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Mat. Zametki, 2006, Volume 79, Issue 1, Pages 60–86 (Mi mz2674)  

This article is cited in 6 scientific papers (total in 6 papers)

Weakly Convex Sets and Their Properties

G. E. Ivanov

Moscow Institute of Physics and Technology

Abstract: In this paper, the notion of a weakly convex set is introduced. Sharp estimates for the weak convexity constants of the sum and difference of such sets are given. It is proved that, in Hilbert space, the smoothness of a set is equivalent to the weak convexity of the set and its complement. Here, by definition, the smoothness of a set means that the field of unit outward normal vectors is defined on the boundary of the set; this vector field satisfies the Lipschitz condition. We obtain the minimax theorem for a class of problems with smooth Lebesgue sets of the goal function and strongly convex constraints. As an application of the results obtained, we prove the alternative theorem for program strategies in a linear differential quality game.

DOI: https://doi.org/10.4213/mzm2674

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English version:
Mathematical Notes, 2006, 79:1, 55–78

Bibliographic databases:

UDC: 517.982.252+517.978.2
Received: 16.01.2004

Citation: G. E. Ivanov, “Weakly Convex Sets and Their Properties”, Mat. Zametki, 79:1 (2006), 60–86; Math. Notes, 79:1 (2006), 55–78

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. G. E. Ivanov, M. V. Balashov, “Lipschitz continuous parametrizations of set-valued maps with weakly convex images”, Izv. Math., 71:6 (2007), 1123–1143  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    2. Balashov M.V. Golubev M.O., “Weak Concavity of the Antidistance Function”, J. Convex Anal., 21:4 (2014), 951–964  mathscinet  zmath  isi
    3. Goncharov V.V. Ivanov G.E., “Strong and Weak Convexity of Closed Sets in a Hilbert Space”, Operations Research, Engineering, and Cyber Security: Trends in Applied Mathematics and Technology, Springer Optimization and Its Applications, 113, ed. Daras N. Rassias T., Springer International Publishing Ag, 2017, 259–297  crossref  mathscinet  zmath  isi  scopus
    4. Lopushanski M.S., “Normal Regularity of Weakly Convex Sets in Asymmetric Normed Spaces”, J. Convex Anal., 25:3 (2018), 737–758  mathscinet  zmath  isi
    5. Ivanov G.E., Golubev M.O., “Strong and Weak Convexity in Nonlinear Differential Games”, IFAC PAPERSONLINE, 51:32 (2018), 13–18  crossref  isi  scopus
    6. V. N. Ushakov, A. A. Ershov, M. V. Pershakov, “Ob odnom dopolnenii k otsenke L.S. Pontryagina geometricheskoi raznosti mnozhestv na ploskosti”, Izv. IMI UdGU, 54 (2019), 63–73  mathnet  crossref  elib
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