I am reading Elliptic Partial Differential Equations by Gilbarg and Trudinger. In section 4.1 'Holder Continuity' it gives an example that $f \in C^1(\overline\Omega)$ is not necessarily Lipschitz if $\Omega$ is cusped and not convex. There is also a remark in the section that merely claims for domains "of interest" in the book $C^1(\overline\Omega)$ functions are always Lipschitz. I am looking for a rigorous proof of this remark.
I think this question is somewhat equal to the problem "whether Riemannian distance can be controlled by Euclidean metric in open subset $\Omega$". Indeed, in convex domains, the two distance functions are exactly the same, and in this case $C^1(\overline\Omega)$ implies Lipschitz is simply a result of the Mean Value Theorem.
I think the claim should be true for $C^1$ or weaker regions that is bounded, and the proof must be very technical. Thank you very much if you can help me with this problem, and it would be better if anyone could give some useful reference.
Actually I found this related answer once I posted this question. It proves the Lipschitz problem I mentioned in the beginning using the extension theorems of Sobolev spaces. However, it doesn't give a (direct) answer to the geometric question in this post.
