# Given a smooth family of strictly convex norms in $\mathbb{R}^n$ ($n\geq 2$), what can we say about the family of dual norms?

Let $$U\subset{\mathbb{R}^n}$$ be an open set (may has compact closure) and $$F:U\times\mathbb{R}^n\rightarrow\mathbb{R}$$ such that $$F(p,\cdot):\mathbb{R}^n\rightarrow\mathbb{R}$$ is, for each $$p\in{U}$$, a strictly convex norm . Furthermore, if we suppose that $$F(\cdot,x):U\rightarrow\mathbb{R}$$ is $$C^1$$ for every $$x\in{\mathbb{R}^n}$$, what can be said of the family of dual norms $$F_*:U\times\mathbb{R}^n\rightarrow\mathbb{R}$$, where, for each $$p\in{U}$$, $$F_*(p,\cdot)=\sup_{F(p,x)\leq{1}}\langle \cdot,x \rangle$$ represents the dual norm of $$F(p,\cdot)$$. Is it $$c^1$$ with respect to $$p$$?