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Uspekhi Mat. Nauk, 1972, Volume 27, Issue 3(165), Pages 127–176 (Mi umn5059)  

This article is cited in 26 scientific papers (total in 27 papers)

Minkowski duality and its applications

S. S. Kutateladze, A. M. Rubinov


Abstract: This article is an account of problems grouped around the concept of Minkowski duality – one of the central constructions in convex analysis. The article consists of an introduction, four sections, and a commentary.
In § 1 we set out the main facts about $H$-convex elements and introduce the Minkowski–Fenchel and the Minkowski–Moreau schemes; we consider the space of $H$-convex sets. Here we collect together the main examples, namely the convex and sublinear functions, and the stable, normal, and convex sets in the sense of Fan, amongst others.
§ 2 is concerned mainly with representations of positive functionals over continuous $H$-convex functions and sets. Here we also establish the links between such constructions and the Choquet theory.
In § 3 we introduce various characterizations of $H$-convexity in the form of theorems on supremal generators. In particular, we consider in detail theorems on the definability of convergence of sequences of operators in terms of their convergence on a cone. Other applications of supremal generators are also given.
In § 4 problems of isoperimetric type (with an arbitrary number of constraints) in the geometry of convex surfaces are analyzed as problems of programming in a space of convex sets. We examine as particular examples exterior and interior isoperimetric problems, the Uryson problem, and others.

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English version:
Russian Mathematical Surveys, 1972, 27:3, 137–191

Bibliographic databases:

UDC: 513.88
Received: 03.11.1971

Citation: S. S. Kutateladze, A. M. Rubinov, “Minkowski duality and its applications”, Uspekhi Mat. Nauk, 27:3(165) (1972), 127–176; Russian Math. Surveys, 27:3 (1972), 137–191

Citation in format AMSBIB
\Bibitem{KutRub72}
\by S.~S.~Kutateladze, A.~M.~Rubinov
\paper Minkowski duality and its applications
\jour Uspekhi Mat. Nauk
\yr 1972
\vol 27
\issue 3(165)
\pages 127--176
\mathnet{http://mi.mathnet.ru/umn5059}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=394117}
\zmath{https://zbmath.org/?q=an:0261.26010}
\transl
\jour Russian Math. Surveys
\yr 1972
\vol 27
\issue 3
\pages 137--191
\crossref{https://doi.org/10.1070/RM1972v027n03ABEH001380}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. V. L. Levin, “Convex integral functionals and the theory of lifting”, Russian Math. Surveys, 30:2 (1975), 119–184  mathnet  crossref  mathscinet  zmath
    2. S. S. Kutateladze, “Choquet boundaries in $K$-spaces”, Russian Math. Surveys, 30:4 (1975), 115–155  mathnet  crossref  mathscinet  zmath
    3. A. M. Rubinov, “Ergodicheskaya teorema dlya markovskikh operatorov i granitsa Shilova”, UMN, 30:6(186) (1975), 183–183  mathnet  mathscinet  zmath
    4. Horst Schirmeier, Ursula Schirmeier, “Einige Bemerkungen über den Satz von Keldych”, Math. Ann, 236:3 (1978), 245  crossref
    5. Francesco Altomare, “Korovkin-type theorems for positive functionals in spaces of continuous affine functions”, Rend Circ Mat Palermo, 36:2 (1987), 167  crossref  mathscinet  zmath
    6. Francesco Altomare, “Positive linear forms and their determining subspaces”, Annali di Matematica, 154:1 (1989), 243  crossref  mathscinet  zmath  isi
    7. Hubertus Th. Jongen, Gerhard -W. Weber, “On parametric nonlinear programming”, Ann Oper Res, 27:1 (1990), 253  crossref  mathscinet  zmath
    8. A. M. Rubinov, V. Jeyakumar, B. M. Glover, “Generalized convex relations with applications to optimization and models of economic dynamics”, Set-Valued Anal, 4:1 (1996), 67  crossref  mathscinet  zmath  isi  elib
    9. A. E. Aban'kin, “Unconstrained minimization of $H$-hyperdifferentiable functions”, Comput. Math. Math. Phys., 38:9 (1998), 1439–1446  mathnet  mathscinet  zmath
    10. Juan-Enrique Martínez-Legaz, Ivan Singer, “On φ-convexity of convex functions”, Linear Algebra and its Applications, 278:1-3 (1998), 163  crossref
    11. C.E.M. Pearce, A.M. Rubinov, “P-functions, Quasi-convex Functions, and Hadamard-type Inequalities”, Journal of Mathematical Analysis and Applications, 240:1 (1999), 92  crossref
    12. A.M Rubinov, “Abstract convexity:examples and applications*”, Optimization, 47:1-2 (2000), 1  crossref
    13. A. M. Rubinov, X. X. Huang, X. Q. Yang, “The Zero Duality Gap Property and Lower Semicontinuity of the Perturbation Function”, moor, 27:4 (2002), 775  crossref  mathscinet  zmath  isi  elib
    14. A.M. Rubinov, J. Dutta, “Hadamard type inequality for quasiconvex functions in higher dimensions”, Journal of Mathematical Analysis and Applications, 270:1 (2002), 80  crossref
    15. A. Rubinov, Z. Dzalilov, “Abstract convexity of positively homogeneous functions”, Journal of Statistics and Management Systems, 5:1-3 (2002), 1  crossref
    16. A.M. Rubinov, R.N. Gasimov, “Strictly Increasing Positively Homogeneous Functions with Application to Exact Penalization”, Optimization, 52:1 (2003), 1  crossref
    17. Dante Carrasco-Olivera, Fabián Flores-Bazán, “On the Representation of Approximate Subdifferentials for a Class of Generalized Convex Functions”, Set-Valued Anal, 13:2 (2005), 151  crossref  mathscinet  zmath  isi
    18. A. E. Gutman, A. G. Kusraev, Yu. G. Reshetnyak, “K voprosu ob $n$-letii Semena Samsonovicha Kutateladze dlya sluchaya $n=60$”, Sib. elektron. matem. izv., 2 (2005), 12–33  mathnet  mathscinet  zmath
    19. Jean-Paul Penot, Alexander M. Rubinov, “Multipliers and general Lagrangians”, Optimization, 54:4-5 (2005), 443  crossref  elib
    20. S. S. Kutateladze, “Upper semilattices of finite-dimensional gauges”, Vladikavk. matem. zhurn., 8:4 (2006), 58–69  mathnet  mathscinet  elib
    21. Jean-Paul Penot, “Critical duality”, J Global Optim, 40:1-3 (2008), 319  crossref  mathscinet  zmath  isi
    22. KLAUS KEIMEL, GORDON D. PLOTKIN, “Predicate transformers for extended probability and non-determinism”, Math Struct Comp Sci, 2009, 1  crossref  mathscinet  isi
    23. S. S. Kutateladze, “Multiobjective problems of convex geometry”, Siberian Math. J., 50:5 (2009), 887–897  mathnet  crossref  mathscinet  isi  elib
    24. Jean-Paul Penot, “Are dualities appropriate for duality theories in optimization?”, J Global Optim, 2009  crossref
    25. Demyanov V.F., “Nonsmooth Optimization”, Lecture Notes in Mathematics, 2010, 55–163  mathscinet  isi  elib
    26. S. S. Kutateladze, “Multiple criteria problems over Minkowski balls”, J. Appl. Ind. Math, 7:2 (2013), 209  crossref
    27. S. S. Kutateladze, “Math-selfie”, Vladikavk. matem. zhurn., 17:3 (2015), 93–100  mathnet
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