Smallest Lipschitz constant on non-convex domains - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T00:09:54Z http://mathoverflow.net/feeds/question/115458 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/115458/smallest-lipschitz-constant-on-non-convex-domains Smallest Lipschitz constant on non-convex domains dima 2012-12-05T01:27:07Z 2012-12-05T01:44:49Z <p>It is well known that if a function $f:U\to \mathbb C^n$, $U\subset \mathbb C^m$ satisfies $\sup_{x\in U}\|Df(x)\|_{\infty} = C &lt; \infty$ uniformly on $U$ and $U$ is compact and convex, then $f$ is Lipschitz with smallest possible constant $C$.</p> <p>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 $f$?</p> http://mathoverflow.net/questions/115458/smallest-lipschitz-constant-on-non-convex-domains/115459#115459 Answer by Robert Israel for Smallest Lipschitz constant on non-convex domains Robert Israel 2012-12-05T01:44:49Z 2012-12-05T01:44:49Z <p>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.</p>