Let X be a Banach space, S(X) - {x ε X : ‖#x02016; = 1} be the unit sphere of X.The parameter, modulus of W * -convexity, W * (ε) = inf{<(x − y)/2, f x > : x, y ɛ S(X), ‖x − y‖ ≥ ε, f x ɛ Δ x }, where 0 ≤ ε ≤ 2 and Δ x ⊆ S(X * ) be the set of norm 1 supporting functionals of S(X) at x, is investigated_ The relationship among uniform nonsquareness, uniform normal structure and the parameter W * (ε) are studied, and a known result is improved. The main result is that for a Banach space X, if there is ε, where 0 < ε < 1/2, such that W * (1 + ε) > ε/2 where W ∗ (1 + ε) = lim α→ε W * (1 + α), then X has normal structure.