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Mat. Sb., 1996, Volume 187, Number 2, Pages 103–130 (Mi msb111)  

This article is cited in 31 scientific papers (total in 31 papers)

Strongly convex analysis

E. S. Polovinkin

Moscow Institute of Physics and Technology

Abstract: Properties of strongly convex sets (that is, of sets that can be represented as intersections of balls of radius fixed for each particular set) are investigated. A connection between strongly convex sets and strongly convex functions is established. The concept of a strongly convex $R$-hull of a set (the minimal strongly convex set containing the given set) is introduced; an explicit formula for the srongly convex $R$-hull of a set is obtained. The behaviour of the strongly convex $R$-hull under the variation of $R$ and of the sets is considered. An analogue of the Carathéodory theorem for strongly convex sets is obtained. The concept of a strongly extreme point is introduced, and a generalization of the Krein–Mil'man theorem for strongly convex sets is proved. Polyhedral approximations of convex and, in particular, of strongly convex compact sets are considered. Sharp error estimates for polyhedral and strongly convex approximations of such sets from inside and outside are established.

DOI: 10.4213/sm111

Full text (in Russian): PDF file (404 kB)
References (in Russian): PDF file   HTML файл

English version:
Sbornik: Mathematics, 1996, 187:2, 259–286

Bibliographic databases:

UDC: 517.977
MSC: Primary 52A20, 52A27; Secondary 90D25
Received: 13.06.1995

Citation: E. S. Polovinkin, “Strongly convex analysis”, Mat. Sb., 187:2 (1996), 103–130

