Precise solutions of the one-dimensional Monge–Kantorovich problem

The Monge–Kantorovich problem on finding a measure realizing the transportation of mass from $\mathbb R$ to $\mathbb R$ 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 $\mathbb R_+$. 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.