Multi- to one-dimensional transportation

Pierre-André Chiappori, Robert J McCann, Brendan Pass

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…