OPTIMIZATION PROBLEMS IN MASS TRANSPORTATION THEORY ...
OPTIMIZATION PROBLEMS IN MASS TRANSPORTATION THEORY ...
OPTIMIZATION PROBLEMS IN MASS TRANSPORTATION THEORY ...
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We consider an abstract framework wherea metric space X with distance d is given.We assume that closed bounded subsets ofX are compact. Fix two points x 0 and x 1in X and consider the path functionalJ (γ) =∫ 10J(γ(t))|γ ′ |(t) dtwhere γ : [0, 1] → X ranges among allLipschitz curves such that γ(0) = x 0 andγ(1) = x 1 . Here:• J : X → [0, +∞] is a given mapping onX;• |γ ′ |(t) is the metric derivative of γ at thepoint t, i.e.|γ ′ |(t) = lim sups→td ( γ(s), γ(t) )|s − t|.52