The decomposition of optimal transportation problems with convex cost

Given a positive l.s.c. convex function $\mathtt c : \mathbb R^d \to \mathbb R^d$ 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 $\mathbb R^d$ into a family of disjoint sets $\{S^h_\mathfrak a\}_{h,\mathfrak a}$ together with the projection $\{O^h_\mathfrak a\}_{h,\mathfrak a}$ on $\mathbb R^d$ of proper extremal faces of $\mathrm{epi}\, \mathtt c$, $h = 0,\dots,d$ and $\mathfrak a \in \mathfrak A^h \subset \mathbb R^{d-h}$, such that - $S^h_\mathfrak a$ is relatively open in its affine span, and has affine dimension $h$; \item $O^h_\mathfrak a$ has affine dimension $h$ and is parallel to $S^h_\mathfrak a$; - $\mathcal L^d(\mathbb R^d \setminus \cup_{h,\mathfrak a} S^h_\mathfrak a) = 0$, and the disintegration of $\mathcal L^d$, $\mathcal L^d = \sum_h \int \xi^h_\mathfrak a \eta^h(d\mathfrak a)$, w.r.t. $S^h_\mathfrak a$ has conditional probabilities $\xi^h_\mathfrak a \ll \mathcal H^h \llcorner_{S^h_\mathfrak a}$; - the sets $S^h_\mathfrak a$ are essentially cyclically connected and cannot be further decomposed. \end{list} The last point is used to prove the existence of an optimal transport map. The main idea is to recast the problem in $(t,x) \in [0,\infty] \times \mathbb R^d$ 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.