Suppose $\Omega_1 \subset \mathbb{R}^2$ is an $n$-sided convex polygon while $\Omega_2 \subset \mathbb{R}^2$ is simply connected with a piecewise smooth Lipschitz boundary comprised of $n$ smooth curves.
My question is: under what circumstances is it possible to find a diffeomorphism in the neighbourhood of $\overline{\Omega}_1$ that, restricted to $\overline{\Omega}_1$, maps $\overline{\Omega}_1$ onto $\overline{\Omega}_2$ (and hence $\partial \Omega_1$ onto $\partial \Omega_2$).
Assigning the $n$ (straight) sides of $\partial \Omega_1$ to the $n$ sides of $\partial \Omega_2$ in, say, counter-clockwise orientation, my intuition tells me that a diffeomorphism could exist when the convex corners of $\partial \Omega_1$ are mapped onto convex corners of $\partial \Omega_2$, i.e., the consecutive sides of $\partial \Omega_2$ must create a convex corner where they attach.
Is my intuition correct ? If so, where can I find a proof ? In case my intuition fails me, is it possible to say something about the existence of a "weak" diffeomorphism, i.e., one that satisfies $\det Du(x) > 0$ almost everywhere in $\overline{\Omega}_1$ ?