Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

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

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

Perhaps you should consider rational functions as well?

share|cite|improve this answer
Yeah, forgot about them. Any norm aside from these "Standard" ones? – Henrique Tyrrell Apr 5 '12 at 16:36
Sure, consider the map $x \mapsto ||A x||_p$, where $A$ is invertible. – copper.hat Apr 5 '12 at 16:39
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

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.