Regularity of optimal transportation between spaces with different dimensions

We study the regularity of solutions to an optimaltransportation problem in which the dimension of the source islarger than that of the target. We prove that,\emph{unless} the cost $c$ has a very special form, (inwhich case we show that the problem can be reduced to anoptimal transportation problem between equal dimensionalspaces), there are smooth marginals for which the optimalmap is discontinuous. If $c$ does not have this specialform, we identify sufficient conditions on the cost and themarginals to ensure that the optimal map is continuous, inthe case where the target is one dimensional.

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Published 12 July 2012