Let $U\subset\mathbb{R}^d$ be open and bounded and $K\subset U$ be a compact set. By the Riemann mapping theorem, I can find a diffeomorphism $Φ:U \to B:= {|x| < 1}$. However, can I choose $\Phi$ such that $\Phi(K)$ is convex?
This question is basically a follow up to Extending a diffeomorphism onto a larger set. Here I had hoped that I could first construct a diffeomorphism $\Psi: K^\circ \to B$ and then extend it to $U$, but that seems unlikely.