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Optimal transportation for the determinant. (English) Zbl 1160.49015

The authors consider the following problem: Given \(d\) probability measures \(\mu_1,\dots,\mu_d\) on \( {\mathbb R}^d \) and an objective function \(H: {\mathbb R}^d \times {\mathbb R}^d \to {\mathbb R} \), find a probability measure \(\gamma\) on \( {\mathbb R}^{d\times d} \) having \(\mu_1,\dots,\mu_d\) as marginals and maximizing \[ \int_{{\mathbb R}^{d\times d} } H (x_1,\dots,x_d) \, d\gamma(x_1,\dots,x_d) \,. \] The authors are interested in the specific objective functions \( H(x) = {\det}\,(x) \) and \(H(x) = |{\det}\,(x)|\). The study of the objective function \(H(x) = |{\det}\,(x)|\) is motivated by the problem of finding the best way to jointly choose three random vectors \(X,Y,Z \in {\mathbb R}^3\) with given marginal probabilities so that the simplex with vertices \(0\), \(X\), \(Y\), and \(Z\) has maximal average volume.
For the objective function \( H(x) = {\det}\,(x) \), the authors prove an existence result by imposing a natural requirement on the marginals. A duality argument underlies their method, but nontrivial constructions and arguments are required to apply the duality method.
When the objective function is the absolute value of the determinant, the authors show that with an additional symmetry hypothesis on the marginals, the solution of the first problem [i.e., \( H(x) = {\det}\,(x) \)] is also a solution of the second.

MSC:

49J55 Existence of optimal solutions to problems involving randomness
49Q20 Variational problems in a geometric measure-theoretic setting
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References:

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