Extremal structure of absolute normalized norms on \(\mathbb R^2\) and the James constant
A norm on \(\mathbb{R}^2\) is absolute if \(\|(x_1, x_2\|=\|(|x_1|, |x_2|)\|\) for all \(x_1\), \(x_2\) in \(\mathbb{R}\) and is normalized if \(\|(1, 0)\|= \|(0, 1)\|= 1\). Let \(AN_2\) be the convex set of all absolute normalized norms on \(\mathbb{R}^2\). It is known that \(AN_2\) is in one-to-one correspondence with the set \(\Psi_2\) of all continuous convex functions \(\psi: [0,1]\to\mathbb{R}\) satisfying \(\psi(0)= \psi(1)= 1\) and \(\max\{1-t,t\}\leq \psi(t)\leq 1\) for \(t\) in \([0,1]\) under the mapping \(\psi\mapsto\|\cdot\|_\psi\), where \(\|(x_1, x_2)\|_\psi= (|x_1|+ |x_2|)\). \(\psi(|x_2|/(|x_1|+ |x_2|))\) for \((x_1,x_2)\neq (0,0)\) and \(0\) for \((x_1, x_2)=(0, 0)\). The extreme points of \(AN_2\) were characterized by the authors before via \(\Psi_2\): they are exactly the ones corresponding to \(\psi_{\alpha,\beta}\), \(0\leq\alpha\leq 1/2\leq\beta\leq 1\), which can be written down explicitly. The James constant of a Banach space \((X,\|\cdot\|)\) is defined by \(J(X)= \sup\{\min\{\| x+ y\|,\| x- y\|\}:\| x\|=\| y\|= 1\}\). The purpose of this paper is to determine the James constant of \((\mathbb{R}^2,\|\cdot\|_{\alpha,\beta})\) for all such values of \(\alpha\) and \(\beta\). This is achieved via a series of lemmas, which involve some elementary but lengthy derivations of specific inequalities.
scientific article; zbMATH DE number 5931357
scientific article