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

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.

DOI: https://doi.org/10.4213/mzm2840

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

Bibliographic databases:

UDC: 517.982.252

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
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• http://mi.mathnet.ru/eng/mz2840
• https://doi.org/10.4213/mzm2840
• http://mi.mathnet.ru/eng/mz/v80/i4/p483

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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
2. Ivanov G. E., “Continuous selections of multifunctions with weakly convex values”, Topology Appl., 155:8 (2008), 851–857
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
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
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
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
7. Lopushanski M.S., “Normal Regularity of Weakly Convex Sets in Asymmetric Normed Spaces”, J. Convex Anal., 25:3 (2018), 737–758
8. M. V. Balashov, “Uslovie Lipshitsa metricheskoi proektsii v gilbertovom prostranstve”, Fundament. i prikl. matem., 22:1 (2018), 13–29
9. M. V. Balashov, “The Pliś metric and Lipschitz stability of minimization problems”, Sb. Math., 210:7 (2019), 911–927
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