The W*-convexity and normal structure in banach spaces

AbstractLet 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, fx> : x, y ɛ S(X), ‖x − y‖ ≥ ε, fx ɛ Δx}, where 0 ≤ ε ≤ 2 and Δx ⊆ 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) = limα→ε W* (1 + α), then X has normal structure