In this paper, we define the modulus of n-dimensional Uflatness as the determinant of an (n+1)×(n+1) matrix. The properties of the modulus are investigated and the relationships between this modulus and other geometric parameters of Banach spaces are studied. Some results on fixed point theory for non-expansive mappings and normal structure in Banach spaces are obtained. 493 494 J. GAO AND S. SAEJUNG Remark 1.2. The following facts are known. • uniform normal structure =⇒ normal structure =⇒

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... k normal structure. • In the setting of reflexive spaces, normal structure ⇐⇒ weak normal structure.