01.01.01 (Real analysis, complex analysis, and functional analysis)

Birth date:

31.03.1938

E-mail:

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.

Biography

Graduated from Faculty of Mathematics and Mechanics of M. V. Lomonosov Moscow State University (MSU) in 1960 (chair of theory of functions and functional analysis). Ph.D. thesis was defended in 1965. D.Sci. thesis was defended in 1988. A list of my works contains more than 85 titles.

Main publications:

Levin V. L. The Monge–Kantorovich problems and stochastic preference relations // Adv. Math. Economics, 2001, 3, 97–124.

V. L. Levin, “Best approximation problems
relating to Monge–Kantorovich duality”, Mat. Sb., 197:9 (2006), 103–114; Sb. Math., 197:9 (2006), 1353–1364

2004

2.

V. L. Levin, “Optimality conditions and exact solutions to the two-dimensional Monge–Kantorovich problem”, Zap. Nauchn. Sem. POMI, 312 (2004), 150–164; J. Math. Sci. (N. Y.), 133:4 (2006), 1456–1463

2002

3.

V. L. Levin, “Optimality Conditions for Smooth Monge Solutions of the Monge–Kantorovich problem”, Funktsional. Anal. i Prilozhen., 36:2 (2002), 38–44; Funct. Anal. Appl., 36:2 (2002), 114–119

1998

4.

V. L. Levin, “Existence and Uniqueness of a Measure-Preserving Optimal Mapping in a General Monge–Kantorovich
Problem”, Funktsional. Anal. i Prilozhen., 32:3 (1998), 79–82; Funct. Anal. Appl., 32:3 (1998), 205–208

1997

5.

V. L. Levin, “On duality theory for non-topological variants of the mass transfer problem”, Mat. Sb., 188:4 (1997), 95–126; Sb. Math., 188:4 (1997), 571–602

1996

6.

V. L. Levin, “Dual Representations of Convex Bodies and Their Polars”, Funktsional. Anal. i Prilozhen., 30:3 (1996), 79–81; Funct. Anal. Appl., 30:3 (1996), 209–210

1992

7.

V. L. Levin, “Measurable selections of multivalued mappings with a bi-analytic graph and $\sigma$-compact values”, Tr. Mosk. Mat. Obs., 54 (1992), 3–28

1990

8.

V. L. Levin, “A problem of complex analysis arising in optimal control theory”, Mat. Zametki, 47:5 (1990), 45–51; Math. Notes, 47:5 (1990), 453–458

9.

V. L. Levin, “A formula for the optimal value in the Monge–Kantorovich problem with a smooth cost function, and a characterization of cyclically monotone mappings”, Mat. Sb., 181:12 (1990), 1694–1709; Math. USSR-Sb., 71:2 (1992), 533–548

1987

10.

V. L. Levin, “Measurable selections of multivalued mappings and the problem of
mass transfer”, Dokl. Akad. Nauk SSSR, 292:5 (1987), 1048–1053

11.

V. L. Levin, “Solution of a problem of convex analysis”, Uspekhi Mat. Nauk, 42:2(254) (1987), 235–236; Russian Math. Surveys, 42:2 (1987), 287–288

1985

12.

V. L. Levin, “Functionally closed preorders and strong stochastic domination”, Dokl. Akad. Nauk SSSR, 283:1 (1985), 30–34

1984

13.

V. L. Levin, “The problem of mass transfer in a topological space and
probability measures with given marginal measures on the product of two
spaces”, Dokl. Akad. Nauk SSSR, 276:5 (1984), 1059–1064

14.

V. L. Levin, “Lipschitz pre-orders and Lipschitz utility functions”, Uspekhi Mat. Nauk, 39:6(240) (1984), 199–200; Russian Math. Surveys, 39:6 (1984), 217–218

1983

15.

V. L. Levin, “Continuous utility theorem for closed preorders on a metrizable $\sigma$-compact space”, Dokl. Akad. Nauk SSSR, 273:4 (1983), 800–804

16.

V. L. Levin, “Measurable utility theorems for closed and lexicographic preference relations”, Dokl. Akad. Nauk SSSR, 270:3 (1983), 542–546

1981

17.

V. L. Levin, “Some applications of duality for the problem of translocation of masses with a lower semicontinuous cost function. Closed preferences and Choquet theory”, Dokl. Akad. Nauk SSSR, 260:2 (1981), 284–288

1980

18.

V. L. Levin, “Measurable selections of multivalued mappings into topological spaces and upper envelopes of Caratheodory integrands”, Dokl. Akad. Nauk SSSR, 252:3 (1980), 535–539

1979

19.

V. L. Levin, A. A. Milyutin, “The problem of mass transfer with a discontinuous cost function and a mass statement of the duality problem for convex extremal problems”, Uspekhi Mat. Nauk, 34:3(207) (1979), 3–68; Russian Math. Surveys, 34:3 (1979), 1–78

1978

20.

V. L. Levin, “Measurable selections of multivalued mappings and projections of measurable sets”, Funktsional. Anal. i Prilozhen., 12:2 (1978), 40–45; Funct. Anal. Appl., 12:2 (1978), 108–112

1977

21.

V. L. Levin, “Duality theorems in the Monge–Kantorovich problem”, Uspekhi Mat. Nauk, 32:3(195) (1977), 171–172

1976

22.

