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Suppose $U\subset \mathbb{R}^n$ is a symmetric convex body. For $x\in\mathbb{R}^n$, the norm of $x$ induced by $U$ is defined as

$$ \|x\|_U= \inf\{r>0:x/r \in U\}. $$

For norm $\|\cdot\|$ defined in $\mathbb{R}^n$, its dual norm is defined as

$$ \|x\|_* = \sup\{\langle z,x\rangle: \|z\|\leq 1\} $$

for $x \in \mathbb{R}^n$. What's the induced norm of $\|x\|_U$? Is it also an induced norm of some convex body?

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    $\begingroup$ The dual norm $\|x\|_*$ is induced by its unit ball: $V:=\{x: \|x\|_*\le 1\}$. Then $\|x\|_*=\|x\|_V$. $\endgroup$
    – daw
    Dec 21, 2021 at 13:54

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The dual norm of $\|\cdot\|_U$ is the norm induced by the polar body of $U$. See Lemma 1.5 in these notes by Rothvoss. The definition of the polar body is in Definition 1.2 in the same notes.

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