OPTIMIZATION PROBLEMS IN MASS TRANSPORTATION THEORY ...

Concentration case. The following factsin the concentration case hold:• If µ 0 and µ 1 are convex combinations(also countable) of Dirac masses, thenthey can be connected by a path γ(t) offinite minimal cost J .• If r > 1 − 1/N then every pair of probabilitiesµ 0 and µ 1 can be connected by apath γ(t) of finite minimal cost J .• The bound above is sharp. Indeed, if r ≤1 − 1/N there are measures that cannotbe connected by a finite cost path (forinstance a Dirac mass and the Lebesguemeasure).57