Any two-dimensional normed space is a generalized Day-James space (Q1951830)

Let \(\Psi\) be the class of continuous and convex functions \(\psi:[0,1]\to\mathbb{R}\) such that \(\psi(0)=\psi(1)=1\) and \(\max\{t,1-t\}\leq\psi(t)\leq1\) for all \(t\in[0,1]\). It was proved by \textit{F. F. Bonsall} and \textit{J. Ducan} [Numerical ranges II. London: Cambridge University Press (1973; Zbl 0262.47001)] that there is a one-to-one correspondence between the class of absolute and normalized norms on \(\mathbb{R}^2\) and the class \(\Psi\). The author proves that there is a one-to-one correspondence of the class of norms on \(\mathbb{R}^2\) and the class of generalized Day-James norms defined by two functions in \(\Psi\). Note that the latter class was introduced by \textit{W. Nilsrakoo} and the reviewer [J. Inequal. Appl. 2006, Article 26265 (2006; Zbl 1104.46007)].