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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.

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    $\begingroup$ See answer to 1) here: mathoverflow.net/questions/223933/… $\endgroup$
    – Deane Yang
    Dec 13, 2015 at 23:34
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    $\begingroup$ Thanks. If I understood correctly your answer, it also implies that if the unit sphere is a $C^k$ submanifold (hypersurface) , the norm is also $C^k$. (and similarly for $C^{\infty}$). $\endgroup$
    – Asaf Shachar
    Dec 14, 2015 at 11:01
  • $\begingroup$ Yes, that's right. $\endgroup$
    – Deane Yang
    Dec 14, 2015 at 12:54

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