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.