Levin, Vladimir L'vovich

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Total publications: 54
Scientific articles: 52

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Senior Researcher
Doctor of physico-mathematical sciences (1988)
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.


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.


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.
List of publications on Google Scholar
List of publications on ZentralBlatt

Publications in Math-Net.Ru
1. V. L. Levin, “Best approximation problems relating to Monge–Kantorovich duality”, Mat. Sb., 197:9 (2006),  103–114  mathnet  mathscinet  zmath  elib; Sb. Math., 197:9 (2006), 1353–1364  isi  elib  scopus
2. V. L. Levin, “Optimality conditions and exact solutions to the two-dimensional Monge–Kantorovich problem”, Zap. Nauchn. Sem. POMI, 312 (2004),  150–164  mathnet  mathscinet  zmath  elib; J. Math. Sci. (N. Y.), 133:4 (2006), 1456–1463  elib
3. V. L. Levin, “Optimality Conditions for Smooth Monge Solutions of the Monge–Kantorovich problem”, Funktsional. Anal. i Prilozhen., 36:2 (2002),  38–44  mathnet  mathscinet  zmath  elib; Funct. Anal. Appl., 36:2 (2002), 114–119  isi  scopus
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  mathnet  mathscinet  zmath  elib; Funct. Anal. Appl., 32:3 (1998), 205–208  isi
5. V. L. Levin, “Semiconical sets, semi-homogeneous functions, and a new duality scheme in convex analysis”, Dokl. Akad. Nauk, 354:5 (1997),  597–599  mathnet  mathscinet  zmath
6. V. L. Levin, “On duality theory for non-topological variants of the mass transfer problem”, Mat. Sb., 188:4 (1997),  95–126  mathnet  mathscinet  zmath  elib; Sb. Math., 188:4 (1997), 571–602  isi  scopus
7. V. L. Levin, “Duality theorems for a nontopological version of the mass transfer problem”, Dokl. Akad. Nauk, 350:5 (1996),  588–591  mathnet  mathscinet  zmath
8. V. L. Levin, “Dual Representations of Convex Bodies and Their Polars”, Funktsional. Anal. i Prilozhen., 30:3 (1996),  79–81  mathnet  mathscinet  zmath; Funct. Anal. Appl., 30:3 (1996), 209–210  isi
9. V. L. Levin, “Exchange models with indivisible goods and the realizability of competitive equilibria in auction-type games”, Dokl. Akad. Nauk, 334:1 (1994),  16–19  mathnet  mathscinet  zmath; Dokl. Math., 49:1 (1994), 15–19
10. 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  mathnet  mathscinet  zmath
11. V. L. Levin, “A problem of complex analysis arising in optimal control theory”, Mat. Zametki, 47:5 (1990),  45–51  mathnet  mathscinet  zmath; Math. Notes, 47:5 (1990), 453–458  isi
12. 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  mathnet  mathscinet  zmath; Math. USSR-Sb., 71:2 (1992), 533–548  isi
13. V. L. Levin, “Measurable selections of multivalued mappings and the problem of mass transfer”, Dokl. Akad. Nauk SSSR, 292:5 (1987),  1048–1053  mathnet  mathscinet  zmath
14. V. L. Levin, “Solution of a problem of convex analysis”, Uspekhi Mat. Nauk, 42:2(254) (1987),  235–236  mathnet  mathscinet  zmath; Russian Math. Surveys, 42:2 (1987), 287–288  isi
15. V. L. Levin, “Functionally closed preorders and strong stochastic domination”, Dokl. Akad. Nauk SSSR, 283:1 (1985),  30–34  mathnet  mathscinet  zmath
16. 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  mathnet  mathscinet  zmath
17. V. L. Levin, “Lipschitz pre-orders and Lipschitz utility functions”, Uspekhi Mat. Nauk, 39:6(240) (1984),  199–200  mathnet  mathscinet  zmath; Russian Math. Surveys, 39:6 (1984), 217–218  isi
18. V. L. Levin, “Continuous utility theorem for closed preorders on a metrizable $\sigma$-compact space”, Dokl. Akad. Nauk SSSR, 273:4 (1983),  800–804  mathnet  mathscinet  zmath
19. V. L. Levin, “Measurable utility theorems for closed and lexicographic preference relations”, Dokl. Akad. Nauk SSSR, 270:3 (1983),  542–546  mathnet  mathscinet  zmath
20. 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  mathnet  mathscinet  zmath
21. 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  mathnet  mathscinet  zmath
22. 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  mathnet  mathscinet  zmath; Russian Math. Surveys, 34:3 (1979), 1–78
23. V. L. Levin, “Measurable selections of multivalued mappings and projections of measurable sets”, Funktsional. Anal. i Prilozhen., 12:2 (1978),  40–45  mathnet  mathscinet  zmath; Funct. Anal. Appl., 12:2 (1978), 108–112
