Modulus of convexity in Banach spaces

AbstractLet X be a Banach space, X2 ⊆ X be a two-dimensional subspace of X, and S(X) = {x ϵ X, ‖x‖ = 1} be the unit sphere of X. Let δ(ϵ) = inf{1 − ‖x + y‖2 : ‖x − y‖ ≤ ϵ}, where x, y ϵ S(X2) and 0 ≤ ϵ ≤ 2 is the modulus of convexity of X. The best results so far about the relationship between normal structure and the modulus of convexity of X are that for any Banach space X either δ(1) > 0 or δ(32) > 14 implies X has normal structure. We generalize the above results in this paper to prove that for any Banach space X, δ(1 + ϵ) > ϵ2 for any ϵ, 0 ≤ ϵ ≤ 1, implies X has uniform normal structure