The decomposition of optimal transportation problems with convex cost

Given a positive l.s.c. convex function $\mathtt c : \R^d \to \R^d$ and an optimal transference plane $\underline{\pi}$ for the transportation problem \begin{equation*}\int \mathtt c(x'-x) \pi(dxdx'), \end{equation*} we show how the results of \cite{biadan} on the existence of a \emph{Sudakov decomposition} for norm cost $\mathtt c= |\cdot|$ can be extended to this case. More precisely, we prove that there exists a partition of $\R^d$ into a family of disjoint sets $\{S^h_\a\}_{h,\a}$ together with the projection $\{O^h_\a\}_{h,\a}$ on $\R^d$ of proper extremal faces of $\epi\, \mathtt c$, $h = 0,\dots,d$ and $a \in \A^h \subset \R^{d-h}$, such that \begin{itemize} \item $S^h_\a$ is relatively open in its affine span, and has affine dimension $h$; \item $O^h_\a$ has affine dimension $h$ and is parallel to $S^h_\a$; \item $\mathcal L^d(\R^d \setminus \cup_{h,\a} S^h_\a) = 0$, and the disintegration of $\mathcal L^d$, $\mathcal L^d = \sum_h \int \xi^h_\a \eta^h(d\a)$, w.r.t. $S^h_\a$ has conditional probabilities $\xi^h_\a \ll \mathcal H^h \llcorner_{S^h_\a}$;\item the sets $S^h_\a$ are essentially cyclically connected and cannot be further decomposed. \end{itemize} The last point is used to prove the existence of an optimal transport map. The main idea is to recast the problem in $(t,x) \in [0,\infty] \times \R^d$ with an $1$-homogeneous norm $\bar{\mathtt c}(t,x) := t \mathtt c(- \frac{x}{t})$ and to extend the regularity estimates of \cite{biadan} to this case