Precise solutions of the one-dimensional Monge-Kantorovich problem

The Monge-Kantorovich problem on finding a measure realizing the transportation of mass from R to R at minimum cost is considered. The initial and resulting distributions of mass are assumed to be the same and the cost of the transportation of the unit mass from a point x to y is expressed by an odd function f (x + y) that is strictly concave on R+. It is shown that under certain assumptions about the distribution of the mass the optimal measure belongs to a certain family of measures depending on countably many parameters. This family is explicitly described: it depends only on the distribution of the mass, but not on f. Under an additional constraint on the distribution of the mass the number of the parameters is finite and the problem reduces to the minimization of a function of several variables. Examples of various distributions of the mass are considered.