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Symmetric Strong Diameter Two Property

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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.

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References

  1. Abrahamsen, T.A., Langemets, J., Lima, V.: Almost square Banach spaces. J. Math. Anal. Appl. 434(2), 1549–1565 (2016)

    Article  MathSciNet  Google Scholar 

  2. Abrahamsen, T.A., Leerand, A., Martiny, A., Nygaard, O.: Two properties of Müntz spaces. Demonstr. Math. 50, 239–244 (2017)

    Article  MathSciNet  Google Scholar 

  3. Abrahamsen, T.A., Lima, V., Nygaard, O.: Remarks on diameter 2 properties. J. Convex Anal. 20(2), 439–452 (2013)

    MathSciNet  MATH  Google Scholar 

  4. Abrahamsen, T.A., Lima, V., Nygaard, O.: Almost isometric ideals in Banach spaces. Glasgow Math. J. 56(2), 395–407 (2014)

    Article  MathSciNet  Google Scholar 

  5. Abrahamsen, T.A., Nygaard, O., Põldvere, M.: New applications of extremely regular function spaces, to appear in Pacific J. Math (2017)

  6. Acosta, M.D., Becerra-Guerrero, J., López-Pérez, G.: Stability results of diameter two properties. J. Convex Anal. 22(1), 1–17 (2015)

    MathSciNet  MATH  Google Scholar 

  7. Becerra Guerrero, J., López-Pérez, G., Rueda Zoca, A.: Octahedrality in Lipschitz-free Banach spaces, to appear in Proc. Roy. Soc. Edinburgh Sect. A

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  10. Bonsall, F.F., Duncan, J.: Numerical ranges. II, Cambridge University Press, New York-London, London Mathematical Society Lecture Notes Series, no. 10 (1973)

  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)

  12. Haller, R., Langemets, J.: Two remarks on diameter 2 properties. Proc. Est. Acad. Sci. 63(1), 2–7 (2014)

    Article  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  14. Haller, R., Langemets, J., Põldvere, M.: On duality of diameter 2 properties. J. Convex Anal. 22, 465–483 (2015)

    MathSciNet  MATH  Google Scholar 

  15. Ivakhno, Y.: Big slice property in the spaces of Lipschitz functions. Visn. Khark. Univ., Ser. Mat. Prykl. Mat. Mekh. 749, 109–118 (2006)

    MATH  Google Scholar 

  16. McShane, E.J.: Extension of range of functions. Bull. Am. Math. Soc. 40, 837–842 (1934)

    Article  MathSciNet  Google Scholar 

  17. Oja, E., Põldvere, M.: Principle of local reflexivity revisited. Proc. Am. Math. Soc. 135(4), 1081–1088 (2007). (electronic)

    Article  MathSciNet  Google Scholar 

  18. Procházka, A., Rueda Zoca, A.: A characterisation of octahedrality in Lipschitz-free Banach spaces, to appear in Ann. Inst. Fourier

  19. Werner, D.: \(M\)-ideals and the ‘basic inequality’. J. Approx. Th. 76, 21–30 (1994)

    Article  MathSciNet  Google Scholar 

  20. Weaver, N.: Lipschitz Algebras. World Scientific, Singapore (1999)

    Book  Google Scholar 

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Acknowledgements

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

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Author notes

  1. Vegard Lima

    Present address: Department of Engineering Sciences, University of Agder, Postboks 422, 4604, Kristiansand, Norway

Authors and Affiliations

  1. Institute of Mathematics, University of Tartu, J. Liivi 2, 50409, Tartu, Estonia

    Rainis Haller, Johann Langemets & Rihhard Nadel

  2. NTNU, Norwegian University of Science and Technology, Aalesund, Postboks 1517, 6025, Ålesund, Norway

    Vegard Lima

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  1. Rainis Haller
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  2. Johann Langemets
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  3. Vegard Lima
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  4. Rihhard Nadel
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Correspondence to Johann Langemets.

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R. Haller, J. Langemets, and R. Nadel were partially supported by institutional research funding IUT20-57 of the Estonian Ministry of Education and Research.

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Haller, R., Langemets, J., Lima, V. et al. Symmetric Strong Diameter Two Property. Mediterr. J. Math. 16, 35 (2019). https://doi.org/10.1007/s00009-019-1306-1

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  • DOI: https://doi.org/10.1007/s00009-019-1306-1

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