  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.
