The decomposition of optimal transportation problems with convex cost

Understand

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.