OPTIMIZATION PROBLEMS IN MASS TRANSPORTATION THEORY ...

and so the value of the cost F (a). The classof admissible controls is taken as{A = a(x) Borel measurable :∫}α ≤ a(x) ≤ β, a(x) dx ≤ m .In the case when f + = δ x and f − = δ yare Dirac masses concentrated on two fixedpoints x, y ∈ Ω, the problem of maximizingF (a) is nothing else than that of provingthe existence of a conformally flat Euclideanmetric whose geodesics joining x and y areas long as possible.This problem has several natural motivations;indeed, in many concrete examples,one can be interested in making as difficultas possible the communication betweenΩsome masses f + and f − .For instance, itis easy to imagine that this situation may41