Speciality:
01.01.01 (Real analysis, complex analysis, and functional analysis)
Birth date:
31.03.1938
Keywords: convex analysis and extremum problems in functional spaces; set-valued analysis; measurable selections of multifunctions; Monge–Kantorovich problem; methods of functional analysis in mathematical economics.
Subject:
In two papers (one of them with D. A. Raikov), an extension to uniform spaces was given of the notion of $B$–completeness and of the Banach closed graph and open mapping theorems. On the algebraic tensor product of a Banach lattice $E$ and a Banach space $X$ a cross-norm was introduced such that, for many concrete lattices of functions or sequences, the completion of $E\otimes X$ by the cross-norm is the space $E(X)$ of the "same" vector functions or sequences with values in $X$. The dual space was described and properties were studied of that tensor product and of two connected classes of linear operators acting between Banach spaces and Banach lattices. Theorems on Lebesgue decomposition were obtained for linear functionals on the space $L^\infty(X)$ (an extension of the Yosida–Hewitt theorem) and on more general spaces of measurable vector functions. A final form for the purification theorem was obtained. It asserts that in a finite dimensional convex extremal problem with a large or even infinite number of constraints all constraints except some $n$ of them (where $n$ is the dimension of the space) can be rejected without decrease of the optimal value. From here, purification theorems follow for a subdifferential of maximum of a family of convex functions and for the minimax and the best approximation problems. A subdifferential calculus of convex functionals on spaces of measurable vector functions with values in an arbitrary Banach space was developed and with its help a complete solution was given of traditional convex analysis' problems on evaluation the subdifferentials of convex functions of integral and of maximum types as well as of a close problem relating to the subdifferential of a composite function. A connection was revealed between the validity in mass setting of regular integral representations for the subdifferentials and the existence of special liftings of $L^\infty$. That connection enables us to treat some topics of measure theory (strong lifting, desintegration and differentiation of measures) as a fragment of convex analysis in function spaces. A cycle of papers and a monograph "Convex analysis in spaces of measurable functions and its applications in mathematics and economics", Moscow: Nauka, 1985, 352 pages, were devoted to these questions. Measurable selection theorems were proved for multifunctions with values in nonseparable and/or nonmetrizable spaces. A number of papers (one of them with A. A. Milyutin) were devoted to the Monge–Kantorovich problem (duality theory; problems with smooth cost functions; existence of the Monge solutions) and to its applications in mathematical economics. Duality theory was developed for two variants of the problem: with fixed marginals and with a fixed marginal difference. Cost functions were completely characterized, for which optimal values of the original and of the dual problems coincide. One of the formulations for a compact space and the problem with a fixed marginal difference is as follows: in a class of cost functions $c(x,y)$ satisfying the triangle inequality the coincidence of optimal values in a mass setting is equivalent to the lower semicontinuity of $c$. In a problem with fixed marginals, one of which is absolutely continuous with respect to the $n$–dimensional Lebesgue measure, theorems on existence and uniqueness of optimal solutions that are the Monge solutions were obtained for three classes of cost functions. In case where the cost function is smooth, optimality conditions for smooth Monge solutions were given. A new duality scheme in convex analysis was proposed for semiconic convex sets and semihomogeneous convex functions.
Main publications:
Levin V. L. The Monge–Kantorovich problems and stochastic preference relations // Adv. Math. Economics, 2001, 3, 97–124.
