OPTIMIZATION PROBLEMS IN MASS TRANSPORTATION THEORY ...

Theorem Assume:• J is lower semicontinuous in X;• J ≥ c with c > 0, or more generally∫ +∞ (infB(r) J ) dr = +∞.0Then, for every x 0 , x 1 ∈ X there exists anoptimal path for the problem}min{J (γ) : γ(0) = x 0 , γ(1) = x 1provided there exists a curve γ 0 , connectingx 0 to x 1 , such that J (γ 0 ) < +∞.The application of the theorem above consistsin taking as X a Wasserstein spaceW p (Ω) where Ω is a compact metric spaceequipped with a distance function c anda positive finite non-atomic Borel measurem (usually a compact of R N with the Euclideandistance and the Lebesgue measure).53