Let $X = \mathbb{R}^m $ and ||.|| be a norm on $X$. The dual norm is defined as $||y||_*:=\sup\{ \langle\,x,y\rangle :||x|| \leq 1\}$
a) Show that $||.||_*$ is also a norm.
b) Construct two norms $||.||^O$ and $||.||^C$ so that:
$\{x:||x||^O=1\}$ is a regular octahedron
and $\{x:||x||^C=1\}$ is a cube.
I have a problem with b).
I'm far away from being an expert with norms. Just starting yet. Now I've read that the 1-norm defines an octahedron and the dual norm of it, is a cube, so $\{x:||x||^O:=||x||_1 = |x|+|y|+|z| = 1\}$.
So from the definition we get the cube $||x||_*=\sup\{\langle\,x,y\rangle :||x||_1 \leq 1\},$ but now I'm not sure how this inner product is defined, since we have 3 entries $x$,$y$, and $z$. Would this work?