Advertisement

Strong Diameter Two Property and Convex Combinations of Slices Reaching the Unit Sphere

  • Ginés López-Pérez
  • Miguel Martín
  • Abraham Rueda ZocaEmail author
Article
  • 26 Downloads

Abstract

We characterise the class of those Banach spaces in which every convex combination of slices of the unit ball intersects the unit sphere as the class of those spaces in which every convex combination of slices of the unit ball contains two points at distance exactly two. Also, we study when the convex combinations of slices of the unit ball are relatively open or have non-empty relative interior for different topologies, studying the relationship between them and studying these properties for \(L_{\infty }\)-spaces and preduals of \(L_1\)-spaces.

Keywords

Diameter two property convex combination of slices relatively weakly open \(L_1\)-predual 

Mathematics Subject Classification

Primary 46B04 Secondary 46B20 

Notes

Acknowledgements

The authors are grateful to the anonymous referee for the valuable suggestions which have improved the exposition of the paper.

References

  1. 1.
    Abrahamsen, T.A., Becerra Guerrero, J., Haller, R., Lima, V., Põldvere, M.: Banach Spaces Where Convex Combinations of Relatively Weakly Open Subsets of the Unit Ball are Relatively Weakly Open, to appear in Studia Math.  https://doi.org/10.4064/sm181016-10-1
  2. 2.
    Abrahamsen, T.A., Lima, V.: Relatively weakly open convex combination of slices. Proc. Am. Math. Soc. 146, 4421–4427 (2018)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Abrahamsen, T.A., Lima, V., Nygaard, O.: Remarks on diameter 2 properties. J. Conv. Anal. 20(2), 439–452 (2013)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Becerra Guerrero, J., López-Pérez, G., Rueda Zoca, A.: Octahedral norms in spaces of operators. J. Math. Anal. Appl. 427, 171–184 (2015)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Becerra Guerrero, J., López-Pérez, G., Rueda Zoca, A.: Subspaces of Banach spaces with big slices. Banach J. Math. Anal. 10, 771–782 (2016)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Ghoussoub, N., Godefroy, G., Maurey, B., Schachermayer, W.: Some topological and geometrical structures in Banach spaces. Mem. Am. Math. Soc. 378 (1987)Google Scholar
  7. 7.
    Haller, R., Kuuseok, P., Põldvere, M.: On Convex Combination of Slices of the Unit Ball in Banach Spaces, Preprint 2017. arXiv:1703.02919
  8. 8.
    Harmand, P., Werner, D., Werner, W.: \(M\)-Ideals in Banach Spaces and Banach Algebras. Lecture Notes in Mathematics, vol. 1547. Springer, Springer (1993)CrossRefGoogle Scholar
  9. 9.
    Langemets, J., Lima, V., Rueda Zoca, A.: Octahedral norms in tensor products of Banach spaces. Q. J. Math. 68, 1247–1260 (2017)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Ryan, R.A.: Introduction to Tensor Products of Banach Spaces, Springer Monographs in Mathematics. Springer, London (2002)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Ginés López-Pérez
    • 1
    • 2
  • Miguel Martín
    • 1
    • 2
  • Abraham Rueda Zoca
    • 1
    Email author
  1. 1.Departamento de Análisis Matemático, Facultad de CienciasUniversidad de GranadaGranadaSpain
  2. 2.Instituto de Matemáticas de la Universidad de Granada (IEMath-GR)GranadaSpain

Personalised recommendations