A Reverse Theorem on the ·-w* Continuity of the Dual Map

Given a Banach space X, x∈X, and Xx=x*∈X*:x*x=1, we define the set X*x of all x*∈X* for which there exist two sequences xnn∈N⊆X∖{x} and xn*n∈N⊆X* such that xnn∈N converges to x, xn*n∈N has a subnet w*-convergent to x*, and xn*xn=1 for all n∈N. We prove that if X is separable and reflexive and X* enjoys the Radon-Riesz property, then X*x is contained in the boundary of Xx relative to X*. We also show that if X is infinite dimensional and separable, then there exists an equivalent norm on X such that the interior of Xx relative to X* is contained in X*x