Symmetric Monge–Kantorovich problems and polar decompositions of vector fields

We address the problem of whether a bounded measurable vector field from a bounded domain Ω into $${\mathbb{R}^d}$$Rd is N-cyclically monotone up to a measure preserving N-involution, where N is any integer larger than 2. Our approach involves the solution of a multidimensional symmetric Monge–Kantorovich problem, which we first study in the case of a general cost function on a product domain ΩN. The polar decomposition described above corresponds to a special cost function derived from the vector field in question (actually N − 1 of them). The problem amounts to showing that the supremum in the corresponding Monge–Kantorovich problem when restricted to those probability measures on ΩN which are invariant under cyclic permutations and with a given first marginal μ, is attained on a probability measure that is supported on a graph of the form x → (x, Sx, S2x,..., SN-1x), where S is a μ-measure preserving transformation on Ω such that SN = I a.e. The proof exploits a remarkable duality between such involutions and those Hamiltonians that are N-cyclically antisymmetric.