OPTIMIZATION PROBLEMS IN MASS TRANSPORTATION THEORY ...

The Monge problem is thenmin{ ∫ déblais |x − T (x)| dx : T admiss. }.It is convenient to consider the Monge problemin the framework of metric spaces:• (X, d) is a metric space;• f + , f − are two probabilities on X(f + = “déblais”, f − = “remblais”);• T is an admissible transport map ifT # f + = f − .The Monge problem is then{ ∫minXd ( x, T (x) ) }dx : T admiss. .In general the problem above does not admita solution, when the measures f + andf − are singular, since the class of admissibletransport maps can be empty.2