Nihonkai Mathematical Journal

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

Hiroyasu Mizuguchi

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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


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