Multi- to one-dimensional transportation

@article{Chiappori2015MultiTO,
  title={Multi- to one-dimensional transportation},
  author={P. Chiappori and R. McCann and B. Pass},
  journal={arXiv: Analysis of PDEs},
  year={2015}
}
Fix probability densities $f$ and $g$ on open sets $X \subset \mathbf{R}^m$ and $Y \subset \mathbf{R}^n$ with $m\ge n\ge1$. 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… Expand

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