Indeed, we can write in the 2-marginals case, det(x1, x2) = 〈x1, Rx2〉, where R: ℝ2 → ℝ2 is the rotation of angle −π/2. Hence, since μ1 and μ2 have finite second moments, up to a change of variable x̃2 = Rx2, the problem (Det) in (5.1) is equivalent to the classical Brenier’s optimal transportation problem:

where

In the sequel, we are going to construct maximizers for (5.1), thanks to some properties of the Kantorovich potentials of the dual problem ...

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