On a new geometric constant related to the modulus of smoothness of a Banach space

We shall introduce a new geometric constant A(X) of a Banach space X, which is closely related to the modulus of smoothness . X (.), and investigate it in relation with the constant A 2(X) by Baronti et al., the von Neumann-Jordan constant C NJ (X) and the James constant J(X). A sequence of recent results on these constants as well as some other geometric constants will be strengthened and improved.