Abstract
Let \({\|\cdot\|_{\psi}}\) be the absolute norm on \({\mathbb{R}^2}\) corresponding to a convex function \({\psi}\) on [0, 1] and \({C_{\text{NJ}}(\|\cdot\|_{\psi})}\) its von Neumann–Jordan constant. It is known that \({\max \{M_1^2, M_2^2\} \leq C_{\text{NJ}}(\| \cdot \|_{\psi}) \leq M_1^2 M_2^2}\), where \({M_1 = \max_{0 \leq t \leq 1} \psi(t)/ \psi_2(t)}\), \({M_2 = \max_{0\leq t \leq 1} \psi_2(t)/ \psi(t)}\) and \({\psi_2}\) is the corresponding function to the ℓ 2-norm. In this paper, we shall present a necessary and sufficient condition for the above right side inequality to attain equality. A corollary, which is valid for the complex case, will cover a couple of previous results. Similar results for the James constant will be presented.
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M. Kato: supported in part by the Grant-in-Aid for Scientific Research, Japan Society for the Promotion of Science No. 23540216.
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Ikeda, T., Kato, M. Notes on von Neumann–Jordan and James Constants for Absolute Norms on \({\mathbb{R}^2}\) . Mediterr. J. Math. 11, 633–642 (2014). https://doi.org/10.1007/s00009-013-0370-1
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DOI: https://doi.org/10.1007/s00009-013-0370-1