The Kantorovich metric: initial history and little-known applications

We remind of the history of the transportation metric (Kantorovich metric) and the Monge–Kantorovich problem. We describe several little-known applications: the first one concerns the theory of decreasing sequences of partitions (tower of measures and iterated metric), the second one concerns Ornstein's theory of Bernoulli automorphisms ($\bar d$-metric), and the third one is the formulation of the strong Monge–Kantorovich problem in terms of matrix distributions.