Modulus of convexity in Banach spaces Original Research Article

Let X be a Banach space, X2 ⊆ X be a two-dimensional subspace of X, and S(X) = {xϵX, ‖x‖ = 1} be the unit sphere of X. Let View the MathML source, where x, yϵS(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 View the MathML source implies X has normal structure. We generalize the above results in this paper to prove that for any Banach space View the MathML source for any ϵ, 0 ≤ ϵ ≤ 1, implies X has uniform normal structure.