Geodesic
Optimality Conditions for Smooth Monge Solutions of the Monge--Kantorovich problem
Funkcionalʹnyj analiz i ego priloženiâ, Tome 36 (2002) no. 2, pp. 38-44.

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The Monge–Kantorovich problem (MKP) with given marginals defined on closed domains $X\subset\mathbb{R}^n$, $Y\subset\mathbb{R}^m$ and a smooth cost function $c\colon X\times Y\to\mathbb{R}$ is considered. Conditions are obtained (both necessary ones and sufficient ones) for the optimality of a Monge solution generated by a smooth measure-preserving map $f\colon X\to Y$. The proofs are based on an optimality criterion for a general MKP in terms of nonemptiness of the sets $Q_0(\zeta)=\{u\in\mathbb{R}^X:u(x)-u(z)\le\zeta(x,z)$ for all $x,z\in X\}$ for special functions $\zeta$ on $X\times X$ generated by $c$ and $f$. Also, earlier results by the author are used when considering the above-mentioned nonemptiness conditions for the case of smooth $\zeta$.
Keywords: Monge–Kantorovich problem, marginal
Mots-clés : Monge solution. 
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V. L. Levin. Optimality Conditions for Smooth Monge Solutions of the Monge--Kantorovich problem. Funkcionalʹnyj analiz i ego priloženiâ, Tome 36 (2002) no. 2, pp. 38-44. http://geodesic.mathdoc.fr/item/FAA_2002_36_2_a3/

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