On new parameters concerning a generalization of the parallelogram law in Banach spaces

We shall introduce a new geometric constant $L^{\prime}_{\mathrm{Y}}(\lambda,X)$ based on a generalization of the parallelogram law. We first investigate some basic properties of this new coefficient. Next, it is shown that, for a Banach space, $L^{\prime}_{\mathrm{Y}}(\lambda,X)$ becomes $1$ for some $\lambda_0\in (0,1)$ if and only if the norm is induced by an inner product. Moreover, some relations between other well-known geometric constants are studied. Finally, a sufficient condition which implies normal structure is presented.