On a Class of Multidimensional Optimal Transportation Problems

We prove existence, uniqueness, duality results and give a characterization of optimal measure-preserving maps for a class of optimal transportation problems with several marginals with compact support in $\mathbb{R}$ under the requirement that the cost function satisfies a so-called monotonicity of order 2 condition. Explicit formulas for the minimizers are given and links with some rearrangement inequalities are sketched