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 discontinu...

We study optimal mass transport problems between two measures with respect to a non-traditional cost function, i.e. c which can attain the value \(+\infty \). define notion of c-compatibility and strong measures, prove that if there is finite-cost plan then must be c-compatible, in addition are strongly an concentrated on c-subgradient c-class function. This function so-called potential plan. g...