Inserted: 12 apr 2013
Last Updated: 15 oct 2013
Journal: Ann. Mat. Pura Appl.
A standard question arising in Optimal Transport Theory is whether the Monge problem and the Kantorovich relaxation have the same infimum; the positive answer means that we can pass to the relaxed problem without loss of information. In the classical case with two marginals, this happens when the cost is positive, continuous, possibly infinite and the first marginal has no atoms. We study a similar multimarginal symmetric problem, arising naturally in density functional theory, motivated by a recent paper by Buttazzo, De Pascale and Gori Giorgi. The cost is the potential interaction between $n$ charged particles (hence it is symmetric, positive, continuous, and infinite whenever $x_i=x_j$) and the marginals are all equal with no atoms. We prove that also in this case there is equality between the infimum in the cyclical Monge problem (the natural Monge problem in this context) and in the classical Kantorovich problem. This result is new even for $2$ marginals, because we consider only transport maps which are involutions. The result is generalized to every symmetric continuous cost function on a Polish space.
Keywords: Optimal transport, Monge–Kantorovich problem, multimarginal problem