The decomposition of optimal transportation problems with convex cost

الأصل · EN

Given a positive l.s.c. convex function c: Rᵈ → Rᵈ and an optimal transference plane π for the transportation problem equation* ∫ c(x'-x) π(dxdx'), equation* we show how the results of biadan on the existence of a Sudakov decomposition for norm cost c= |·| can be extended to this case. More precisely, we prove that there exists a partition of Rᵈ into a family of disjoint sets {Sʰₐ}ₕ,ₐ together with the projection {Oʰₐ}ₕ,ₐ on Rᵈ of proper extremal faces of epi c, h = 0,,d and a ∈ Aʰ ⊂ Rᵈ⁻ʰ, such that - Sʰₐ is relatively open in its affine span, and has affine dimension h; Oʰₐ has affine dimension h and is parallel to Sʰₐ; - Lᵈ(Rᵈ ∪ₕ,ₐ Sʰₐ) = 0, and the disintegration of Lᵈ, Lᵈ = ∑ₕ ∫ ξʰₐ ηʰ(da), w.r.t. Sʰₐ has conditional probabilities ξʰₐ ≪ Hʰ ₛₕₐ; - the sets Sʰₐ are essentially cyclically connected and cannot be further decomposed. list The last point is used to prove the existence of an optimal transport map. The main idea is to recast the problem in (t,x) ∈ [0,∞] × Rᵈ with an 1-homogeneous norm c(t,x):= t c(- x/t) and to extend the regularity estimates of biadan to this case.