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Set-Valued Analysis

, Volume 12, Issue 1–2, pp 61–77 | Cite as

Constructible Convex Sets

  • Jonathan M. Borwein
  • Jon D. Vanderwerff
Article

Abstract

We investigate when closed convex sets can be written as countable intersections of closed half-spaces in Banach spaces. It is reasonable to consider this class to comprise the constructible convex sets since such sets are precisely those that can be defined by a countable number of linear inequalities, hence are accessible to techniques of semi-infinite convex programming. We also explore some model theoretic implications. Applications to set convergence are given as limiting examples.

convex sets countable intersections biorthogonal systems Mosco convergence slice convergence Martin's axiom Kunen's space 

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References

  1. 1.
    Azagra, D. and Ferrera, J.: Every closed convex set is the set of minimizers of some C -smooth convex function, Proc. Amer. Math. Soc. 130 (2002), 3687-3892.Google Scholar
  2. 2.
    Beer, G.: Topologies on Closed and Closed Convex Sets, Kluwer Academic Publishers, Dordrecht, 1993.Google Scholar
  3. 3.
    Borwein, J. M. and Lewis, A. S.: Convergence of decreasing sequences of convex sets in nonreflexive Banach spaces, Set-Valued Anal. 1 (1993), 355-363.Google Scholar
  4. 4.
    Borwein, J. M. and Vanderwerff, J. D.: Banach spaces which admit support sets, Proc. Amer. Math. Soc. 124 (1996), 751-756.Google Scholar
  5. 5.
    Diestel, J.: Sequences and Series in Banach Spaces, Grad. Texts in Math. 92, Springer, New York, 1984.Google Scholar
  6. 6.
    Fabian, M. and Godefroy, G.: The dual of every Asplund space admits a projectional resolution of identity, Studia Math. 91 (1988), 141-151.Google Scholar
  7. 7.
    Fabian, M., Habala, P., Hajek, P., Montesinos Santalucia, V., Pelant, J. and Zizler, V.: Functional Analysis and Infinite-Dimensional Geometry, CMS Books in Math. 8, Springer, New York, 2000.Google Scholar
  8. 8.
    Georgiev, P. G., Granero, A. S., Jiménez Sevilla, M. and Moreno, J. P.: Mazur intersection properties and differentiability of convex functions in Banach spaces, J. London Math. Soc. 61 (2000), 531-542.Google Scholar
  9. 9.
    Godefroy, G.: Banach spaces of continuous functions on compact spaces, to appear.Google Scholar
  10. 10.
    Granero, A. S., Jiménez, M., Montesinos, A., Moreno, J. P. and Plichko, A.: On the Kunen-Shelah properties in Banach spaces, to appear.Google Scholar
  11. 11.
    Jiménez Sevilla, M. and Moreno, J. P.: Renorming Banach spaces with the Mazur intersection property, J. Funct. Anal. 144 (1997), 486-504.Google Scholar
  12. 12.
    Johnson, W. B. and Lindenstraus, J.: Some remarks on weakly compactly generated Banach spaces, Israel J. Math. 17 (1974), 219-230.Google Scholar
  13. 13.
    Negrepontis, S.: Banach spaces and topology, In: K. Kunen and J. E. Vaughan (eds), Handbook of Set-Theoretic Topology, Elsevier Science Publishers, 1984, pp. 1045-1142.Google Scholar
  14. 14.
    Shelah, S.: Uncountable constructions for B.A., e.c. groups and Banach spaces, Israel J. Math. 51 (1985), 273-297.Google Scholar
  15. 15.
    Stegall, C.: The Radon-Nikodým property in conjugate Banach spaces, Trans. Amer. Math. Soc. 206 (1975), 213-223.Google Scholar

Copyright information

© Kluwer Academic Publishers 2004

Authors and Affiliations

  • Jonathan M. Borwein
    • 1
  • Jon D. Vanderwerff
    • 2
  1. 1.Centre for Experimental and Constructive MathematicsSimon Fraser UniversityBurnabyCanada
  2. 2.Department of MathematicsLa Sierra UniversityRiversideUSA

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