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

Mizuguchi, Hiroyasu

Nihonkai Math. J. 27, No. 1-2, 41-58 (2016).

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

Keywords:

isosceles orthogonality; Birkhoff orthogonality; absolute normalized norm; piecewise linear function

Citations:

Zbl 1114.46017; Zbl 1339.46013