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Some theorems on Chebyshev sets

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Abstract

We consider the class of δ-suns which is used in the study of Chebyshev sets. We give sufficient conditions for a set to be a δ-sun. We prove that, in a uniformly smooth Banach space, a weakly closed Chebyshev set is convex.

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Literature cited

  1. N. V. Efimov and S. B. Stechkin, “Some properties of Chebyshev sets,” Dokl. Akad. Nauk SSSR,118, No. 1, 17–19 (1958).

    Google Scholar 

  2. I. Singer, Best Approximation in Normed Linear Spaces by Elements of Linear Subspaces [in Rumanian], Edit. Acad. R. S. Romania, Bucharest (1967).

    Google Scholar 

  3. A. L. Garkavi, “The theory of best approximation in normed linear spaces,” in: Matem. Analiz. (Itogi Nauki), 1967 VINII Akad. Nauk SSSR, Moscow (1969), pp. 75–132.

    Google Scholar 

  4. L. P. Vlasov, “On Chebyshev and approximately convex sets,” Matem. Zametki,2, No. 2, 191–200 (1967).

    Google Scholar 

  5. V. L. Klee, “Convexity of Chebyshev sets,” Math. Ann.,142, 292–304 (1961).

    Google Scholar 

  6. L. P. Vlasov, “Almost convex and Chebyshev sets,” Matem. Zametki,8, No. 5, 545–550 (1970).

    Google Scholar 

  7. J. Blatter, P. D. Morris, and D. E. Wulbert, “Continuity of the set-valued metric projection,” Math. Ann.,178, No. 1, 12–24 (1968).

    Google Scholar 

  8. D. F. Cudia, Rotundity and Convexity, Proc. Sympos. Pure Math., Vol. 7, Amer. Math. Soc., Providence, R. I. (1963), pp. 73–97.

    Google Scholar 

  9. N. V. Efimov and S. B. Stechkin, “Approximate compactness and Chebyshev sets,” Dokl. Akad. Nauk SSSR,140, No. 3, 522–524 (1961).

    Google Scholar 

  10. L. P. Vlasov, “Approximate properties of sets in Banach spaces,” Matem. Zametki,7, No. 5, 593–604 (1970).

    Google Scholar 

  11. E. Asplund, “Chebyshev sets in Hilbert space,” Trans. Amer. Math. Soc.,144, 235–240 (1969).

    Google Scholar 

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Authors and Affiliations

  1. Institute of Mathematics and Mechanics, Academy of Sciences of the USSR, USSR

    L. P. Vlasov

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  1. L. P. Vlasov
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Additional information

Translated from Matematicheskie Zametki, Vol. 11, No. 2, pp. 135–144, February, 1972.

The author expresses his thanks to V. I. Berdyshev and N. I. Chernykh for a lively discussion of this note.

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Vlasov, L.P. Some theorems on Chebyshev sets. Mathematical Notes of the Academy of Sciences of the USSR 11, 87–92 (1972). https://doi.org/10.1007/BF01097922

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  • DOI: https://doi.org/10.1007/BF01097922

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