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Mat. Zametki, 2006, Volume 80, Issue 4, Pages 483–489 (Mi mz2840)  

This article is cited in 9 scientific papers (total in 9 papers)

Properties of the metric projection on weakly vial-convex sets and parametrization of set-valued mappings with weakly convex images

M. V. Balashov, G. E. Ivanov

Moscow Institute of Physics and Technology

Abstract: We continue studying the class of weakly convex sets (in the sense of Vial). For points in a sufficiently small neighborhood of a closed weakly convex subset in Hilbert space, we prove that the metric projection on this set exists and is unique. In other words, we show that the closed weakly convex sets have a Chebyshev layer. We prove that the metric projection of a point on a weakly convex set satisfies the Lipschitz condition with respect to a point and the Hölder condition with exponent $1/2$ with respect to a set. We develop a method for constructing a continuous parametrization of a set-valued mapping with weakly convex images. We obtain an explicit estimate for the modulus of continuity of the parametrizing function.


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English version:
Mathematical Notes, 2006, 80:4, 461–467

Bibliographic databases:

UDC: 517.982.252
Received: 14.03.2005

Citation: M. V. Balashov, G. E. Ivanov, “Properties of the metric projection on weakly vial-convex sets and parametrization of set-valued mappings with weakly convex images”, Mat. Zametki, 80:4 (2006), 483–489; Math. Notes, 80:4 (2006), 461–467

Citation in format AMSBIB
\by M.~V.~Balashov, G.~E.~Ivanov
\paper Properties of the metric projection on weakly vial-convex sets and parametrization of set-valued mappings with weakly convex images
\jour Mat. Zametki
\yr 2006
\vol 80
\issue 4
\pages 483--489
\jour Math. Notes
\yr 2006
\vol 80
\issue 4
\pages 461--467

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    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. Ivanov G. E., “Continuous selections of multifunctions with weakly convex values”, Topology Appl., 155:8 (2008), 851–857  crossref  mathscinet  zmath  isi  elib  scopus
    3. Goncharov V.V., Pereira F.F., “Neighbourhood retractions of nonconvex sets in a Hilbert space via sublinear functionals”, J. Convex Anal., 18:1 (2011), 1–36  mathscinet  zmath  isi  elib
    4. Goncharov V.V., Pereira F.F., “Geometric Conditions for Regularity in a Time-Minimum Problem with Constant Dynamics”, J. Convex Anal., 19:3 (2012), 631–669  mathscinet  zmath  isi  elib
    5. 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, eds. Daras N., Rassias T., Springer International Publishing Ag, 2017, 259–297  crossref  mathscinet  zmath  isi  scopus
    6. 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
    7. Lopushanski M.S., “Normal Regularity of Weakly Convex Sets in Asymmetric Normed Spaces”, J. Convex Anal., 25:3 (2018), 737–758  mathscinet  zmath  isi
    8. M. V. Balashov, “Uslovie Lipshitsa metricheskoi proektsii v gilbertovom prostranstve”, Fundament. i prikl. matem., 22:1 (2018), 13–29  mathnet
    9. M. V. Balashov, “The Pliś metric and Lipschitz stability of minimization problems”, Sb. Math., 210:7 (2019), 911–927  mathnet  crossref  crossref  adsnasa  isi  elib
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