Smallest Lipschitz constant on non-convex domains

Question

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 $f$?

Answer 1

You still have $\|f(x) -f(y)\| \le C L(x,y)$ where $L(x,y)$ is the length of the shortest path in $U$ from $x$ to $y$ (assuming such a path of finite length exists). The basic problem is that there can be points $x,y$ such that $\|x - y\|$ is small but $L(x,y)$ is large. An appropriate "modulus of non-convexity" would be the supremum of $L(x,y)/\|x-y\|$ for $x, y \in U$ with $x \ne y$, assuming that is finite.