Optimal Maps for the Multidimensional Monge-kantorovich Problem

@inproceedings{Gangbo1996OptimalMF,
  title={Optimal Maps for the Multidimensional Monge-kantorovich Problem},
  author={Wilfrid Gangbo and Andrzej Świe},
  year={1996}
}
Let μ1, · · · , μN be Borel, probability measures on R. Denote by Γ(μ1, · · · , μN ) the set of all N -tuples T = (T1, · · · , TN ) such that Ti : R → R (i = 1 · · · ,N) are Borel measurable and satisfy μ1[T −1 i (V )] = μi[V ], for all Borel V ⊂ R . The Multidimensional Monge-Kantorovich Problem investigated in this paper consists of finding S = (S1, · · · , SN ) ∈ Γ(μ1, · · · , μN ) minimizing 
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