On a Class of Multidimensional Optimal Transportation Problems

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Optimal transportation with infinitely many marginals (2012, background): “Now, using the result of Agueh and Carlier, we know that µ is absolutely continuous, ie dµ = g(x)dx, and it is well known that for each i ∈ B, ui solves the Monge-Ampere equation almost everywhere, detD2ui(x) = g(x) gi(Dui(x)) .”
Rearrangement inequalities for functionals with monotone integrands (2005, unknown): “2) on Ω = R [18].”
Multi-marginal optimal transport: theory and applications (2014, background): “The arguments in [12, 61] essentially showed that a splitting set S must be c-monotone, in the sense that if both (x2, .”
Multi-marginal optimal transport and multi-agent matching problems: uniqueness and structure of solutions (2012, result): “Unlike the results in [17][18][8] and [28], our argument here does not rely on a dual formulation of (MK).”
An Explicit Martingale Version of the One-Dimensional Brenier's Theorem with Full Marginals Constraint (2016, unknown): “The theory has been extended to the multiple marginals case by Gangbo and Świȩch [18], Carlier [8], Olkin and S.T. Rachev [41], Knott and Smith [36], Rüschendorf and Uckelmann [48], Heinich [20], and Pass [43, 44, 45], etc.”