Constructing a cube with a Norm

[RiotRick]

Homework Statement

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 cube<

I have a problem with b)

Homework Equations

Definition of Norm

The Attempt at a Solution

Now I've read that the One-Norm defines a Octahedron and the dual Norm a cube.
So {$x:||x||^O:=||x||_1 = |x|+|y|+|z| = 1$}
Now I have a problem to construct the dual norm since I don't fully understand dual norms.
But from the definition we get the cube $||x||_*=sup(\langle\,x,y\rangle :||x||_1 \leq 1)$
But how do I do this since I have 3 entries and this inner product doesn't seem to fit?

 

[member 587159]

But how do I do this since I have 3 entries and this inner product doesn't seem to fit?

What do you mean here?