Let us call a norm on $\mathbb{R}^n$ smooth if its restriction $\| \cdot \|:\mathbb{R}^n\setminus \{ 0 \} \to \mathbb{R}$ is a smooth map.
Suppose the unit sphere of a norm $\| \cdot \|$ is an embedded submanifold of $\mathbb{R}^n$. Is $\| \cdot \|$ necessarily smooth?
Remark:
The converse statement is easy:
Assume $\| \cdot \|$ is smooth. Then it's a submersion (considered as a map $\mathbb{R}^n\setminus \{ 0 \} \to \mathbb{R}$). To see it's differential is nonzero just note that $df_x(x)=\|x\|$. So, the unit sphere is an inverse image of a regular value.