Pass, Brendan Regularity of optimal transportation between spaces with different dimensions. (English) Zbl 1270.49048 Math. Res. Lett. 19, No. 2, 291-307 (2012). Summary: 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 the marginals to ensure that the optimal map is continuous, in the case where the target is one dimensional. Cited in 6 Documents MSC: 49Q20 Variational problems in a geometric measure-theoretic setting 49N60 Regularity of solutions in optimal control Keywords:optimal transportation; regularity of solutions; smooth marginals PDFBibTeX XMLCite \textit{B. Pass}, Math. Res. Lett. 19, No. 2, 291--307 (2012; Zbl 1270.49048) Full Text: DOI arXiv