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Mat. Zametki, 2006, Volume 80, Issue 2, Pages 163–170 (Mi mz2795)  

This article is cited in 11 scientific papers (total in 11 papers)

On Farthest Points of Sets

M. V. Balashov, G. E. Ivanov

Moscow Institute of Physics and Technology

Abstract: For a convex closed bounded set in a Banach space, we study the existence and uniqueness problem for a point of this set that is the farthest point from a given point in space. In terms of the existence and uniqueness of the farthest point, as well as the Lipschitzian dependence of this point on a point in space, we obtain necessary and sufficient conditions for the strong convexity of a set in several infinite-dimensional spaces, in particular, in a Hilbert space. A set representable as the intersection of closed balls of a fixed radius is called a strongly convex set. We show that the condition “for each point in space that is sufficiently far from a set, there exists a unique farthest point of the set” is a criterion for the strong convexity of a set in a finite-dimensional normed space, where the norm ball is a strongly convex set and a generating set.

Keywords: farthest point, existence and uniqueness problem, strong convexity, Hilbert space, reflexive Banach space, proximity function

DOI: https://doi.org/10.4213/mzm2795

Full text: PDF file (376 kB)
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English version:
Mathematical Notes, 2006, 80:2, 159–166

Bibliographic databases:

UDC: 517.982.252, 517.982.256
Received: 28.03.2005

Citation: M. V. Balashov, G. E. Ivanov, “On Farthest Points of Sets”, Mat. Zametki, 80:2 (2006), 163–170; Math. Notes, 80:2 (2006), 159–166

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, “Farthest Points and Strong Convexity of Sets”, Math. Notes, 87:3 (2010), 355–366  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    2. Balashov M.V., “Uslovie lipshitsa dlya naibolee udalennoi tochki v gilbertovom prostranstve”, Trudy moskovskogo fiziko-tekhnicheskogo instituta, 2012, 8–14  elib
    3. Mirmostafaee A.K., Mirzavaziri M., “Uniquely Remotal Sets in C(0)-Sums and l(Infinity)-Sums of Fuzzy Normed Spaces”, Iran. J. Fuzzy. Syst., 9:6 (2012), 113–122  mathscinet  zmath  isi
    4. Balashov M.V., Golubev M.O., “Weak Concavity of the Antidistance Function”, J. Convex Anal., 21:4 (2014), 951–964  mathscinet  zmath  isi
    5. Khademzadeh H.R., Mazaheri H., “Monotonicity and the Dominated Farthest Points Problem in Banach Lattice”, Abstract Appl. Anal., 2014, 616989  crossref  mathscinet  isi  elib  scopus
    6. Balashov M.V., “Antidistance and Antiprojection in the Hilbert Space”, J. Convex Anal., 22:2 (2015), 521–536  mathscinet  zmath  isi
    7. 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
    8. 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
    9. 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
    10. Alimov A.R., “Solarity of Sets in Max-Approximation Problems”, J. Fixed Point Theory Appl., 21:3 (2019), UNSP 76  crossref  isi
    11. 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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