V. L. Levin, “On subdifferentials and continuous extensions with preservation of a measurable dependence on a parameter”, Funktsional. Anal. i Prilozhen., 10:3 (1976), 84–85; Funct. Anal. Appl., 10:3 (1976), 235–237

1975

23.

V. L. Levin, “Extremal problems with convex functionals that are lower semicontinuous with respect to convergence in measure”, Dokl. Akad. Nauk SSSR, 224:6 (1975), 1256–1259

24.

V. L. Levin, “On the mass transfer problem”, Dokl. Akad. Nauk SSSR, 224:5 (1975), 1016–1019

25.

V. L. Levin, “Convex integral functionals and the theory of lifting”, Uspekhi Mat. Nauk, 30:2(182) (1975), 115–178; Russian Math. Surveys, 30:2 (1975), 119–184

1974

26.

V. L. Levin, “The Lebesgue decomposition for functionals on the vector-function space $L_{\mathfrak{X}}^\infty$”, Funktsional. Anal. i Prilozhen., 8:4 (1974), 48–53; Funct. Anal. Appl., 8:4 (1974), 314–317

1973

27.

V. L. Levin, “Subdifferentials of convex integral functionals and liftings that are the identity on subspaces of $\mathscr{L}^\infty$”, Dokl. Akad. Nauk SSSR, 211:5 (1973), 1046–1049

1972

28.

A. D. Ioffe, V. L. Levin, “Subdifferentials of convex functions”, Tr. Mosk. Mat. Obs., 26 (1972), 3–73

29.

V. I. Arkin, V. L. Levin, “Convexity of values of vector integrals, theorems on measurable choice and variational problems”, Uspekhi Mat. Nauk, 27:3(165) (1972), 21–77; Russian Math. Surveys, 27:3 (1972), 21–85

1971

30.

V. I. Arkin, V. L. Levin, “A variational problem with functions of several variables and operator restrictions: The maximum principle and existence theorem”, Dokl. Akad. Nauk SSSR, 200:1 (1971), 9–12

31.

V. I. Arkin, V. L. Levin, “Extreme points of a certain set of measurable vector functions of several variables and convexity of the values of vector integrals”, Dokl. Akad. Nauk SSSR, 199:6 (1971), 1223–1226

1970

32.

V. L. Levin, “The subdifferential of a composite functional”, Dokl. Akad. Nauk SSSR, 194:2 (1970), 268–269

33.

V. L. Levin, “Subdifferentials of convex functionals”, Uspekhi Mat. Nauk, 25:4(154) (1970), 183–184

1969

34.

V. L. Levin, “Tensor products and functors in categories of Banach spaces defined by $KB$-lineals”, Tr. Mosk. Mat. Obs., 20 (1969), 43–82

35.

V. L. Levin, “Application of E. Helly's theorem to convex programming, problems of best approximation and related questions”, Mat. Sb. (N.S.), 79(121):2(6) (1969), 250–263; Math. USSR-Sb., 8:2 (1969), 235–247

1968

36.

V. L. Levin, “Some properties of support functionals”, Mat. Zametki, 4:6 (1968), 685–696; Math. Notes, 4:6 (1968), 900–906

37.

V. L. Levin, “Infinite dimensional analogs of a linear programming problem, and the saddle point theorem”, Uspekhi Mat. Nauk, 23:3(141) (1968), 181–182

1965

38.

V. L. Levin, “Tensor products and functors in Banach space categories defined by $KB$-lineals”, Dokl. Akad. Nauk SSSR, 163:5 (1965), 1058–1060

39.

V. L. Levin, “Functors in categories of Banach spaces, defined by KV-spaces”, Dokl. Akad. Nauk SSSR, 162:2 (1965), 262–265

40.

V. L. Levin, “The open mapping theorem for uniform spaces”, Izv. Vyssh. Uchebn. Zaved. Mat., 1965, 2, 86–90

1963

41.

V. L. Levin, D. A. Raikov, “Closed-graph theorems for uniform spaces”, Dokl. Akad. Nauk SSSR, 150:5 (1963), 981–983

1962

42.

V. L. Levin, “$B$-completeness conditions for ultrabarrelled and barrelled spaces”, Dokl. Akad. Nauk SSSR, 145:2 (1962), 273–275

43.

V. L. Levin, “On a class of locally convex spaces”, Dokl. Akad. Nauk SSSR, 145:1 (1962), 35–37

1961

44.

V. L. Levin, “On a theorem of A. I. Plessner”, Uspekhi Mat. Nauk, 16:5(101) (1961), 177–179

1960

45.

V. L. Levin, “Non-degenerate spectra of locally convex spaces”, Dokl. Akad. Nauk SSSR, 135:1 (1960), 12–15

2002

46.

V. L. Bodneva, V. G. Boltyanskii, I. M. Gel'fand, V. V. Dicusar, A. V. Dmitruk, A. D. Ioffe, V. L. Levin, Ya. M. Kazhdan, N. P. Osmolovskii, V. M. Tikhomirov, G. M. Henkin, “Aleksei Alekseevich Milyutin (obituary)”, Uspekhi Mat. Nauk, 57:3(345) (2002), 137–140; Russian Math. Surveys, 57:3 (2002), 577–580

1980

47.

V. L. Levin, A. A. Milyutin, “Correction to the paper: “The problem of mass transfer with a discontinuous cost function and
a mass statement of the duality problem for convex extremal problems””, Uspekhi Mat. Nauk, 35:2(212) (1980), 275