A~formula for the optimal value in the Monge--Kantorovich problem with a~smooth cost function, and a~characterization of cyclically monotone mappings

V. L. Levin

Sbornik. Mathematics, Tome 71 (1992) no. 2, pp. 533-548

Résumé

The general Monge–Kantorovich problem consists in the computation of the optimal value $$ \mathscr A(c,\rho):=\inf\biggl\{\int_{X\times X}c(x,y)\mu(d(x,y))\colon\mu\in V_+(X\times X),\ (\pi_1-\pi_2)\mu=\rho\biggr\}, $$ where the cost function $c\colon X\times X\to \mathbf R^1$ and the measure $\rho$ on $X$ with $\rho X=0$ are assumed to be given, $V_+(X\times X)$ is the cone of finite positive Borel measures on $X\times X$, and $\pi_1$ and $\pi_2$ are the projections on the first and second coordinates, which assign to a measure $\mu$ the corresponding marginal measures. An explicit formula is obtained for $\mathscr A(c,\rho)$ in the case when $X$ is a domain in $\mathbf R^n$ and $c$ is bounded, vanishes on the diagonal, and is continuously differentiable in a neighborhood of the diagonal. Conditions for the set $$ Q_0(c):=\{u\colon X\to\mathbf R^1:u(x)-u(y)\leqslant c(x,y)\ \ \forall\,x,y\in X\} $$ to be nonempty are investigated, and with their help new characterizations of cyclically monotone mappings are obtained.