2014/09/01 by Stefano Bianchini, Bianchini, Stefano, Mauro Bardelloni +1
Computer Science · Engineering · #28A50 #49Q20 #Aerospace Engineering and Control Systems #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Optimization and Mathematical Programming #Optimization and Variational Analysis
paper · pdf · doi:10.48550/arxiv.1409.0515
openalex publication_date 2014/09/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Given a positive l.s.c. convex function \mathtt c : \mathbb Rd → \mathbb Rd and an optimal transference plane \underlineπ for the transportation problem ∫ \mathtt c(x'-x) π(dxdx'), we show how the results of \citebiadan on the existence of a Sudakov decomposition for norm cost \mathtt c= |⋅| can be extended to this case. More precisely, we prove that there exists a partition of \mathbb Rd into a family of disjoint sets \Sh_\mathfrak a\h,\mathfrak a together with the projection \Oh_\mathfrak a\h,\mathfrak a on \mathbb Rd of proper extremal faces of epi \mathtt c, h = 0,…,d and \mathfrak a ∈ \mathfrak Ah ⊂ \mathbb Rd-h, such that - Sh_\mathfrak a is relatively open in its affine span, and has affine dimension h; \item Oh_\mathfrak a has affine dimension h and is parallel to Sh_\mathfrak a; - \mathcal Ld(\mathbb Rd ∖ ∪h,\mathfrak a Sh_\mathfrak a) = 0, and the disintegration of \mathcal Ld, \mathcal Ld = ∑h ∫ ξh_\mathfrak a ηh(d\mathfrak a), w.r.t. Sh_\mathfrak a has conditional probabilities ξh_\mathfrak a ≪ \mathcal Hh \llcornerSh_\mathfrak a; - the sets Sh_\mathfrak a are essentially cyclically connected and cannot be further decomposed. \endlist 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,∞] × \mathbb Rd with an 1-homogeneous norm \mathtt c(t,x) := t \mathtt c(- (x)/(t)) and to extend the regularity estimates of \citebiadan to this case.