A function $f$ is said to be strongly convex with respect to a norm $\|\cdot\|$ at a point $y$ if $f(x) \geq f(y) + \nabla f(y)^T(x-y) + \frac{1}{2}\|x-y\|^2.$
It is said to be strongly smooth with respect to a norm $\|\cdot\|$ at a point $x$ if $f(y) \leq f(x) + \nabla f(x)^T (y-x) + \frac{1}{2} \|y-x\|^2$.
The Fenchel dual of a convex function $f: \mathbb{R}^n \to \mathbb{R} $ is defined as: $f^*(x) = \max_y x^T y - f(y)$.
Now, it's a general fact that if $f$ is strongly convex with respect to some norm $\|\cdot\|$ everywhere, then $f^*$ is strongly smooth with respect to the dual norm $\|\cdot\|^*$ everywhere.
However, I was wondering if it is also true pointwise. Specifically, if $f$ is strongly convex at its minimum $x_0$, can one say that $f^*$ is strongly smooth at 0?