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Constructing a cube with a Norm

  • #1
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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?
 

Answers and Replies

  • #2
Math_QED
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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?
 

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