Diffeomorphism between two norms If I consider the $L^2$ and $L^4$ norms on $\mathbb R^n$, i.e. $||x||_2 = (\sum_{i=1}^n x_i^2)^{1/2}$ and $||x||_4 = (\sum_{i=1}^n x_i^4)^{1/4}$, I know these two norms are equivalent in the sense we can find constants $c, C$ such that $c||x||_2 \leq ||x||_4 \leq C||x||_2$, but are they equivalent as smooth (away from zero) functions in the sense I can find a change of coordinates/diffeomorphism sending one norm to the other?
I can definitely find a composition which sends one norm to the other: if
$$\varphi(x_1, \cdots, x_n) = (x_1^2, \cdots, x_n^2), \quad \psi(x) = \sqrt{x}$$
then $\psi \circ ||x||_2 \circ \varphi = ||x||_4$.
I would like to know whether I can find a single function $\Phi: \mathbb R^n \rightarrow \mathbb R^n$ such that $||x||_2 \circ \Phi = ||x||_4$ and whether $\Phi$ can have bounded Jacobian. Also, in general, can I do this for any pair of norms on $\mathbb R^n$ and not necessarily $L^p$? If not, can we do it with two functions $\varphi, \psi$ instead?
I have some intuition: I think I "want to send a $L^2$-ball to an $L^4$-ball" in order for $||x||_2\circ\Phi=||x||_4$ to hold. And if I have a map $\Phi_0$ sending one ball to another $B_{L^2}(1) \rightarrow B_{L^4}(1)$ I think I can extend this to a map $\Phi$ on the entire space by taking a point, scaling it back to the unit ball, mapping it under $\varphi$, then rescaling. The only downside of this is that I imagine is the scaling: I think I would end up needing to divide (and multiply) by a norm and I can't imagine this having a nice derivative. This makes me think maybe I'm hoping too much.
The idea of this application might be the integration of radial functions - is there a nice change of coordinates (i.e. bounded Jacobian) from one norm to another?
 A: Yes, note that the map
$$\Psi_i:\Bbb R_{>0}\times S^{n-1}_i\to \Bbb R^n-\{0\}, \qquad (r,\omega)\mapsto r\cdot \omega$$
is a diffeomorphism. Here $S^{n-1}_i$ is the unit sphere of $\Bbb R^n$ with respect to the norm $\|\cdot\|_i$. You must check that these are all diffeomorphic to one another, let $\Phi_{ij}:S^{n-1}_j\to S^{n-1}_i$ be such a diffeomorphism.
Note that $\|\Psi_i(r,\omega)\|_i = r$, so clearly
$$\|\cdot\|_i\circ \Psi_i\circ \Phi_{ij}\circ \Psi_{j}^{-1}=\|\cdot\|_j.$$
Obviously there is still some work to show that the spheres are diffeomorphic, but if you draw some pictures it becomes clear.
To be more careful about it note that they are boundary of some bounded + convex + open set, and that this boundary is smooth (using implicit function theorem). If you apply some theorems about such sets this gives the diffeomorphism.
As a remark, there are norms where the unit sphere is not a smooth manifold, eg $x\mapsto \sup_i |x_i|$ has this property. Here you only gain a homeomorphism. Similarly the $\ell^p$ "norms" for $p\in(0,1)$ do not give convex unit balls (since they are not norms) and again you only get homeomorphism.
