Modulus of convexity in Banach spaces

Abstract

Let X be a Banach space, X 2 ⊆ 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( X 2) 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( 3 2 ) > 1 4 implies X has normal structure. We generalize the above results in this paper to prove that for any Banach space X, δ(1 + ϵ) > ϵ 2 for any ϵ, 0 ≤ ϵ ≤ 1, implies X has uniform normal structure.