An Improved Inequality between the von Neumann-Jordan and James Constants

We provide a strengthened answer to the questions posed by Alonso, Martín, and Papini [Studia Math. 188 (2008), 135–150] concerning the comparison between the von Neumann–Jordan and James constants . The classical inequality $\text{CNJ}(X)\le J(X)$ between the von Neumann–Jordan and James constants was established by Takahashi and Kato [J. Math. Anal. Appl. 359 (2009), 602–609], Wang [Proc. Amer. Math. Soc. 138 (2010), 695–701], and Yang and Li [Appl. Math. Lett. 23 (2010), 277–281], while related sharp estimates were obtained by Kato and Takahashi [Rend. Circ. Mat. Palermo (2) Suppl. 82 (2010), 75–91]. We sharpen this inequality to $\text{CNJ}(X)\le \text{JB}(X)$, where $\text{JB}(X)$ is the Birkhoff–James orthogonal James constant introduced by Baronti and Papini [Constr. Math. Anal. 5 (2022), 37–45]. Our proof employs optimal distance ellipses and alternating contact directions.