The Decomposition of Optimal Transportation Problems with Convex Cost

Given a positive l.s.c. convex function c : Rd → R and an optimal transference plane π for the transportation problem ∫ c(x′ − x)π(dxdx′), we show how the results of [6] 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 {Sh a }h,a together with the projection {Oh a }h,a on Rd of proper extremal faces of epi c, h = 0, . . . , d and a ∈ Ah ⊂ Rd−h, such that • Sh a is relatively open in its affine span, and has affine dimension h; • Oh a has affine dimension h and is parallel to Sh a ; • Ld(Rd \∪h,aS a ) = 0, and the disintegration of Ld w.r.t. Sh a , Ld = ∑ h ∫ ξh a η h(da), has conditional probabilities ξh a HxSh a ; • the sets Sh a are essentially cyclically connected and cannot be further decomposed. By standard techniques, the last point can be 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 c̄(t, x) := tc(− t ) and to extend the regularity estimates of [6] to this case. Preprint SISSA 45/2014/MATE