24. V. L. Levin, “Borel cross sections of many-valued mappings”, Sibirsk. Mat. Zh., 19:3 (1978),  617–623  mathnet  mathscinet  zmath; Siberian Math. J., 19:3 (1978), 434–438  isi
25. V. L. Levin, “Duality theorems in the Monge–Kantorovich problem”, Uspekhi Mat. Nauk, 32:3(195) (1977),  171–172  mathnet  mathscinet  zmath
26. 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  mathnet  mathscinet  zmath; Funct. Anal. Appl., 10:3 (1976), 235–237
27. 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  mathnet  mathscinet  zmath
28. V. L. Levin, “On the mass transfer problem”, Dokl. Akad. Nauk SSSR, 224:5 (1975),  1016–1019  mathnet  mathscinet  zmath
29. V. L. Levin, “Convex integral functionals and the theory of lifting”, Uspekhi Mat. Nauk, 30:2(182) (1975),  115–178  mathnet  mathscinet  zmath; Russian Math. Surveys, 30:2 (1975), 119–184
30. 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  mathnet  mathscinet  zmath; Funct. Anal. Appl., 8:4 (1974), 314–317
31. 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  mathnet  mathscinet  zmath
32. V. L. Levin, “On the duality of certain classes of linear operators that act between Banach spaces and Banach lattices”, Sibirsk. Mat. Zh., 14:3 (1973),  599–608  mathnet  mathscinet  zmath; Siberian Math. J., 14:3 (1973), 416–422
33. A. D. Ioffe, V. L. Levin, “Subdifferentials of convex functions”, Tr. Mosk. Mat. Obs., 26 (1972),  3–73  mathnet  mathscinet  zmath
34. 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  mathnet  mathscinet  zmath; Russian Math. Surveys, 27:3 (1972), 21–85
35. V. L. Levin, “Subdifferentials of convex mappings and of composite functions”, Sibirsk. Mat. Zh., 13:6 (1972),  1295–1303  mathnet  mathscinet  zmath; Siberian Math. J., 13:6 (1972), 903–909
36. 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  mathnet  mathscinet  zmath
37. 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  mathnet  mathscinet  zmath
38. V. L. Levin, “The subdifferential of a composite functional”, Dokl. Akad. Nauk SSSR, 194:2 (1970),  268–269  mathnet  mathscinet  zmath
39. V. L. Levin, “Subdifferentials of convex functionals”, Uspekhi Mat. Nauk, 25:4(154) (1970),  183–184  mathnet  mathscinet  zmath
40. V. L. Levin, “Tensor products and functors in categories of Banach spaces defined by $KB$-lineals”, Tr. Mosk. Mat. Obs., 20 (1969),  43–82  mathnet  mathscinet  zmath
41. 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  mathnet  mathscinet  zmath; Math. USSR-Sb., 8:2 (1969), 235–247
42. V. L. Levin, “Two classes of linear mappings which operate between Banach spaces and Banach lattices”, Sibirsk. Mat. Zh., 10:4 (1969),  903–909  mathnet  mathscinet  zmath; Siberian Math. J., 10:4 (1969), 664–668
43. V. L. Levin, “Some properties of support functionals”, Mat. Zametki, 4:6 (1968),  685–696  mathnet  mathscinet  zmath; Math. Notes, 4:6 (1968), 900–906
44. 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  mathnet  mathscinet  zmath
45. V. L. Levin, “Tensor products and functors in Banach space categories defined by $KB$-lineals”, Dokl. Akad. Nauk SSSR, 163:5 (1965),  1058–1060  mathnet  mathscinet  zmath
46. V. L. Levin, “Functors in categories of Banach spaces, defined by KV-spaces”, Dokl. Akad. Nauk SSSR, 162:2 (1965),  262–265  mathnet  mathscinet  zmath
47. V. L. Levin, “The open mapping theorem for uniform spaces”, Izv. Vyssh. Uchebn. Zaved. Mat., 1965, 2,  86–90  mathnet  mathscinet  zmath
48. V. L. Levin, D. A. Raikov, “Closed-graph theorems for uniform spaces”, Dokl. Akad. Nauk SSSR, 150:5 (1963),  981–983  mathnet  mathscinet  zmath
49. V. L. Levin, “$B$-completeness conditions for ultrabarrelled and barrelled spaces”, Dokl. Akad. Nauk SSSR, 145:2 (1962),  273–275  mathnet  mathscinet  zmath
50. V. L. Levin, “On a class of locally convex spaces”, Dokl. Akad. Nauk SSSR, 145:1 (1962),  35–37  mathnet  mathscinet  zmath
51. V. L. Levin, “On a theorem of A. I. Plessner”, Uspekhi Mat. Nauk, 16:5(101) (1961),  177–179  mathnet  mathscinet  zmath
52. V. L. Levin, “Non-degenerate spectra of locally convex spaces”, Dokl. Akad. Nauk SSSR, 135:1 (1960),  12–15  mathnet  mathscinet  zmath

53. 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  mathnet  mathscinet  zmath; Russian Math. Surveys, 57:3 (2002), 577–580  isi
54. 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  mathnet

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