) implies 1). We now prove that 2) and 3) are equivalent. Assuming first that S is measure preserving such that S 2 = I a.e, then for every anti-symmetric H in hence , which clearly yields that S is an involution µ-almost everywhere.
We now give the following variational formulation for monotone operators, which shows that they are in some way orthogonal to involutions. Denote by S 2 (Ω, µ) the set of all µ-measure preserving involutions on Ω. It is an easy exercise to show that S 2 (Ω, µ) is a closed subset of a sphere of L 2 (Ω, R d ). In order to simplify the exposition, we shall assume that dµ is Lebesgue measure dx normalized to be a probability on the bounded open set Ω, and µ can and will then be dropped from all notation. We shall also assume that the boundary of Ω has measure zero.
Proposition 2.2 Let u : Ω → R d be a vector field in L 2 (Ω, R d ). The following properties are then equivalent: 1. u is monotone a.e. on Ω, that is there exists a measure zero set N such that the restriction of u to Ω \ N is monotone.
Proof: Assume u is a monotone map and use Proposition 2.1 to write it -modulo an obvious abuse in notation-as u(x) = ∇ 2 H(x, x) a.e. on Ω, where H is anti-symmetric and convex in the second variable. We can write for any measurable map S,

It follows that
Note that Ω H(x, S(x))dx = 0 since S is measure preserving and S 2 = I. Similarly, Ω H(x, x)dx = 0 and it then follows that Ω u(x), x − Sx dx ≥ 0. Finally, by taking Sx = x, one can then see that the supremum in 2) is equal to zero. Suppose now that 2) holds. In order to show 1) i.e., that u is monotone, consider a pair x 1 , x 2 in Ω and R small enough so that B(x i , R) ⊂ Ω for i ∈ {1, 2}. Define the measure preserving involution S R via Since Ω u(x), x − S R (x) dx ≥ 0, we have Since u is Lebesgue integrable, almost every point x ∈ Ω is a Lebesgue point, which means that u(x) = lim R→0 | B | −1 B u(x + Ry)dy. This leads to x 1 − x 2 , u(x 1 ) − u(x 2 ) ≥ 0 for a.e. x 1 , x 2 ∈ Ω. By developing the square, it is clear that property 2) is equivalent to 3), which says that the identity map is the projection of u on the closed subset S 2 (Ω, µ) of the sphere of L 2 (Ω, R d ), that is dist(u, S 2 (Ω, µ)) = u − I 2 .
Finally, note that 4) implies 2) by considering for any S ∈ S 2 (Ω, µ) the symmetric measure dπ on Ω × Ω that is the image of Lebesgue measure by the map x → (x, Sx). Note that Back to the case of a general vector field, the above then shows that the variational problem used in the polar decomposition sup Ω u(x), Sx dx; S measure preserving involution on Ω , is nothing but a symmetric Monge-Kantorovich problem sup{ Ω×Ω u(x), y dπ; π ∈ P sym (Ω × Ω, dx)}, where the cost function is given by c(x, y) = u(x), y . In the case where u is monotone then the involution where the supremum is attained is simply the identity.

Multidimensional Monge-Kantorovich Theorems
In this section, we are interested in relating the Gangbo-Świȩch solution of the multidimensional Monge-Kantorovich with quadratic cost to the following recent result of Ghoussoub-Moameni [14].
Theorem 3.1 Given a probability measure µ on Ω and bounded vector fields u 1 , u 2 , ...., u m−1 from Ω to R d that are µ-non-degenerate, then 1. The symmetric Monge-Kantorovich problem attains its maximum at a measure of the formθ = (I, S, S 2 , .., S m−1 ) # µ, where S is a µ-measure preserving transformation on Ω such that S m = I a.e.
2. There exists a Hamiltonian H ∈ H m (Ω) such that for i = 1, ..., m − 1, Moreover, H could be replaced by a Hamiltonian that is concave in the first variable and convex in the other variables, though only m-cyclically sub-antisymmetric.
and which maximizes the following dual problem In their seminal paper, Gangbo andŚwiȩch [7] dealt with the case of a quadratic cost function, and established the following remarkable result. 2. Each T i is one-to-one µ i -almost everywhere, is uniquely determined, and has the form and ϕ i is a convex function which is related to the solutions (u i ) m−1 i=0 of the dual problem (D) by the The above result was clarified further by Agueh and Carlier [1], who essentially established the following.

Proposition 3.1 (Agueh-Carlier)
Under the conditions of the Theorem 3.2 and with the same notation, we have for each i = 0, ..., m − 1, that 1 m ∇f i is the Brenier map that pushes the measure µ i onto the measure ν on R d which is the image of the optimal measureθ by the "barycentric map" (x 0 , ..., Moreover, ν is the unique minimizer of the functional ν → We now describe the situation in the case where the measures µ i are obtained from one measure µ by cyclic permutations.

m-cyclically monotone operators
We shall now apply the above corollary to the case where σ is the cyclic permutation σ(x 0 , ...., x m−1 ) = (x 1 , ..., x m−1 , x 0 ) on a product space X m . But first, we point to the connection with m-cyclically monotone operators studied recently by Galichon-Ghoussoub [6]. They proved the following which is obviously an extension of Krauss' theorem to the case when m ≥ 3. It is easy to see that the above is also equivalent to the statement that and that the sup is attained at the image of µ by the map x → (x, x, ..., x), which is nothing but a particular case of the symmetric Monge-Kantorovich problem, when the cost function is given by c(x 0 , ..., x m−1 ) = u(x 0 ), x m−1 , where u is an m-cyclically monotone operator.
Let now H be the Legendre transform of L with respect to the last two variables, that is H(x, y, z) = sup{ y, p + z, q − L(x, p, q); p ∈ R d , q ∈ R d }.