Gangbo, Wilfried; McCann, Robert J. Optimal maps in Monge’s mass transport problem. (English. Abridged French version) Zbl 0858.49002 C. R. Acad. Sci., Paris, Sér. I 321, No. 12, 1653-1658 (1995). Summary: Choose a cost function \(c({\mathbf x})\geq0\) which is either strictly convex in \(\mathbb{R}^d\), or a strictly concave function of the distance \(|{\mathbf x}|\). Given two nonnegative functions \(f,g\in L^1(\mathbb{R}^d)\) with the same total mass, we assert the existence and uniqueness of a map which is measure-preserving between \(f\) and \(g\), and minimizes the mass transport cost measured against \(c({\mathbf x}-{\mathbf y})\). An analytical proof based on the Euler-Lagrange equation of a dual problem is outlined. It assumes \(f,g\) to be compactly supported, and disjointly supported in the concave case. Cited in 1 ReviewCited in 32 Documents MSC: 49J10 Existence theories for free problems in two or more independent variables 49Q20 Variational problems in a geometric measure-theoretic setting Keywords:optimal maps; Monge’s mass transport problem; cost function; strictly convex; strictly concave; mass transport cost PDF BibTeX XML Cite \textit{W. Gangbo} and \textit{R. J. McCann}, C. R. Acad. Sci., Paris, Sér. I 321, No. 12, 1653--1658 (1995; Zbl 0858.49002) OpenURL