cyclical monotonicity is studied for a multivalued operator F : X → L, where L $ \subseteq$RX. A criterion for F to be L-cyclically monotone is obtained and connections with the notions of L-convex function and of its L-subdifferentials are established. Applications are given to the general Monge–Kantorovich problem with fixed marginals. In particular, we show that in some cases the optimal measure is unique and generated by a unique (up to the a.e. equivalence) optimal solution (measure preserving map) for the corresponding Monge problem.

Abstract Cyclical Monotonicity and Monge Solutions for the General Monge–Kantorovich Problem

We obtain optimality conditions for Monge solutions of the Monge— Kantorovich problem with a smooth cost function. Also we give explicit solutions to Monge problems and to Monge—Kantorovich problems for several natural classes of cost functions.

Optimal solutions of the Monge problem

CONVEXITY IN MEASURE THEORY AND IN CONVEX ANALYSIS V. L. Levin UDC 519.853

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Abstract Convexity in Measure Theory and in Convex Analysis

In the present survey, we reveal links between abstract convex analysis and two variants of the Monge-Kantorovich problem (MKP), with given marginals and with a given marginal difference. It includes: (1) the equivalence of the validity of duality theorems for MKP and appropriate abstract convexity of the corresponding cost functions; (2) a characterization of a (maximal) abstract cyclic monotone map F: X → L ⊂ IRX in terms connected with the constraint set 

$$ Q_0 (\varphi ): = \{ u \in \mathbb{R}^z :u(z_1 ) - u(z_2 ) \leqslant \varphi (z_1 ,z_2 ){\text{ }}\forall z_1 ,z_1 \in Z = dom{\text{ }}F\} $$

of a particular dual MKP with a given marginal difference and in terms of L-subdifferentials of L-convex functions; (3) optimality criteria for MKP (and Monge problems) in terms of abstract cyclic monotonicity and non-emptiness of the constraint set Q 0(ϕ), where ϕ is a special cost function on X × X determined by the original cost function c on X × Y. The Monge-Kantorovich duality is applied then to several problems of mathematical economics relating to utility theory, demand analysis, generalized dynamics optimization models, and economics of corruption, as well as to a best approximation problem.

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Abstract Convexity and the Monge-Kantorovich Duality

A method in demand analysis based on the Monge—Kantorovich duality is developed. We characterize (insatiate) demand functions that are rationalized, in different meanings, by concave utility functions with some additional properties such as upper semi-continuity, continuity, non-decrease, strict concavity, positive homogeneity and so on. The characterizations are some kinds of abstract cyclic monotonicity strengthening revealed preference axioms, and also they may be considered as an extension of the Afriat—Varian theory to an arbitrary (infinite) set of ‘observed data’. Particular attention is paid to the case of smooth functions.

A method in demand analysis connected with the Monge—Kantorovich problem

Duality theory for nontopological versions of mass transfer problem extending the classical Monge-Kantorovich problem is developed. Applications to dynamic optimization and approximation theory are outlined.

V. L. Levin

Papers

Topics in the Duality Theory for Mass Transfer Problems