Boundaries and generation of convex sets

We introduce a notion which is intermediate between that of taking the w*-closed convex hull of a set and taking the norm closed convex hull of this set. This notion helps to streamline the proof (given in [FLP]) of the famous result of James in the separable case. More importantly, it leads to stronger results in the same direction. For example: 1. Assume X is separable and non-reflexive and its unit sphere is covered by a sequence of balls {C _i}_i=1^∞ of radius a 1. Then for every sequence of positive numbers {ε_i}_i=1 ^∞ tending to 0 there is an f ∈ X*, such that ∥ f ∥ = 1 and f(x) ≤ 1 - ε_i, whenever x ∈ C _i, i = 1,2,.. .. 2. Assume X is separable and non-reflexive and let T: Y → X* be a bounded linear non-surjective operator. Then there is an f ∈ X* which does not attain its norm on B_X such that f ∉ T (Y).