I'm trying to understand the strong convexity of the convex conjugate defined on a Banach space. Assume $B$ is a Banach space with the norm $||\cdot||_{B}$. Consider a smooth (wrt $||\cdot||_B$) function $\phi: B \to \mathbb{R}$ and denote its convex conjugate by $\phi^*: B^* \to \mathbb{R}$.
For finite-dimensional Banach spaces, it has been established that if $\phi$ is smooth then $\phi^*$ is strongly convex (wrt $||\cdot||_{B^*}$) (cf https://arxiv.org/pdf/0809.0813.pdf by Anatoli B. Juditsky and Arkadi S.Nemirovski). I'm trying to understand if this property holds for $\phi$ defined on a general Banach space $B$ (e.g. infinite dimensional Banach spaces).