On Sudakov's Type Decomposition of Transference Plans with Norm Costs

The authors consider the original strategy proposed by Sudakov for solving the Monge transportation problem with norm cost      

            with  ,   probability measures in     and   absolutely continuous w.r.t.    . The key idea in this approach is to decompose (via disintegration of measures) the Kantorovich optimal transportation problem into a family of transportation problems in      , where         are disjoint regions such that the construction of an optimal map         is simpler than in the original problem, and then to obtain   by piecing together the maps    . When the norm       is strictly convex, the sets     are a family of  -dimensional segments determined by the Kantorovich potential called optimal rays, while the existence of the map     is straightforward provided one can show that the disintegration of     (and thus of  ) on such segments is absolutely continuous w.r.t. the  -dimensional Hausdorff measure. When the norm       is not strictly convex, the main problems in this kind of approach are two: first, to identify a suitable family of regions       on which the transport problem decomposes into simpler ones, and then to prove the existence of optimal maps.

In this paper the authors show how these difficulties can be overcome, and that the original idea of Sudakov can be successfully implemented.

The results yield a complete characterization of the Kantorovich optimal transportation problem, whose straightforward corollary is the solution of the Monge problem in each set     and then in    . The strategy is sufficiently powerful to be applied to other optimal transportation problems.