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We study the regularity of solutions to an optimal transportation problem in which the dimension of the source is larger than that of the target. We prove that, unless the cost c has a very special form, (in which case we show that the problem can be reduced to an optimal transportation problem between equal dimensional spaces), there are smooth marginals for which the optimal map is discontinuous. If c does not have this special form, we identify sufficient conditions on the cost and thedoi:10.4310/mrl.2012.v19.n2.a3 fatcat:4f643cavwbecjmydiqp5ogkedy