It is well known that if a function $f:U\to \mathbb C^n$, $U\subset \mathbb C^m$ satisfies $\sup_{x\in U}\|Df(x)\|_{\infty} = C < \infty$ uniformly on $U$ and $U$ is compact and convex, then $f$ is Lipschitz with smallest possible constant $C$.

What if $U$ is non-convex, but still compact and connected? Is there any reasonable "measure of non-convexity" which can be used to bound the Lipschitz constant of such $f$?