2. Basic questions 9 with respect to Lebesgue measure: dμ(x) = f(x) dx, dν(y) = g(y) dy. Make the ansatz (in general unrealistic) that the T which we are looking for is a C1 diffeomorphism (from Rn to Rn, or more generally between smooth manifolds). Then, the reader can check, as an exercise, that the change of variable formula enables one to rewrite formula (10) as (15) f(x) = g(T (x)) | det ∇T (x)|. By usual standards this is a “highly nonlinear” constraint on T , very difficult to handle by the classical tools of calculus of variations. This is quite in contrast with the constraint (1) for the Kantorovich problem, which is linear. In the terminology of the calculus of variation, Kantorovich’s problem can be seen as a relaxed version of Monge’s problem. Relaxation here means that, starting from the Monge problem, one extends the class of objects on which the infimum is taken, by embedding the space of measurable functions T : X Y into the space L0(X P (Y )) of measurable functions with values in P (Y ). Explicitly, this embedding is defined by (16) I : T −→ [x δT (x)], with δT (x) standing for the Dirac mass at T (x) Y . Under this procedure, the nonlinear constraint on T formally turns into a linear constraint on I(T ) one can use this fact to define a minimization problem on L0(X P (Y )) with a linear constraint, and this new problem turns out to be Kantorovich’s problem. The very same idea has been popularized in the calculus of variations, and the embedding (16) is the starting point of the theory of Young mea- sures [255, 256, 257]. We refer to [14, 139] for a user-friendly comparison between the ideas of Kantorovich and those of Young. Using approxima- tion arguments inspired by the theory of Young measures, Gangbo [139, Appendix A] and Ambrosio [11, Theorem 2.1] proved under quite general assumptions (continuous cost function) that the values of the infima in both the Monge and the Kantorovich problems have to coincide, if μ has no atom, i.e. there is no x X with μ[{x}] 0. Their proof is based on the fact that the range of I is dense in L1(X P (Y )), equipped with the measure μ. Probably because of the evolution of mathematical feelings, nowadays it is Kantorovich’s problem which appears most natural (at least to the author), which is why we prefer to present Monge’s problem as a particular case of Kantorovich’s, rather than to introduce Kantorovich’s problem as a relaxed version of Monge’s.
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