# What is the dual norm of the norm induced by a convex body?

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?

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

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.