Let $X$ be a Banach space with continuous dual $X^{*}$. The so-called duality set of $x\in X$ is the collection
\begin{equation} L(x)=\{\psi\in X^{*} \mid \|\psi\|^{2}_{\text{op}}=\psi(x)=\|x\|^{2}\} \end{equation}
which is a closed, convex, and (by Hahn-Banach) nonempty subset of $X^{*}$. Suppose $x_{1},\ldots,x_{N}\in X$ are distinct. What (if any) assumptions are sufficient to guarantee that $\bigcap_{j=1}^{N}L(x_{j})\neq\emptyset$? I am assuming that $X$ is reflexive and has a (Schauder) basis, and so far, have attempted a ``partition of unity" style argument to construct a member of the intersection in question.
To be more specific, we may find $\epsilon>0$ so that $B_{\epsilon}(x_{i})\cap B_{\epsilon}(x_{j})=\emptyset$ if $i\neq j$, and \begin{equation} B_{\epsilon}(x_{1})\cup\dots\cup B_{\epsilon}(x_{N})\cup\underbrace{\left\{x\in X \;\middle\vert\; \text{dist}\left(x,\bigcup_{j=1}^{N}B_{\frac{\epsilon}{2}}(x_{j})\right)>\frac{\epsilon}{4}\right\}}_{G} \end{equation} is an open cover of $X$ and we may let $\{\alpha_{j}\}_{j=0}^{N}$ be a partition of unity subordinate to this open cover where $\text{supp}(\alpha_{0})\subset G$ and $\text{supp}(\alpha_{j})\subset B_{\epsilon}(x_{j})$ Morally, I want to define some function like \begin{equation} \varphi(x)=\sum_{j=1}^{N}\alpha_{j}(x)\psi_{j}(x) \end{equation} where $\psi_{j}\in L(x_{j})$ to be a member of the intersection in question. Obviously, $\varphi$ as above does not satisfy the requirements I need - the most pressing issue being that it's not necessarily linear. Any suggestions about how to modify my argument or about how to show that this intersection is nonempty in a different way would be much appreciated!
A closing remark: this is for a personal project only (and certainly not a hw problem). Moreover, I am not at all sure that the claim $\bigcap_{j=1}^{N}L(x_{j})\neq\emptyset$ should hold for an arbitrary finite set of distinct vectors and any intuition in this direction would be greatly appreciated as well! For instance, the fact that $X$ has a basis might be used to choose specific vectors.