The decomposition of optimal transportation problems with convex cost

Abstract

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) π(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 $\{S_{\mathfrak{a}}^h{\}}_{h,\mathfrak{a}}$ together with the projection $\{O_{\mathfrak{a}}^h{\}}_{h,\mathfrak{a}}$ 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 - $S_{\mathfrak{a}}^h$ is relatively open in its affine span, and has affine dimension $h$; \item $O_{\mathfrak{a}}^h$ has affine dimension $h$ and is parallel to $S_{\mathfrak{a}}^h$; - ${\mathcal{L}}^d({\mathbb{R}}^d\setminus {\cup }_{h,\mathfrak{a}}{S}_{\mathfrak{a}}^h)=0$, and the disintegration of ${\mathcal{L}}^d$, ${\mathcal{L}}^d={\sum }_h\int {\xi }_{\mathfrak{a}}^h{\eta }^h(d\mathfrak{a})$, w.r.t. $S_{\mathfrak{a}}^h$ has conditional probabilities ${\xi }_{\mathfrak{a}}^h\ll {\mathcal{H}}^h{└}_{S_{\mathfrak{a}}^h}$; - the sets $S_{\mathfrak{a}}^h$ 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.

48 pages, 16 figures