Uniform Convexity Property 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 $E^\star$.

*

*It suffices to ask about $E^*$. If $E^*$ admits an equivalent locally uniformly convex (LUR) norm, then there is an equivalent norm on $E$ which makes both $E$ and $E^*$ LUR (Richard Haydon). 

*As you probably know, every separable space has an equivalent LUR norm (Kadec renorming, 1959).

*$\ell_\infty$ does not admit an equivalent LUR norm. I saw this credited to Lindenstrauss ("Weakly compact sets—their topological properties and the Banach spaces they generate") but did not read the paper.

