We consider the original strategy proposed by Sudakov for solving the Monge transportation problem with norm cost | . |Dmin { |T(x) - x|D∗dμ(x), T : Rd → Rd, v = T#μ } , with μ, v probability measures in Rd and μ absolutely continuous w.r.t. 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 Zα Rd, where {Zα}αA Rd are disjoint regions such that the construction of an optimal map Tα : Zα → Rd is simpler than in the original problem, and then to obtain T by piecing together the maps Tα. When the norm ||D∗ is strictly convex, the sets Zα are a family of 1-dimensional segments determined by the Kantorovich potential called optimal rays, while the existence of the map Tα is straightforward provided one can show that the disintegration of Ld (and thus of μ) on such segments is absolutely continuous w.r.t. The 1-dimensional Hausdorff measure. When the norm ||D∗ is not strictly convex, the main problems in this kind of approach are two: first, to identify a suitable family of regions {Zα}αA on which the transport problem decomposes into simpler ones, and then to prove the existence of optimal maps. In this paper we 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 Zα and then in Rd. The strategy is sufficiently powerful to be applied to other optimal transportation problems. The analysis requires (1) the study of the transportation problem on directed locally affine partitions {Zkα , Ckα }k,α of Rd, i.e. sets Zkα Rd which are relatively open in their k-dimensional affine hull and on which the transport occurs only along directions belonging to a cone Ckα ; (2) the proof of the absolute continuity w.r.t. The suitable k-dimensional Hausdorff measure of the disintegration of Ld on these directed locally affine partitions; (3) the definition of cyclically connected sets w.r.t. a family of transportation plans with finite cone costs; (4) the proof of the existence of cyclically connected directed locally affine partitions for transport problems with cost functions which are indicator functions of cones and no potentials can be constructed.
On Sudakov's type decomposition of transference plans with norm costs
Daneri S
2018-01-01
Abstract
We consider the original strategy proposed by Sudakov for solving the Monge transportation problem with norm cost | . |Dmin { |T(x) - x|D∗dμ(x), T : Rd → Rd, v = T#μ } , with μ, v probability measures in Rd and μ absolutely continuous w.r.t. 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 Zα Rd, where {Zα}αA Rd are disjoint regions such that the construction of an optimal map Tα : Zα → Rd is simpler than in the original problem, and then to obtain T by piecing together the maps Tα. When the norm ||D∗ is strictly convex, the sets Zα are a family of 1-dimensional segments determined by the Kantorovich potential called optimal rays, while the existence of the map Tα is straightforward provided one can show that the disintegration of Ld (and thus of μ) on such segments is absolutely continuous w.r.t. The 1-dimensional Hausdorff measure. When the norm ||D∗ is not strictly convex, the main problems in this kind of approach are two: first, to identify a suitable family of regions {Zα}αA on which the transport problem decomposes into simpler ones, and then to prove the existence of optimal maps. In this paper we 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 Zα and then in Rd. The strategy is sufficiently powerful to be applied to other optimal transportation problems. The analysis requires (1) the study of the transportation problem on directed locally affine partitions {Zkα , Ckα }k,α of Rd, i.e. sets Zkα Rd which are relatively open in their k-dimensional affine hull and on which the transport occurs only along directions belonging to a cone Ckα ; (2) the proof of the absolute continuity w.r.t. The suitable k-dimensional Hausdorff measure of the disintegration of Ld on these directed locally affine partitions; (3) the definition of cyclically connected sets w.r.t. a family of transportation plans with finite cone costs; (4) the proof of the existence of cyclically connected directed locally affine partitions for transport problems with cost functions which are indicator functions of cones and no potentials can be constructed.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
