I am researching different generalizations of the familiar open mapping theorem from functional analysis. Every "proof" I attempt while simply assuming positive-homogeneity, even in the finite-dim case, has hit a giant brick wall, making me think it is not true, but I cannot seem to write down a counter-example.


Is there an example in which a map $f\colon\mathbb{R}^m\longrightarrow\mathbb{R}^n$ is positively-homogeneous, onto, and not open?


For simplicity, why not take $f$ to be analytic and homogeneous of degree $k\in\mathbb{N}$ and $m\geq n$?



An open mapping theorem for homogeneous functions?
Here is an example. In complex notation, $f(x+iy):=(x^2+iy^2)^4$ defines a homogeneous polynomial map from $\mathbb{R}^2$ to $\mathbb{R}^2$, which is clearly surjective, but not open (for instance, the image of the open set $\{(x,y)\in\mathbb{R}^2: |y|<|x|\}$ is the upper open half-plane plus the half-line $\{(x,0)\in\mathbb{R}^2: x>0\}$, not open).


