On some geometric parameters in Banach spaces

Let X be a normed linear space and S(X) = {x ∈ X: x =1} be the unit sphere of X. Let δ( ) : [0, 2]→ [0, 1], ρX( ) : [0,+∞)→[0,+∞), and J(X) = sup{ x + y ∧ x − y }, x and y ∈ S(X) be the modulus of convexity, the modulus of smoothness, and the modulus of squareness of X, respectively. Let E(X) = sup{ x + y 2 + x − y 2: x,y ∈ S(X)}. In this paper we proved some sufficient conditions on δ( ), ρX( ), J(X), E(X), and w(X) = sup{λ>0: λ ·lim infn→∞ xn +x lim infn→∞ xn −x }, where the supremum is taken over all the weakly null sequence xn in X and all the elements x of X for the uniform normal structure. © 2007 Elsevier Inc. All rights reserved