On new moduli related to the generalization of the parallelogram law
In the last few decades, several geometric constants have been defined and studied in the literature, which make it easier for us to deal with some problems in Banach spaces, because it can describe the geometric properties of the space quantitatively, and these geometric constants have mathematical beauty, and there are countless relationships between different geometric constants. Special attention is paid to the von Neumann-Jordan constant and the James constant, which are rigorously studied based on their importance. The extreme point set plays an important role in Banach space geometry because it can largely reflect the geometric properties of a space \(X\), such as the well-known convexity. \textit{M. S. Moslehian} and \textit{J. M. Rassias} obtained a new equivalent characterization of inner product spaces [Commun. Math. Anal. 8, No.~2, 16--21 (2010; Zbl 1194.46037)]. The authors introduce a new geometric constant \(L'_{ YJ}(\lambda, μ, X)\) based on a generalization of the parallelogram law, which was proposed by Moslehian and Rassias [loc.\,cit.]. They start with some basic properties of this new coefficient. Next, it is shown that, for a Banach space, \(L'_{ YJ}(\lambda, \mu, X)\) becomes 1 if and only if the norm is induced by an inner product. Moreover, some sufficient conditions which imply normal structure are presented. At the end, some relations between the well-known geometric constants and \(L'_{ YJ}(\lambda, \mu, X)\) through some inequalities are described. The paper opens many useful avenues for future research in this area.
scientific article; zbMATH DE number 7453230
scientific article