Nihonkai Mathematical Journal

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

Hiroyasu Mizuguchi

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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

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

Received: 16 December 2015
Revised: 25 April 2016
First available in Project Euclid: 14 September 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

isosceles orthogonality Birkhoff orthogonality absolute normalized norm piecewise linear function


Mizuguchi, Hiroyasu. The constants related to isosceles orthogonality in normed spaces and its dual. Nihonkai Math. J. 27 (2016), no. 1-2, 41--58.

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