Citation in format AMSBIB
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\paper Strongly convex analysis
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\yr 1996
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\pages 259--286
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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. М. В. Балашов, Е. С. Половинкин, “$M$-сильно выпуклые подмножества и их порождающие множества”, Матем. сб., 191:1 (2000), 27–64  mathnet  crossref  mathscinet  zmath; M. V. Balashov, E. S. Polovinkin, “$M$-strongly convex subsets and their generating sets”, Sb. Math., 191:1 (2000), 25–60  crossref
    2. С. И. Дудов, И. В. Златорунская, “Равномерная оценка выпуклого компакта шаром произвольной нормы”, Матем. сб., 191:10 (2000), 13–38  mathnet  crossref  mathscinet  zmath; S. I. Dudov, I. V. Zlatorunskaya, “Uniform estimate of a compact convex set by a ball in an arbitrary norm”, Sb. Math., 191:10 (2000), 1433–1458  crossref
    3. Е. С. Половинкин, Г. Е. Иванов, М. В. Балашов, Р. В. Константинов, А. В. Хорев, “Об одном алгоритме численного решения линейных дифференциальных игр”, Матем. сб., 192:10 (2001), 95–122  mathnet  crossref  mathscinet  zmath; E. S. Polovinkin, G. E. Ivanov, M. V. Balashov, R. V. Konstantinov, A. V. Khorev, “An algorithm for the numerical solution of linear differential games”, Sb. Math., 192:10 (2001), 1515–1542  crossref
    4. М. В. Балашов, “Об аналоге теоремы Крейна–Мильмана для сильно выпуклой оболочки в гильбертовом пространстве”, Матем. заметки, 71:1 (2002), 37–42  mathnet  crossref  mathscinet  zmath; M. V. Balashov, “An Analog of the Krein–Mil'man Theorem for Strongly Convex Hulls in Hilbert Space”, Math. Notes, 71:1 (2002), 34–38  crossref  elib
    5. М. В. Балашов, “О $P$-свойстве выпуклых компактов”, Матем. заметки, 71:3 (2002), 323–333  mathnet  crossref  mathscinet  zmath; M. V. Balashov, “On the $P$-Property of Compact Convex Sets”, Math. Notes, 71:3 (2002), 295–304  crossref  elib
    6. Pichard, K, “Unified treatment of algebraic and geometric difference by a new difference scheme and its continuity properties”, Set-Valued Analysis, 11:2 (2003), 111  crossref  mathscinet  zmath  elib
    7. Rublev, IV, “Attainability sets in cascade control systems”, Differential Equations, 40:12 (2004), 1716  crossref  mathscinet  zmath  elib
    8. Polovinkin, ES, “Convex bodies of constant width”, Doklady Mathematics, 70:1 (2004), 560  mathscinet  elib
    9. Г. Е. Иванов, “Множества, слабо выпуклые по Виалю и по Ефимову–Стечкину”, Изв. РАН. Сер. матем., 69:6 (2005), 35–60  mathnet  crossref  mathscinet  zmath  elib; G. E. Ivanov, “Weak convexity in the senses of Vial and Efimov–Stechkin”, Izv. Math., 69:6 (2005), 1113–1135  crossref
    10. Veliov, VM, “Error analysis of discrete approximations to bang-bang optimal control problems: the linear case”, Control and Cybernetics, 34:3 (2005), 967  mathscinet  zmath  elib
    11. А. С. Дудова, “Об устойчивости решения задачи наилучшего приближения выпуклого компакта шаром”, Изв. вузов. Матем., 2006, № 7, 25–33  mathnet  mathscinet  elib; A. S. Dudova, “On the stability of the solution of the best approximation of a convex compact set by a ball”, Russian Math. (Iz. VUZ), 50:7 (2006), 22–30
    12. Veliov, VM, “Approximations with error estimates for optimal control problems for linear systems”, Large-Scale Scientific Computing, 3743 (2006), 263  crossref  mathscinet  zmath
    13. С. И. Дудов, А. С. Дудова, “Об устойчивости решения задач о внешней и внутренней оценке выпуклого компакта шаром”, Ж. вычисл. матем. и матем. физ., 47:10 (2007), 1657–1671  mathnet  mathscinet  elib; S. I. Dudov, A. S. Dudova, “On the stability of inner and outer approximations of a convex compact set by a ball”, Comput. Math. Math. Phys., 47:10 (2007), 1589–1602  crossref  elib
    14. Barrett, DE, “The spectrum of the Leray transform for convex Reinhardt domains in C-2”, Journal of Functional Analysis, 257:9 (2009), 2780  crossref  mathscinet  zmath
    15. Rusmevichientong P., Tsitsiklis J.N., “Linearly Parameterized Bandits”, Mathematics of Operations Research, 35:2 (2010), 395–411  crossref  mathscinet  zmath
    16. Balashov M.V., Repovs D., “Polyhedral approximations of strictly convex compacta”, Journal of Mathematical Analysis and Applications, 374:2 (2011), 529–537  crossref  mathscinet  zmath
    17. Nikodem K., Pales Z., “Characterizations of Inner Product Spaces by Strongly Convex Functions”, Banach Journal of Mathematical Analysis, 5:1 (2011), 83–87  crossref  mathscinet  zmath
    18. Ger R., Nikodem K., “Strongly convex functions of higher order”, Nonlinear Analysis-Theory Methods & Applications, 74:2 (2011), 661–665  crossref  mathscinet  zmath
    19. Maxim V. Balashov, Dušan Repovš, “Uniformly convex subsets of the Hilbert space with modulus of convexity of the second order”, Journal of Mathematical Analysis and Applications, 377:2 (2011), 754  crossref  mathscinet  zmath
    20. Jacek Tabor, Józef Tabor, Marek Żołdak, “On ω-strongly quasiconvex and ω-strongly quasiconcave sequences”, Aequat. Math, 2011  crossref
    21. Reissig G., “Computing Abstractions of Nonlinear Systems”, IEEE Trans Automat Control, 56:11 (2011), 2583–2598  crossref  mathscinet
    22. Maxim V. Balashov, Maxim O. Golubev, “About the Lipschitz property of the metric projection in the Hilbert space”, Journal of Mathematical Analysis and Applications, 2012  crossref  mathscinet
    23. Tabor J., Tabor J., Zoldak M., “On Omega-Quasiconvex Functions”, Math. Inequal. Appl., 15:4 (2012), 845–857  crossref  mathscinet  zmath
    24. Mako J., Nikodem K., Pales Z., “On Strong (Alpha,F)-Convexity”, Math. Inequal. Appl., 15:2 (2012), 289–299  mathscinet  zmath
    25. Ипатов Д.Е., “Разрешимость дифференциального включения для многослойной модели общей циркуляции океана с многозначной правой частью”, Альманах современной науки и образования, 2012, 52–55  elib
    26. José Pedro Moreno, Rolf Schneider, “Lipschitz selections of the diametric completion mapping in Minkowski spaces”, Advances in Mathematics, 233:1 (2013), 248  crossref  mathscinet  zmath
    27. Hugo Leiva, Nelson Merentes, Kazimierz Nikodem, José Luis Sánchez, “Strongly convex set-valued maps”, J Glob Optim, 2013  crossref
    28. Andrej V. Plotnikov, Natalia V. Skripnik, “Existence and Uniqueness Theorem for Set-Valued Volterra Integral Equations”, AJAMS, 1:3 (2013), 41  crossref  mathscinet
    29. Tabor J., Tabor J., Zoldak M., “Strongly Midquasiconvex Functions”, J. Convex Anal., 20:2 (2013), 531–543  mathscinet  zmath
    30. Е. С. Половинкин, “Об одном контрпримере в анализе”, Матем. заметки, 95:1 (2014), 123–128  mathnet; E. S. Polovinkin, “On a Counterexample in Analysis”, Math. Notes, 95:1 (2014), 111–115  crossref
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