On the Upper Bound of Some Geometric Constants of Absolute Normalized Norms on $${\mathbb{R}^2}$$ R 2

Abstract

This is a continuation of the paper (Mizuguchi and Saito, Ann Funct Anal 2:22–33, 2011). We consider the Banach space $${X=(\mathbb{R}^2, \|\cdot\|)}$$ with a normalized, absolute norm. We treat three geometric constants; the von Neumann–Jordan constant C NJ(X), the modified von Neumann–Jordan constant $${C^{\prime}_{\rm NJ}(X)}$$ and the Zbăganu constant C Z (X). We consider the conditions in which these constants coincide with their upper bound.