## The constants related to isosceles orthogonality in normed spaces and its dual.(English)Zbl 1381.46015

Summary: We consider isosceles orthogonality and Birkhoff orthogonality, which are the most used notions of generalized orthogonality. D. Ji and S. Wu [J. Math. Anal. Appl. 323, No. 1, 1–7 (2006; Zbl 1114.46017)] 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 $$D(X)=D(X^\ast)$$ holds for any symmetric Minkowski plane. On the other hand, for the James constant $$J(X)$$, K.-S. Saito et al. [Acta Math. Sin., Engl. Ser. 31, No. 8, 1303–1314 (2015; Zbl 1339.46013)] recently showed that, if the norm of a two-dimensional space $$X$$ is absolute and symmetric, then $$J(X)=J(X^\ast)$$ 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 $$D(X,\lambda)=D(X^\ast,\lambda)$$ holds for any $$\lambda\in(0,1)$$.

### MSC:

 46B20 Geometry and structure of normed linear spaces

### Citations:

Zbl 1114.46017; Zbl 1339.46013
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