Banach spaces which are nearly uniformly convex

A property which generalizes uniform convexity is defined in terms of sequences.Its relationships to uniform convexity and to weak and norm convergence on spheres are investigated. Introduction.Let X be a (real) banach space with norm ||-||, let B ô (x) (respectively, B ô (x)) denote the open (closed) ball with center x and radius <?, and let co(^) (co(^4)) denote the convex hull (closed convex hull) of a set A.We will say that the norm is a Kadec-Klee (KK-)norm provided on the unit sphere sequences converge in norm whenever they converge weakly.(This is property (H) in [2].)An equivalent formulation is the following.(KK): x n -x wkly I => ||x|| < 1. (*»)JS=i not norm CauchyJ For notation, given a sequence (x n ) we let sepCO = inf {\\x n -x m \\ : m # n}.If (x n ) is not norm-Cauchy, then for some subsequence (y n ) we must have SQ p(yn) > 0. The above definition can be reformulated as follows.(x n ) a B x mThis formulation suggests the following two successively stronger notions.The norm will be called uniformly Kadec-Klee (UKK) if for every e > 0 there exists ö < 1 such that (UKK): x n -> x wkly \ =>xe B d (0).