  We study the regularity of solutions to an optimal transportation problem
where the dimension of the source is larger than that of the target. We
demonstrate that if the target is $c$-convex, then the source has a canonical
foliation whose co-dimension is equal to the dimension of the target and the
problem reduces to an optimal transportation problem between spaces with equal
dimensions. If the $c$-convexity condition fails, we do not expect regularity
for arbitrary smooth marginals, but, in the case where the source is
2-dimensional and the target is 1 dimensional, we identify sufficient
conditions on the marginals and cost to ensure that the optimal map is
continuous.
