Given a Banach spaceX,x∈𝖲X, and𝖩Xx=x*∈𝖲X*:x*x=1, we define the set𝖩X*xof allx*∈𝖲X*for which there exist two sequencesxnn∈N⊆𝖲X∖{x}andxn*n∈N⊆𝖲X*such thatxnn∈Nconverges tox,xn*n∈Nhas a subnetw*-convergent tox*, andxn*xn=1for alln∈N. We prove that ifXis separable and reflexive andX*enjoys the Radon-Riesz property, then𝖩X*xis contained in the boundary of𝖩Xxrelative to𝖲X*. We also show that ifXis infinite dimensional and separable, then there exists an equivalent norm onXsuch that the interior of𝖩Xxrelative to𝖲X*is contained in𝖩X*x.