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We shall introduce a new geometric constant CZ(λ, μ, 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) 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, X has the fixed point property. Also, a sufficient condition...