  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.
