## Tokyo Journal of Mathematics

### A Sufficient Condition That $J(X^*)=J(X)$ Holds for a Banach Space $X$

#### Abstract

It is shown that the James constant of the space $\mathbb{R}^2$ endowed with a $\pi/2$-rotation invariant norm coincides with that of its dual space. As a corollary, we have the same statement on symmetric absolute norms on $\mathbb{R}^2$.

#### Article information

Source
Tokyo J. Math., Volume 41, Number 1 (2018), 219-223.

Dates
First available in Project Euclid: 18 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.tjm/1513566019

Mathematical Reviews number (MathSciNet)
MR3830815

Zentralblatt MATH identifier
06966865

Subjects
Primary: 46B20: Geometry and structure of normed linear spaces

#### Citation

KOMURO, Naoto; SAITO, Kichi-Suke; TANAKA, Ryotaro. A Sufficient Condition That $J(X^*)=J(X)$ Holds for a Banach Space $X$. Tokyo J. Math. 41 (2018), no. 1, 219--223. https://projecteuclid.org/euclid.tjm/1513566019

#### References

• J. Alonso, Uniqueness properties of isosceles orthogonality in normed linear spaces, Ann. Sci. Math. Québec 18 (1994), 25–38.
• J. Alonso, P. Martín and P. L. Papini, Wheeling around von Neumann-Jordan constant in Banach spaces, Studia Math. 188 (2008), 135–150.
• J. Gao and K.-S. Lau, On the geometry of spheres in normed linear spaces, J. Aust. Math. Soc. Ser. A 48 (1990), 101–112.
• M. Kato, L. Maligranda and Y. Takahashi, On James and Jordan-von Neumann constants and the normal structure coefficient of Banach spaces, Studia Math. 144 (2001), 275–295.
• N. Komuro, K.-S. Saito and K.-I. Mitani, Convex property of James and von Neumann-Jordan constant of absolute norms on $\mathbb{R}^2$, Proceedings of the $8$th International Conference on Nonlinear Analysis and Convex Analysis, 301–308, Yokohama Publ., Yokohama, 2015.
• N. Komuro, K.-S. Saito and R. Tanaka, On the class of Banach spaces with James constant $\sqrt{2}$, Math. Nachr. 289 (2016), 1005–1020.
• N. Komuro, K.-S. Saito and R. Tanaka, On the class of Banach spaces with James constant $\sqrt{2}$: Part II, Mediterr. J. Math. 13 (2016), 4039–4061.
• K.-S. Saito, M. Sato and R. Tanaka, When does the equality $J(X^*)=J(X)$ hold for a two-dimensional Banach space $X$?, Acta Math. Sin. (Engl. Ser.) 31 (2015), 1303–1314.
• Y. Takahashi and M. Kato, von Neumann-Jordan constant and uniformly non-square Banach spaces, Nihonkai Math. J. 9 (1998), 155–169.
• F. Wang, On the James and von Neumann-Jordan constants in Banach spaces, Proc. Amer. Math. Soc. 138 (2010), 695–701.
• C. Yang and H. Li, An inequality between Jordan-von Neumann constant and James constant, Appl. Math. Lett. 23 (2010), 277–281.