A reverse theorem on the \(||\cdot||-w^{\ast}\) continuity of the dual map (Q492600)

Summary: Given a Banach space \(X\), \(x\in S_X\), and \(J_X(x)= \{x^*\in S_{X^*}: x^*(x)= 1\}\), we define the set \(J^*_X(x)\) of all \(x^*\in S_X\), for which there exist two sequences \((x_n)_{n\in\mathbb{N}}\subseteq S_X\setminus\{x\}\) and \((x^*_n)_{n\in\mathbb{N}}\subseteq S_X\), such that \((x_n)_{n\in\mathbb{N}}\) converges to \(x\), \((x^*_n)_{n\in\mathbb{N}}\) has a subnet \(w^*\)-convergent to \(x^*\), and \(x^*_n(x_n)= 1\) for all \(n\in\mathbb{N}\). We prove that if \(X\) is separable and reflexive and \(X^*\) enjoys the Radon-Riesz property, then \(J^*_X(x)\) is contained in the boundary of \(J_X(x)\) relative to \(S_{X^*}\). We also show that if \(X\) is infinite-dimensional and separable, then there exists an equivalent norm on \(X\) such that the interior of \(J_X(x)\) relative to \(S_{X^*}\) is contained in \(J^*_X(x)\).