OPTIMIZATION PROBLEMS IN MASS TRANSPORTATION THEORY ...

Problem 4 - Optimal Riemannian MetricsHere the domain Ω and the probabilities f +and f − are given, whereas the distance dis supposed to be conformally flat, that isgenerated by a coefficient a(x) through theformula{ ∫ 1d a (x, y) = inf a ( γ(t) ) |γ ′ (t)| dt :0}γ ∈ Lip(]0, 1[; Ω), γ(0) = x, γ(1) = y.We can then consider the cost functionalF (a) = MK(f + , f − , d a ).The goal is to prevent as much as possiblethe transportation of f + onto f − by maximizingthe cost F (a) among the admissiblecoefficients a(x). Of course, increasing a(x)would increase the values of the distance d a40