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Izv. RAN. Ser. Mat., 2009, Volume 73, Issue 3, Pages 23–66 (Mi izv2646)  

This article is cited in 34 scientific papers (total in 34 papers)

Weakly convex and proximally smooth sets in Banach spaces

M. V. Balashov, G. E. Ivanov

Moscow Institute of Physics and Technology

Abstract: We establish interconnections between the conditions of weak convexity in the sense of Vial, weak convexity in the sense of Efimov–Stechkin, and proximal smoothness of sets in Banach spaces. We prove a theorem on the separation by a sphere of two disjoint sets, one of which is weakly convex in the sense of Vial and the other is strongly convex. We also prove that weakly convex and proximally smooth sets are locally connected, and study questions related to the preservation of the conditions of weak convexity and proximal smoothness under passage to the limit.

Keywords: proximal smoothness, weak convexity, uniform convexity, uniform smoothness, generating set, separation by a sphere, supporting ball.

DOI: https://doi.org/10.4213/im2646

Full text: PDF file (908 kB)
References: PDF file   HTML file

English version:
Izvestiya: Mathematics, 2009, 73:3, 455–499

Bibliographic databases:

UDC: 517.982.252
MSC: 46B07, 46B20, 52A30
Received: 04.04.2007

Citation: M. V. Balashov, G. E. Ivanov, “Weakly convex and proximally smooth sets in Banach spaces”, Izv. RAN. Ser. Mat., 73:3 (2009), 23–66; Izv. Math., 73:3 (2009), 455–499

