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Journal of Mathematical Sciences

, Volume 197, Issue 4, pp 447–454 | Cite as

Local Solarity of Suns in Normed Linear Spaces

  • A. R. Alimov
Article

Abstract

The paper is concerned with solarity of intersections of suns with bars (in particular, with closed balls and extreme hyperstrips) in normed linear spaces. A sun in a finite-dimensional (BM)-space (in particular, in 1(n)) is shown to be monotone path connected. A nonempty intersection of an m-connected set (in particular, a sun in a two-dimensional space or in a finite-dimensional (BM)-space) with a bar is shown to be a monotone path-connected sun. Similar results are obtained for boundedly compact subsets of infinite-dimensional spaces. A nonempty intersection of a monotone path-connected subset of a normed space with a bar is shown to be a monotone path-connected α-sun.

Keywords

Closed Ball Normed Linear Space Separable Banach Space Nonempty Intersection Converse Assertion 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    A. R. Alimov, “Connectedness of suns in the space c 0,” Izv. Math., 69, No. 4, 651–666 (2005).CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    A. R. Alimov, “The geometric structure of Chebyshev sets in (n),” Funct. Anal. Appl., 39, No. 1, 1–8 (2005).CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    A. R. Alimov, “Monotone path-connectedness of Chebyshev sets in the space C(Q),” Sb. Math., 197, No. 9, 1259–1272 (2006).CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    A. R. Alimov, “Preservation of approximative properties of subsets of Chebyshev sets and suns in (n),” Izv. Math., 70, No. 5, 857–866 (2006).CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    A. R. Alimov, “Preservation of approximative properties of Chebyshev sets and suns in a plane,” Moscow Univ. Math. Bull., 63, No. 5, 198–201 (2008).CrossRefMathSciNetGoogle Scholar
  6. 6.
    A. R. Alimov, “A monotone path connected Chebyshev set is a sun,” Math. Notes, 91, No. 2, 290–292 (2012).CrossRefGoogle Scholar
  7. 7.
    A. R. Alimov, “Bounded strict solarity of strict suns in the space C(Q),” Moscow Univ. Math. Bull. (2012).Google Scholar
  8. 8.
    A. R. Alimov and V. Yu. Protasov, “Separation of convex sets by extreme hyperplanes,” Fundam. Prikl. Mat., 17, No. 4, 3–12 (2011/2012).MathSciNetGoogle Scholar
  9. 9.
    V. G. Boltyanskii and P. S. Soltan, Combinatorial Geometry and Convexity Classes [in Russian], Shtiintsa, Kishinev (1978).Google Scholar
  10. 10.
    B. Brosowski and F. Deutsch, “On some geometric properties of suns,” J. Approx. Theory, 10, No. 3, 245–267 (1974).CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    B. Brosowski, F. Deutsch, J. Lambert, and P. D. Morris, “Chebyshev sets which are not suns,” Math. Ann., 212, No. 1, 89–101 (1974).CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    A. L. Brown, “Suns in normed linear spaces which are finite dimensional,” Math. Ann., 279, 87–101 (1987).CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    A. L. Brown, “On the connectedness properties of suns in finite dimensional spaces,” Proc. Cent. Math. Anal. Aust. Natl. Univ., 20, 1–15 (1988).Google Scholar
  14. 14.
    A. L. Brown, “Suns in polyhedral spaces,” in: D. G. Álvarez, G. Lopez Acedo, and R. V. Caro, eds., Seminar of Math. Analysis. Univ. Malaga and Seville (Spain), Sept. 2002—Feb. 2003. Proceedings, Univ. Sevilla, Sevilla (2003), pp. 139–146.Google Scholar
  15. 15.
    E. Dancer and B. Sims, “Weak star separability,” Bull. Aust. Math. Soc., 20, No. 2, 253–257 (1979).CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    C. Franchetti and S. Roversi, Suns, M-connected sets and P-acyclic sets in Banach spaces, Preprint No. 50139, Inst. di Matematica Applicata G. Sansone (1988).Google Scholar
  17. 17.
    J. R. Giles, “The Mazur intersection problem,” J. Convex Anal., 13, No. 3–4, 739–750 (2006).zbMATHMathSciNetGoogle Scholar
  18. 18.
    J. R. Giles, D. A. Gregory, and B. Sims, “Characterisation of normed linear spaces with Mazur’s intersection property,” Bull. Aust. Math. Soc., 18, 105–123 (1978).CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    A. S. Granero, M. Jiménez-Sevilla, and J. P. Moreno, “Intersections of closed balls and geometry of Banach spaces,” Extracta Math., 19, No. 1, 55–92 (2004).zbMATHMathSciNetGoogle Scholar
  20. 20.
    V. A. Koshcheev, “Connectedness and solar properties of sets in normed linear spaces,” Math. Notes, 19, No. 2, 158–164 (1976).CrossRefzbMATHGoogle Scholar
  21. 21.
    J. P. Moreno and R. Schneider, “Continuity properties of the ball hull mapping,” Nonlinear Anal., 66, 914–925 (2007).CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    R. R. Phelps, “A representation theorem for bounded convex sets,” Proc. Amer. Math. Soc., 11, 976–983 (1960).CrossRefMathSciNetGoogle Scholar
  23. 23.
    I. G. Tsar’kov, “Bounded Chebyshev sets in finite-dimensional Banach spaces,” Math. Notes, 36, No. 1, 530–537 (1984).CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    A. A. Vasil’eva, “Closed spans in vector-valued function spaces and their approximative properties,” Izv. Math., 68, No. 4, 709–747 (2004).CrossRefzbMATHMathSciNetGoogle Scholar
  25. 25.
    L. P. Vlasov, “Approximative properties of sets in normed linear spaces,” Russ. Math. Surv., 28, No. 6, 1–66 (1973).CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Faculty of Mechanics and MathematicsMoscow State UniversityMoscowRussia

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