OPTIMIZATION PROBLEMS IN MASS TRANSPORTATION THEORY ...

Diffusion case. The following facts in thediffusion case hold:• If µ 0 and µ 1 are in L q (Ω), then they canbe connected by a path γ(t) of finite minimalcost J . The proof uses the displacementconvexity (McCann 1997) which,for a functional F and every µ 0 , µ 1 , is theconvexity of the map t ↦→ F (T (t)), beingT (t) = [(1 − t)Id + tT ] # µ 0 and T an optimaltransportation between µ 0 and µ 1 .• If q < 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 q ≥1 + 1/N there are measures that cannotbe connected by a finite cost path (forinstance a Dirac mass and the Lebesguemeasure).59