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Diameter two properties and polyhedrality

  • Ginés López-Pérez
  • Abraham Rueda ZocaEmail author
Original Paper
  • 84 Downloads

Abstract

We study the (possible) relation between diameter two properties and various notions of polyhedrality. I-polyhedral spaces (being non-reflexive, M-embedded) have the strong diameter two property; however it turns out that even in II-polyhedral spaces the unit ball may contain slices of small diameter.

Keywords

Polyhedral Banach spaces Diameter two property Slices 

Mathematics Subject Classification

46B20 

Notes

Acknowledgements

The authors want to thank an anonymous referee for suggestions that improved the exposition.

References

  1. 1.
    Abrahamsen, T., Lima, V., Nygaard, O., Troyanski, S.: Diameter two properties, convexity and smoothness. Milan J. Math. 84, 231–242 (2016)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Becerra Guerrero, J., López-Pérez, G., Rueda Zoca, A.: Big slices versus big relatively weakly open subsets in Banach spaces. J. Math. Anal. Appl. 428, 855–865 (2015)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Becerra Guerrero, J., López-Pérez, G., Rueda Zoca, A.: Extreme differences between weakly open subsets and convex combinations of slices in Banach spaces. Adv. Math. 269, 56–70 (2015)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Fabian, M., Habala, P., Hájek, P., Montesinos, V., Pelant, J., Zizler, V.: Functional Analysis and Infinite-Dimensional Geometry. CMS Books in Mathematics. Springer, New York (2001)zbMATHGoogle Scholar
  5. 5.
    Fonf, V.P., Veselý, L.: Infinite-dimensional polyhedrality. Can. J. Math. 56(3), 472–494 (2004)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Harmand, P., Werner, D., Werner, W.: \(M\)-ideals in Banach spaces and Banach algebras Lecture Notes in Math, vol. 1547. Springer, Berlin (1993)zbMATHGoogle Scholar
  7. 7.
    López-Pérez, G.: The big slice phenomena in \(M\)-embedded and \(L\)-embedded spaces. Proc. Am. Math. Soc. 134, 273–282 (2005)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Italia S.r.l. 2017

Authors and Affiliations

  1. 1.Departamento de Análisis Matemático, Facultad de CienciasUniversidad de GranadaGranadaSpain
  2. 2.Instituto de Matemáticas de la Universidad de Granada (IEMath-GR)GranadaSpain

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