When is the closed unit ball $B^*$ in the dual space strictly convex?
I'm finding the conditions (on the primal normed space $X$ or on the closed unit ball $B$ of $X$) to ensure that the closed unit ball $B^*$ in the dual space $X^*$ is strictly convex. Anyone can help me?
The unit ball $B^*$ of dual space $X^*$ is strictly convex iff each functional defined on subspace of $X$ have unique extension to the whole $X$.
If a functional $f$ has two Hahn-Banach extensions $g$ and $h$ then $(g+h)/2$ is also a Hahn-Banach extension, so $X^*$ isn't strictly convex. The other direction is due to Foguel, see here.