## Nihonkai Mathematical Journal

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

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

#### 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**

Received: 16 December 2015

Revised: 25 April 2016

First available in Project Euclid: 14 September 2017

**Permanent link to this document**

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

**Keywords**

isosceles orthogonality Birkhoff orthogonality absolute normalized norm piecewise linear function

#### 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