2013/11/08 by Stefano Bianchini, Bianchini, Stefano, Sara Daneri +1
Mathematics · #28A50 #49Q20 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Point processes and geometric inequalities
paper · pdf · doi:10.48550/arxiv.1311.1918
openalex publication_date 2013/11/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider the original strategy proposed by Sudakov for solving the Monge transportation problem with norm cost |⋅|D^* min \∫ |\mathtt T(x) - x|D^* dμ(x), \mathtt T : \mathbb Rd → \mathbb Rd, ν= \mathtt T_# μ\, with μ, ν probability measures in \mathbb Rd and μ absolutely continuous w.r.t. \mathcal Ld. 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 Z_\mathfrak a×\mathbb Rd, where \Z_\mathfrak a\\mathfrak a∈\mathfrak A ⊂ \mathbb Rd are disjoint regions such that the construction of an optimal map \mathtt T_\mathfrak a : Z_\mathfrak a → \mathbb Rd is simpler than in the original problem, and then to obtain \mathtt T by piecing together the maps \mathtt T_\mathfrak a. In this paper we show how 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 Z_\mathfrak a and then in \mathbb Rd. The strategy is sufficiently powerful to be applied to other optimal transportation problems.