Author： Levin V.L.
Publisher： Springer Publishing Company
ISSN： 0016-2663
Source： Functional Analysis and Its Applications, Vol.36, Iss.2, 2002-04, pp. : 114-119
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Abstract
The Monge–Kantorovich problem (MKP) with given marginals defined on closed domains Xsubsetmathbb{R}^n, Ysubsetmathbb{R}^m and a smooth cost function ccolon Xtimes Ytomathbb{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 fcolon Xto Y. The proofs are based on an optimality criterion for a general MKP in terms of nonemptiness of the sets Q_0(zeta)={uinmathbb{R}^X:u(x)-u(z)lezeta(x,z) for all x,zin X} for special functions zeta on Xtimes 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.
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