Precise solutions of the one-dimensional Monge--Kantorovich problem
Sbornik. Mathematics, Tome 195 (2004) no. 9, pp. 1291-1307
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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.
@article{SM_2004_195_9_a3,
author = {A. Yu. Plakhov},
title = {Precise solutions of the one-dimensional {Monge--Kantorovich} problem},
journal = {Sbornik. Mathematics},
pages = {1291--1307},
publisher = {mathdoc},
volume = {195},
number = {9},
year = {2004},
language = {en},
url = {http://geodesic.mathdoc.fr/item/SM_2004_195_9_a3/}
}
A. Yu. Plakhov. Precise solutions of the one-dimensional Monge--Kantorovich problem. Sbornik. Mathematics, Tome 195 (2004) no. 9, pp. 1291-1307. http://geodesic.mathdoc.fr/item/SM_2004_195_9_a3/