Given an open bounded convex polytope $C$ in $\mathbb{R}^n$, is there a way to construct a smooth function from $\mathbb{R}^m$ to $\mathbb{R}^n$ whose image is $C$?
For instance, if $C$ is the unit cube, we can use the logistic function in each coordinate; thus we can get any parallelepiped. If $C$ is a simplex, you can use the stick-breaking process. What other polytopes work?
I am actually interested in one particular polytope that comes from a problem in statistics. My polytope is entirely contained in the unit cube and its defining matrix is all 0's and 1's. I mention this just in case it proves relevant, even though I don't see why it would. I suspect if there is a solution for my case, then it would work more generally.
Would appreciate any leads!