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Abstract. We study the boundary structure for w * -compact subsets of dual Banach spaces. To be more precise, for a Banach space X, 0 < ε < 1 and a subset T of the dual space X * such that S {B(t, ε) : t ∈ T } contains a James boundary for BX * we study different kinds of conditions on T , besides T being countable, which ensure thatWe analyze two different non-separable cases where the equality (SP) holds: (a) if J : X → 2 B X * is the duality mapping and there exists a σ-fragmented map f : X → X * such that B(f (x), ε) ∩ J(x) = ∅ for every x ∈ X, then (SP) holds for T = f (X) and in this case X is Asplund; (b) if T is weakly countably K-determined then (SP) holds, X * is weakly countably K-determined and moreover for every James boundary B of BX * we have BX * = co(B)· . Both approaches use Simons' inequality and ideas exploited by Godefroy in the separable case (i.e., when T is countable). While proving (a) we show that X is Asplund if, and only if, the duality mapping has an ε-selector, 0 < ε < 1, that sends separable sets into separable ones. A consequence is that the dual unit ball BX * is norm fragmented if, and only if, it is norm ε-fragmented for some fixed 0 < ε < 1. Our analysis is completed by a characterization of those Banach spaces (not necessarily separable) without copies of 1 via the structure of the boundaries of w * -compact sets of their duals. Several applications and complementary results are proved. Our results extend to the non-separable case results by Godefroy, Contreras-Payá and Rodé.