Let $E$ be a normed vector space with norm $\|\ \|$. Is it possible to find a equivalent norm $|\ |$ in $E$ such that $E$ and $E^\star$ are locally uniform convex spaces?
Note: Im assuming the norm $\|f\|_{E^\star}=\displaystyle\sup_{|x|\leq 1}|f(x)|$ in $E^\star$.