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 $p$?


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