| Summary: | We shall introduce a new geometric constant CZ(λ,μ,X){C}_{Z}\left(\lambda ,\mu ,X) based on a generalization of the parallelogram law, which was proposed by Moslehian and Rassias. First, it is shown that, for a Banach space, CZ(λ,μ,X){C}_{Z}\left(\lambda ,\mu ,X) is equal to 1 if and only if the norm is induced by an inner product. Next, a characterization of uniformly non-square is given, that is, XX has the fixed point property. Also, a sufficient condition which implies weak normal structure is presented. Moreover, a generalized James constant J(λ,X)J\left(\lambda ,X) is also introduced. Finally, some basic properties of this new coefficient are presented.
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