Multi- to one-dimensional transportation

Fix probability densities $f$ and $g$ on open sets $X\subset {\mathbf{R}}^m$ and $Y\subset {\mathbf{R}}^n$ with $m\ge n\ge 1$. Consider transporting $f$ onto $g$ so as to minimize the cost $-s(x,y)$. We give a non-degeneracy condition (a) on $s\in C^{1,1}$ which ensures the set of $x$ paired with [$g$-a.e.] $y\in Y$ lie in a codimension $n$ submanifold of $X$. Specializing to the case $m>n=1$, we discover a nestedness criteria relating $s$ to $(f,g)$ which allows us to construct a unique optimal solution in the form of a map $F:X \longrightarrow \overline Y$\mathrm{.}When$s\in C^2 \cap W^{3,1}$and$\log f$and$\log g$arebounded,theKantorovichdualpotentials$(u,v)$satisfy$v \in C^{1,1}{loc}(Y)$,andthenormalvelocity$V$of$F^{-1}(y)$ with respect to changes in $y$ is given by $V(x)=v\mathrm{"}(f(x))-s_{yy}(x,f(x))$. Positivity (b) of $V$ locally implies a Lipschitz bound on $f$; moreover, $v\in C^2$ if $F^{-1}(y)$ intersects $\mathrm{\partial }X\in C^1$ transversally ©. On subsets where (a)-© can be be quantified, for each integer $r\ge 1$ the norms of $u,v \in C^{r+1,1}$and$F \in C^{r,1}$arecontrolledbythesebounds,$||\log f,\log g, \partial X ||{C^{r-1,1}}, ||\partial X||{C^{1,1}}$,$||s||{C^{r+1,1}}$,andthesmallnessof$F^{-1}(y)$. We give examples showing regularity extends from $X$ to part of $\bar{X}$, but not from $Y$ to $\bar{Y}$. We also show that when $s$ remains nested for all $(f,g)$, the problem in ${\mathbf{R}}^m\times \mathbf{R}$ reduces to a supermodular problem in $\mathbf{R}\times \mathbf{R}$.