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Are there $C^\infty$ norms in $\mathbb{R^n}$ aside from the 2 norm(Euclidean norm)? $C^\infty$ in $\mathbb{R^n} - 0$, of course.

The question I'm working on is the following: "Prove that there is a $C^\infty$ function $f: \mathbb{R^n} - {0} \rightarrow \mathbb{R}$ which is positive and positively homogeneous of degree k. Prove also that are such functions which are not polynomials."

Def.: A function $f:U \rightarrow \mathbb{R}$ is said positively homogeneous of degree $k$, $k \in \mathbb{R},\ k \neq0$, when $\forall t > 0\ , \forall x \in \mathbb{R^n}$ we have $f(tx) = t^k f(x)$. We also require that $U$ satisfy: $\forall x \in U, tx \in U\ \forall t>0 $ in order for the definition to make sense. e.g. Linear functions on $\mathbb{R^n}$ are positively homogeneous of degree 1.

My progress so far: declare $f(x) = |x|^k$ (euclidean norm), whence $f(tx) = |tx|^k = |t|^k|x|^k = t^k f(x)$ since $t > 0$. It solves the first part. But if k is even, $f$ would be a polynomial in n indeterminates. That's why I'm looking for a $C^\infty$ norm in $\mathbb{R^n} - 0$ aside from the Euclidean.

So... Any lights?

Thanks in advance. Henrique.

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p-norms (1 < p < $\infty$) are smooth. Any aside from those? –  Henrique Tyrrell Apr 5 '12 at 16:36

1 Answer 1

All the $p$-norms (with $1 < p < \infty$) are smooth on $\mathbb{R^n} \setminus {0}$.

Perhaps you should consider rational functions as well?

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Yeah, forgot about them. Any norm aside from these "Standard" ones? –  Henrique Tyrrell Apr 5 '12 at 16:36
1  
Sure, consider the map $x \mapsto ||A x||_p$, where $A$ is invertible. –  copper.hat Apr 5 '12 at 16:39
1  
Look at the Minkowski functional of suitably 'nice' sets. –  copper.hat Apr 5 '12 at 16:43
    
Thanks man. These are a great class of examples –  Henrique Tyrrell Apr 5 '12 at 16:48

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