A generalization of the isosceles constant in Banach spaces (Q6552246)

Motivated from the notion of `rectangular constant' \textit{M. Baronti} et al. [J. Math. Anal. Appl. 536, No. 1, Article ID 128251, 8 p. (2024; Zbl 1537.46013)] defined another constant, called `isosceles constant' to study the structure of the unit sphere of a Banach space \(X.\) It is defined as follows:\N\[\NH(X)= \sup\bigg\{\frac{1 +\lambda}{\|x+\lambda y\|}: \|x\|=\|y\|=1, x \perp_I y, \lambda \geq 0\bigg\}.\N\]\NThey proved that \(H(X)=\sqrt{2}\) if and only if \(X\) is a Hilbert space.\N\NIn this article, the authors introduce a generalization of the `isosceles constant' defined as follows:\N\[\NH_p(X)= \sup\bigg\{\frac{(1 +\lambda^p)^{\frac{1}{p}}}{\|x+\lambda y\|}: \|x\|=\|y\|=1, x \perp_I y, \lambda \geq 0\bigg\}.\N\]\NThey obtain a sharp bound for the above constant, in particular,\N\[\N\max\{1, 2^{\frac{2-p}{2p}}\} \leq H_p(X) \leq \max_{0 \leq a \leq 1}\frac{((1+a)^p)+(1-a)^p)^{\frac{1}{p}}}{1+a^2}.\N\]\NUsing this boundary values of they obtain that \(X\) is a Hilbert space if and only if \(H_p(X)= \max\{1, 2^{\frac{2-p}{2p}}\}.\) Also, they provid a necessary condition for non-uniformly non-square spaces via this newly introduced constant \(H_p(X).\)