On Sudakov’s type decomposition of transference plans with norm costs

Abstract

Abstract. We consider the original strategy proposed by Sudakov for solving the Monge transportation problem with norm cost | · | D * min |T(x) − x| D * dµ(x), T :with µ, ν probability measures in R d and µ absolutely continuous w.r.t. L d . 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 Za × R d , where {Za} a∈A ⊂ R d are disjoint regions such that the construction of an optimal map Ta : Za → R d is simpler than in the original problem, and then to obtain T by piecing together the maps Ta. When the norm | · | D * is strictly convex [25], the sets Za are a family of 1-dimensional segments determined by the Kantorovich potential called optimal rays, while the existence of the map Ta is straightforward provided one can show that the disintegration of L d (and thus of µ) on such segments is absolutely continuous w.r.t. the 1-dimensional Hausdorff measure [12]. 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 {Za} a∈A 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 Za and then in R d . 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 partitionswhich are relatively open in their k-dimensional affine hull and on which the transport occurs only along directions belonging to a cone C k a ; (2) the proof of the absolute continuity w.r.t. the suitable k-dimensional Hausdorff measure of the disintegration of L d 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.

Leaves decompositions in Euclidean spaces

“…Let us note that Theorem 6.2, described above, proves in particular that if we disintegrate the Lebesgue measure with respect to the partition obtained from a 1-Lipschitz map, then (intS, • , µ S ) will satisfy the curvature-dimension condition CD(0, n) for leaves S of dimension m. This complements the results of [1], [47]; see also [14], [15] and [10]. Note that our result tells in particular that the conditional measures are equivalent to the m-dimensional Hausdorff measure, which provides a strengthening of the previously known results.…”

“…The flaw has been remedied by Ambrosio in [1] and later by Trudinger and Wang in [48] for the Euclidean distance and by Caffarelli, Feldman and McCann in [13] for distances induced by norms that satisfy certain smoothness and convexity assumptions. In [14] Caravenna has carried out the original strategy of Sudakov for general strictly convex norms and eventually Bianchini and Daneri in [10] accomplished the plan of a proof of Sudakov for general norms on finitedimensional normed spaces. Let us note here also a paper [15], which deals with a related problem in the context of faces of convex functions.…”

Rectifiability of entropy defect measures in a micromagnetics model

“…The next theorem from [BD18] provides the existence of an L 1 -optimal map with respect to quite general distances on R N . Theorem 5.…”

Optimal transport of vector measures

“…The flaw has been remedied by Ambrosio in [1] and later by Trudinger and Wang in [24] for the Euclidean distance and by Caffarelli, Feldman and McCann in [5] for distances induced by norms that satisfy certain smoothness and convexity assumptions. In [6] Caravenna has carried out the original strategy of Sudakov for general strictly convex norms and eventually Bianchini and Daneri in [4] accomplished the plan of a proof of Sudakov for general norms on finite-dimensional normed spaces.…”