Abstract
The class of two-dimensional normed spaces with James constant \({\sqrt{2}}\) is studied. It is shown that, if the norm is \({\pi/2}\)-rotation invariant, then its James constant is \({\sqrt{2}}\) if and only if the norm is \({\pi/4}\)-rotation invariant. We also present a characterization of \({\pi/4}\)-rotation invariant norms using some properties of certain convex functions on the unit interval, which allow us to easily construct norms the James constant of which are \({\sqrt{2}}\). Moreover, two important examples are given, which show that neither absoluteness, symmetry nor \({\pi/2}\)-rotation invariance can be a characteristic property of the norms with James constant \({\sqrt{2}}\).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Alonso J.: Uniqueness properties of isosceles orthogonality in normed linear spaces. Ann. Sci. Math. Québec. 18, 25–38 (1994)
Alonso, J.: Any two-dimensional normed space is a generalized Day–James space. J. Inequal. Appl. 2011, 2 (2011) (3 pp)
Alonso J., Martini H., Wu S.: On Birkhoff orthogonality and isosceles orthogonality in normed linear spaces. Aequationes Math. 83, 153–189 (2012)
Bonsall F.F., Duncan J.: Numerical Ranges II. Cambridge University Press, Cambridge (1973)
Gao J., Lau K.-S.: On the geometry of spheres in normed linear spaces. J. Aust. Math. Soc. Ser. A 48, 101–112 (1990)
James R.C.: Orthogonality in normed linear spaces. Duke Math. J. 12, 291–302 (1945)
James Y.: Orthogonality and linear functionals in normed linear spaces. Trans. Am. Math. Soc. 61, 265–292 (1947)
Komuro, N., Saito, K.-S., Mitani, K.-I.: Convex property of James and von Neumann–Jordan constant of absolute norms on \({\mathbb{R}^2}\). In: Proceedings of the 8th International Conference on Nonlinear Analysis and Convex Analysis, pp. 301–308. Yokohama Publ., Yokohama (2015)
Komuro, N., Saito, K.-S., Tanaka, R.: On the class of Banach spaces with James constant \({\sqrt{2}}\). Math. Nachr. Published online
Mitani K.-I., Saito K.-S.: The James constant of absolute norms on \({\mathbb{R}^2}\). J. Nonlinear Convex Anal. 4, 399–410 (2003)
Mitani K.-I., Saito K.-S., Suzuki T.: Smoothness of absolute norms on \({\mathbb{C}^n}\). J. Convex Anal. 10, 89–107 (2003)
Nilsrakoo, W., Saejung, S.: The James constant of normalized norms on \({\mathbb{R}^2}\). J. Inequal. Appl. Art. ID 26265 (2006) (12 pp)
Saito K.-S., Kato M., Takahashi Y.: Von Neumann–Jordan constant of absolute normalized norms on \({\mathbb{C}^2}\). J. Math. Anal. Appl. 244, 515–532 (2000)
Author information
Authors and Affiliations
Corresponding author
Additional information
The second and third authors were supported in part Grand-in-Aid for Scientific Research (No. 15K04920 and No. 16J01162), Japan Society of the Promotion of Science, respectively.
Rights and permissions
About this article
Cite this article
Komuro, N., Saito, KS. & Tanaka, R. On the Class of Banach Spaces with James Constant \({\sqrt{2}}\) : Part II. Mediterr. J. Math. 13, 4039–4061 (2016). https://doi.org/10.1007/s00009-016-0731-7
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00009-016-0731-7