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. Adly S., Nacry F., Thibault L., “New Metric Properties For Prox-Regular Sets”, Math. Program.  crossref  isi
    2. Adly S., Nacry F., Thibault L., “Prox-Regular Sets and Legendre-Fenchel Transform Related to Separation Properties”, Optimization  crossref  isi
    3. Balashov M.V., Polyak B.T., Tremba A.A., “Gradient Projection and Conditional Gradient Methods For Constrained Nonconvex Minimization”, Numer. Funct. Anal. Optim.  crossref  mathscinet  isi
    4. Balashov M.V., “The Gradient Projection Algorithm For Smooth Sets and Functions in Nonconvex Case”, Set-Valued Var. Anal.  crossref  isi
    5. G. E. Ivanov, “Farthest Points and Strong Convexity of Sets”, Math. Notes, 87:3 (2010), 355–366  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    6. Balashov M.V., Repovš D., “Weakly convex sets and modulus of nonconvexity”, J. Math. Anal. Appl., 371:1 (2010), 113–127  crossref  mathscinet  zmath  isi  elib
    7. A. R. Alimov, “Monotone path-connectedness of $R$-weakly convex sets in the space $C(Q)$”, J. Math. Sci., 185:3 (2012), 360–366  mathnet  crossref
    8. Balashov M.V., Golubev M.O., “About the Lipschitz property of the metric projection in the Hilbert space”, J. Math. Anal. Appl., 394:2 (2012), 545–551  crossref  mathscinet  zmath  isi  elib
    9. A. R. Alimov, “Monotone path-connectedness of $R$-weakly convex sets in spaces with linear ball embedding”, Eurasian Math. J., 3:2 (2012), 21–30  mathnet  mathscinet  zmath
    10. Ivanov G.M., “Uklonenie vypukloi obolochki ogranichennykh mnozhestv”, Tr. Moskovskogo fiziko-tekhnicheskogo instituta, 2012, no. 4-16, 105–112  elib
    11. Ivanov G.E., Lopushanski M.S., “Approksimativnye svoistva slabo vypuklykh mnozhestv v prostranstvakh s nesimmetrichnoi polunormoi”, Tr. Moskovskogo fiziko-tekhnicheskogo instituta, 2012, no. 4-16, 94–104  elib
    12. M. V. Balashov, “Proximal smoothness of a set with the Lipschitz metric projection”, J. Math. Anal. Appl., 406:1 (2013), 360–363  crossref  mathscinet  zmath  isi  elib
    13. Balashov M.V., “Weak Convexity of the Distance Function”, J. Convex Anal., 20:1 (2013), 93–106  mathscinet  zmath  isi  elib
    14. G. E. Ivanov, M. S. Lopushanski, “Well-posedness of approximation and optimization problems for weakly convex sets and functions”, J. Math. Sci., 209:1 (2015), 66–87  mathnet  crossref  mathscinet
    15. M. V. Balashov, “Maximization of a function with Lipschitz continuous gradient”, J. Math. Sci., 209:1 (2015), 12–18  mathnet  crossref  mathscinet
    16. A. R. Alimov, “Monotone path-connectedness and solarity of Menger-connected sets in Banach spaces”, Izv. Math., 78:4 (2014), 641–655  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    17. Balashov M.V., Golubev M.O., “Weak Concavity of the Antidistance Function”, J. Convex Anal., 21:4 (2014), 951–964  mathscinet  zmath  isi
    18. A. R. Alimov, I. G. Tsar'kov, “Connectedness and other geometric properties of suns and Chebyshev sets”, J. Math. Sci., 217:6 (2016), 683–730  mathnet  crossref  mathscinet
    19. G. E. Ivanov, “Sharp estimates for the moduli of continuity of metric projections onto weakly convex sets”, Izv. Math., 79:4 (2015), 668–697  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    20. Ivanov G.E., “Continuity and Selections of the Intersection Operator Applied To Nonconvex Sets”, J. Convex Anal., 22:4 (2015), 939–962  mathscinet  isi
    21. Ivanov G.E., “Weak Convexity of Sets and Functions in a Banach Space”, J. Convex Anal., 22:2 (2015), 365–398  mathscinet  zmath  isi
    22. A. R. Alimov, I. G. Tsar'kov, “Connectedness and solarity in problems of best and near-best approximation”, Russian Math. Surveys, 71:1 (2016), 1–77  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    23. 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
    24. G. E. Ivanov, M. C. Lopushanski, “Teorema ob otdelimosti dlya nevypuklykh mnozhestv i eë prilozheniya”, Fundament. i prikl. matem., 21:4 (2016), 23–66  mathnet  mathscinet
    25. Ivanov G.E., Lopushanski M.S., “Separation Theorems For Nonconvex Sets in Spaces With Non-Symmetric Seminorm”, Math. Inequal. Appl., 20:3 (2017), 737–754  crossref  mathscinet  zmath  isi
    26. 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
    27. Lopushanski M.S., “Normal Regularity of Weakly Convex Sets in Asymmetric Normed Spaces”, J. Convex Anal., 25:3 (2018), 737–758  mathscinet  zmath  isi
    28. A. R. Alimov, “Ogranichennaya styagivaemost strogikh solnts v trëkhmernykh prostranstvakh”, Fundament. i prikl. matem., 22:1 (2018), 3–11  mathnet  mathscinet
    29. M. V. Balashov, “Uslovie Lipshitsa metricheskoi proektsii v gilbertovom prostranstve”, Fundament. i prikl. matem., 22:1 (2018), 13–29  mathnet
    30. R. A. Khachatryan, “O nekotorykh svoistvakh pochti vypuklykh funktsii i mnozhestv”, Vestnik Tambovskogo universiteta. Seriya: estestvennye i tekhnicheskie nauki, 23:124 (2018), 824–837  mathnet  crossref  elib
    31. Alimov A.R., Shchepin V E., “Convexity of Suns in Tangent Directions”, J. Convex Anal., 26:4 (2019), 1071–1076  isi
    32. M. V. Balashov, “The gradient projection algorithm for a proximally smooth set and a function with Lipschitz continuous gradient”, Sb. Math., 211:4 (2020), 481–504  mathnet  crossref  crossref  mathscinet  isi  elib
    33. Balashov V M., “Lipschitz Stability of Extremal Problems With a Strongly Convex Set”, J. Convex Anal., 27:1 (2020), 103–116  mathscinet  isi
    34. Ivanov G.E., “Nonlinear Images of Sets. i: Strong and Weak Convexity”, J. Convex Anal., 27:1 (2020), 361–380  mathscinet  isi
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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