Let $\ell_\infty$ be the space of bounded sequences equipped with the supremum norm.
Are there any strictly convex subsets of $\ell_\infty$ that have non-empty interior? (The last condition is to rule out trivial examples such as singletons.)
The unit ball of $\ell_\infty$ is not strictly convex, and $\ell_\infty$ is not separable, so I can't use this result.
A strictly convex set $C$, by the way, is a convex set whose boundary contains no open line segment, i.e. $x,y \in \partial C$ implies $\{\alpha x + (1-\alpha)y: \alpha \in (0,1)\} \subset$ int$C$.
Define $f(x):= \sum_n \frac1{n^2}x_n^2$. Then $f:l^\infty\to \mathbb R$ is continuous and strictly convex. And the set $\{x\in l^\infty: \ f(x) <1\}$ is strictly convex with non-empty interior. Note that $x\mapsto \sqrt{f(x)}$ is a strictly convex norm on $l^\infty$.