OPTIMIZATION PROBLEMS IN MASS TRANSPORTATION THEORY ...

is a delicate matter. Indeed, maximizingsequences {a n } ⊂ A could develop an oscillatorybehavior producing only a relaxedsolution. This phenomenon is well known;basically what happens is that the class A isnot closed with respect to the natural convergencea n → a ⇐⇒ d an → d a uniformlyand actually it can be proved that A is densein the class of all geodesic distances (in particular,in all the Riemannian ones).Nevertheless, we were able to prove the followingexistence result.Theorem The maximization problem aboveadmits a solution in a opt ∈ A.43