On a generalized modulus of convexity and uniform normal structure

In this article, the authors study a generalized modulus of convexity, δα)(ɛ). Certain related geometrical properties of this modulus are analyzed. Their main result is that Banach space X has uniform normal structure if there exists ɛ, 0 ≤ ɛ ≤ 1, such that δ(α) (1 + ɛ) > (1-α) ɛ.