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The Decomposition of Optimal Transportation Problems with Convex Cost
by Mauro Bardelloni   Stefano Bianchini

Vol. 11 No. 2 (2016) P.401~P.484

ABSTRACT

Given a non-negative 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 [6] 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 epi $\mathtt c$, $h = 0,\dots,d$ and $a \in \mathfrak{A}^h \subset \mathbb{R}^{d-h}$, such that

● $S^h_\mathfrak a$ is relatively open in its affine span, and has affine dimension $h$;
● $O^h_\mathfrak a$ 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 $\xi^h_\mathfrak a \ll \mathcal H^h \llcorner_{S^h_\mathfrak a}$;
(1) the sets $S^h_\mathfrak a$ are essentially cyclically connected and cannot be further decomposed.

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 [6] to this case.

KEYWORDS
Optimal transport, Sudakov decomposition, Monge problem.

MATHEMATICAL SUBJECT CLASSIFICATION 2010
Primary: 28A50, 49Q20.

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