Advertisement

Symmetric Strong Diameter Two Property

  • Rainis Haller
  • Johann LangemetsEmail author
  • Vegard Lima
  • Rihhard Nadel
Article
  • 22 Downloads

Abstract

We study Banach spaces with the property that, given a finite number of slices of the unit ball, there exists a direction such that all these slices contain a line segment of length almost 2 in this direction. This property was recently named the symmetric strong diameter two property by Abrahamsen, Nygaard, and Põldvere. The symmetric strong diameter two property is not just formally stronger than the strong diameter two property (finite convex combinations of slices have diameter 2). We show that the symmetric strong diameter two property is only preserved by \(\ell _\infty \)-sums, and working with weak star slices we show that \(\text {Lip}_0(M)\) have the weak star version of the property for several classes of metric spaces M.

Keywords

Strong diameter two property almost square spaces Lipschitz spaces 

Mathematics Subject Classification

Primary 46B20 46B22 

Notes

Acknowledgements

The authors wish to express their thanks to Indrek Zolk for his collaboration in proving Lemma 5.4

References

  1. 1.
    Abrahamsen, T.A., Langemets, J., Lima, V.: Almost square Banach spaces. J. Math. Anal. Appl. 434(2), 1549–1565 (2016)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Abrahamsen, T.A., Leerand, A., Martiny, A., Nygaard, O.: Two properties of Müntz spaces. Demonstr. Math. 50, 239–244 (2017)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Abrahamsen, T.A., Lima, V., Nygaard, O.: Remarks on diameter 2 properties. J. Convex Anal. 20(2), 439–452 (2013)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Abrahamsen, T.A., Lima, V., Nygaard, O.: Almost isometric ideals in Banach spaces. Glasgow Math. J. 56(2), 395–407 (2014)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Abrahamsen, T.A., Nygaard, O., Põldvere, M.: New applications of extremely regular function spaces, to appear in Pacific J. Math (2017)Google Scholar
  6. 6.
    Acosta, M.D., Becerra-Guerrero, J., López-Pérez, G.: Stability results of diameter two properties. J. Convex Anal. 22(1), 1–17 (2015)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Becerra Guerrero, J., López-Pérez, G., Rueda Zoca, A.: Octahedrality in Lipschitz-free Banach spaces, to appear in Proc. Roy. Soc. Edinburgh Sect. AGoogle Scholar
  8. 8.
    Becerra Guerrero, J., López-Pérez, G., Rueda Zoca, A.: Some results on almost square Banach spaces. J. Math. Anal. Appl 438(2), 1030–1040 (2016)MathSciNetCrossRefGoogle Scholar
  9. 9.
    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
  10. 10.
    Bonsall, F.F., Duncan, J.: Numerical ranges. II, Cambridge University Press, New York-London, London Mathematical Society Lecture Notes Series, no. 10 (1973)Google Scholar
  11. 11.
    Ghoussoub, N., Godefroy, G., Maurey, B., Schachermayer, W.: Some topological and geometrical structures in Banach spaces. Mem. Am. Math. Soc. 70(378), iv+116 (1987)Google Scholar
  12. 12.
    Haller, R., Langemets, J.: Two remarks on diameter 2 properties. Proc. Est. Acad. Sci. 63(1), 2–7 (2014)CrossRefGoogle Scholar
  13. 13.
    Haller, R., Langemets, J., Nadel, R.: Stability of average roughness, octahedrality, and strong diameter 2 properties of Banach spaces with respect to absolute sums. Banach J. Math. Anal. 12(1), 222–239 (2018)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Haller, R., Langemets, J., Põldvere, M.: On duality of diameter 2 properties. J. Convex Anal. 22, 465–483 (2015)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Ivakhno, Y.: Big slice property in the spaces of Lipschitz functions. Visn. Khark. Univ., Ser. Mat. Prykl. Mat. Mekh. 749, 109–118 (2006)zbMATHGoogle Scholar
  16. 16.
    McShane, E.J.: Extension of range of functions. Bull. Am. Math. Soc. 40, 837–842 (1934)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Oja, E., Põldvere, M.: Principle of local reflexivity revisited. Proc. Am. Math. Soc. 135(4), 1081–1088 (2007). (electronic)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Procházka, A., Rueda Zoca, A.: A characterisation of octahedrality in Lipschitz-free Banach spaces, to appear in Ann. Inst. FourierGoogle Scholar
  19. 19.
    Werner, D.: \(M\)-ideals and the ‘basic inequality’. J. Approx. Th. 76, 21–30 (1994)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Weaver, N.: Lipschitz Algebras. World Scientific, Singapore (1999)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Institute of MathematicsUniversity of TartuTartuEstonia
  2. 2.NTNU, Norwegian University of Science and TechnologyÅlesundNorway
  3. 3.Department of Engineering SciencesUniversity of AgderKristiansandNorway

Personalised recommendations