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Mat. Sb., 2000, Volume 191, Number 1, Pages 27–64 (Mi msb447)  

This article is cited in 23 scientific papers (total in 24 papers)

$M$-strongly convex subsets and their generating sets

M. V. Balashov, E. S. Polovinkin

Moscow Institute of Physics and Technology

Abstract: For subsets of a Banach space the notions of a generating set $M$ and an $M$-strongly convex set are introduced. The latter can be represented as the intersection of sets of the form $M+x$, which are translates of the generating set $M$. A generating set must satisfy a condition that ensures a special support principle, as shown in the paper. Using this support principle a new area of convex analysis is constructed enabling one to strengthen classical results of the type of the Caratheodory and Krein–Milman theorems. Various classes of generating sets are described and the properties of $M$-strongly convex sets are studied.


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English version:
Sbornik: Mathematics, 2000, 191:1, 25–60

Bibliographic databases:

UDC: 517.977
MSC: Primary 90C25, 49J52, 52A07; Secondary 46B20, 46N10
Received: 18.02.1999

Citation: M. V. Balashov, E. S. Polovinkin, “$M$-strongly convex subsets and their generating sets”, Mat. Sb., 191:1 (2000), 27–64; Sb. Math., 191:1 (2000), 25–60

Citation in format AMSBIB
\by M.~V.~Balashov, E.~S.~Polovinkin
\paper $M$-strongly convex subsets and their generating sets
\jour Mat. Sb.
\yr 2000
\vol 191
\issue 1
\pages 27--64
\jour Sb. Math.
\yr 2000
\vol 191
\issue 1
\pages 25--60

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    This publication is cited in the following articles:
    1. 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  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    2. M. V. Balashov, “On the $P$-Property of Compact Convex Sets”, Math. Notes, 71:3 (2002), 295–304  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    3. Polovinkin E.S., “Convex bodies of constant width”, Dokl. Math., 70:1 (2004), 560  mathnet  mathscinet  zmath  isi  elib
    4. Moreno J.P., Papini P.L., Phelps R.R., “New families of convex sets related to diametral maximality”, J. Convex Anal., 13:3-4 (2006), 823–837  mathscinet  zmath  isi  elib
    5. G. E. Ivanov, “A criterion of smooth generating sets”, Sb. Math., 198:3 (2007), 343–368  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    6. M. V. Balashov, G. E. Ivanov, “Weakly convex and proximally smooth sets in Banach spaces”, Izv. Math., 73:3 (2009), 455–499  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    7. Balashov M.V., Repovs D., “Uniformly convex subsets of the Hilbert space with modulus of convexity of the second order”, J Math Anal Appl, 377:2 (2011), 754–761  crossref  mathscinet  zmath  isi  elib  scopus  scopus  scopus
    8. 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  isi  scopus  scopus  scopus
    9. Pedro Moreno J., Schneider R., “Diametrically Complete Sets in Minkowski Spaces”, Isr. J. Math., 191:2 (2012), 701–720  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    10. Balashov M.V., Repovs D., “On Plis Metric on the Space of Strictly Convex Compacta”, J. Convex Anal., 19:1 (2012), 171–183  mathscinet  zmath  isi  elib
    11. Moreno J.P., Schneider R., “Local Lipschitz Continuity of the Diametric Completion Mapping”, Houst. J. Math., 38:4 (2012), 1207–1223  mathscinet  zmath  isi
    12. Pedro Moreno J., Schneider R., “Lipschitz Selections of the Diametric Completion Mapping in Minkowski Spaces”, Adv. Math., 233:1 (2013), 248–267  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    13. Andrej V. Plotnikov, Natalia V. Skripnik, “Existence and Uniqueness Theorem for Set-Valued Volterra Integral Equations”, AJAMS, 1:3 (2013), 41  crossref  mathscinet
    14. Balashov M.V., Golubev M.O., “Weak Concavity of the Antidistance Function”, J. Convex Anal., 21:4 (2014), 951–964  mathscinet  zmath  isi
    15. Moreno J.P., Schneider R., “Some Geometry of Convex Bodies in C(K) Spaces”, J. Math. Pures Appl., 103:2 (2015), 352–373  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    16. F. S. Stonyakin, “Sequential analogues of the Lyapunov and Krein–Milman theorems in Fréchet spaces”, Journal of Mathematical Sciences, 225:2 (2017), 322–344  mathnet  crossref
    17. A. R. Alimov, “Prostranstva Mazura i 4.3-svoistvo peresecheniya $(BM)$-prostranstv”, Izv. Sarat. un-ta. Nov. ser. Ser. Matematika. Mekhanika. Informatika, 16:2 (2016), 133–137  mathnet  crossref  mathscinet  elib
    18. M. V. Balashov, O. V. Besov, B. I. Golubov, V. V. Goryainov, V. N. Diesperov, S. I. Dudov, G. E. Ivanov, S. P. Konovalov, R. V. Konstantinov, A. B. Kurzhanskii, S. R. Nasyrov, A. G. Sergeev, V. V. Starkov, V. M. Tikhomirov, M. I. Shabunin, “Evgenii Sergeevich Polovinkin (on his 70th birthday)”, Russian Math. Surveys, 71:5 (2016), 983–987  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    19. Holmsen A.F., Karasev R., “Colorful theorems for strong convexity”, Proc. Amer. Math. Soc., 145:6 (2017), 2713–2726  crossref  mathscinet  zmath  isi  scopus
    20. Balashov M.V., “About the Gradient Projection Algorithm For a Strongly Convex Function and a Proximally Smooth Set”, J. Convex Anal., 24:2 (2017), 493–500  mathscinet  zmath  isi
    21. Jahn T., Martini H., Richter Ch., “Ball Convex Bodies in Minkowski Spaces”, Pac. J. Math., 289:2 (2017), 287–316  crossref  mathscinet  zmath  isi  scopus
    22. Plotnikov A.V., Komleva T.A., Molchanyuk I.V., “Existence and Uniqueness Theorem For Set-Valued Volterra-Hammerstein Integral Equations”, Asian-Eur. J. Math., 11:3 (2018), 1850036  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    23. Balashov M.V., Ivanov G.E., “The Farthest and the Nearest Points of Sets”, J. Convex Anal., 25:3 (2018), 1019–1031  mathscinet  zmath  isi
    24. M. V. Balashov, “Uslovie Lipshitsa metricheskoi proektsii v gilbertovom prostranstve”, Fundament. i prikl. matem., 22:1 (2018), 13–29  mathnet
  • Математический сборник - 1992–2005 Sbornik: Mathematics (from 1967)
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