## Nihonkai Mathematical Journal

### The constants related to isosceles orthogonality in normed spaces and its dual

Hiroyasu Mizuguchi

#### Abstract

We consider isosceles orthogonality and Birkhoff orthogonality, which are the most used notions of generalized orthogonality. In 2006, Ji and Wu introduced a geometric constant $D(X)$ to give a quantitative characterization of the difference between these two orthogonality types. From their results, we have that $D(X)=D(X^*)$ holds for any symmetric Minkowski plane. On the other hand, for the James constant $J(X)$, Saito, Sato and Tanaka recently showed that if the norm of a two-dimensional space $X$ is absolute and symmetric then $J(X)=J(X^*)$ holds. In this paper, we consider the constant $D(X,\lambda)$ such that $D(X)=\inf_{\lambda \in \mathbb{R}}D(X,\lambda)$ and obtain that in the same situation $D(X,\lambda)=D(X^*,\lambda)$ holds for any $\lambda \in (0,1)$.

#### Article information

Source
Nihonkai Math. J., Volume 27, Number 1-2 (2016), 41-58.

Dates
Revised: 25 April 2016
First available in Project Euclid: 14 September 2017

https://projecteuclid.org/euclid.nihmj/1505419740

Mathematical Reviews number (MathSciNet)
MR3698240

Zentralblatt MATH identifier
1381.46015

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

#### Citation

Mizuguchi, Hiroyasu. The constants related to isosceles orthogonality in normed spaces and its dual. Nihonkai Math. J. 27 (2016), no. 1-2, 41--58. https://projecteuclid.org/euclid.nihmj/1505419740

#### References

• J. Alonso, H. Martini and S. Wu, On Birkhoff orthogonality and isosceles orthogonality in normed linear spaces, Aequat. Math. 83 (2012) 153–189.
• G. Birkhoff, Orthogonality in linear metric spaces, Duke Math. J. 1 (1935) 169–172.
• F. F. Bonsall and J. Duncan, Numerical Ranges II, London Math. Soc. Lecture Note Series 10, Cambridge University Press, Cambridge, 1973.
• J. Gao and K.-S. Lau, On the geometry of spheres in normed linear spaces, J. Austral. Math. Soc. Ser. A 48 (1990), 101–112.
• C. Hao and S. Wu, Homogeneity of isosceles orthogonality and related inequalities, J. Inequal. Appl. 2011, 9pp.
• R. C. James, Orthogonality in normed linear spaces, Duke Math. J. 12 (1945), 291–302.
• R. C. James, Orthogonality and linear functionals in normed linear spaces, Trans. Amer. Math. Soc. 61 (1947), 265–292.
• D. Ji and S. Wu, Quantitative characterization of the difference between Birkhoff orthogonality and isosceles orthogonality, J. Math. Anal. Appl. 323 (2006), 1–7.
• K.-I. Mitani and K.-S. Saito, The James constant of absolute norms on $\mathbb{R}^2$, J. Nonlinear Convex Anal. 4 (2003), 399–410.
• K.-I. Mitani and K.-S. Saito, Dual of two dimensional Lorentz sequence spaces, Nonlinear Anal. 71 (2009), 5238–5247.
• K.-I. Mitani, K.-S. Saito and T. Suzuki, Smoothness of absolute norms on $\mathbb{C}^n$, J. Convex Anal. 10 (2003), 89–107.
• K.-I. Mitani, S. Oshiro and K.-S. Saito, Smoothness of $\psi$-ditect sums of Banach spaces, Math. Inequal. Appl. 8 (2005), 147–157.
• H. Mizuguchi, The constants to measure the differences between Birkhoff and isosceles orthogonalities, Filomat, to appear.
• P. L. Papini and S. Wu, Measurements of differences between orthogonality types, J. Math. Anal. Appl. 397 (2013) 285–291.
• K.-S. Saito, M. Kato and Y. Takahashi, Von Neumann–Jordan constant of absolute normalized norms on $\mathbb{C}^2$, J. Math. Anal. Appl. 244 (2000), 515–532.
• K.-S. Saito, M. Sato and R. Tanaka, When does the equality $J(X)=J(X^*)$ hold?, Acta Math. Sin. (Engl. Ser.) 31 (2015), 1303–1314.