rectifiability

How to add shortcuts in the plane Imagine you have a network of roads. One way to judge the efficiency of the roads is in terms of how much longer it takes to travel between two points in your network than if you could just drive off-road in a straight line. This measure of efficiency is called quasiconvexity. Definition: A set $E$ is $C$-quasiconvex meaning that for every $x,y\in \tilde{\Gamma}$, there is a curve $\gamma\subseteq \mathbb{R}^{2}$ connecting $x$ and $y$ so that $\ell(\gamma)\leq C|x-y|$. Not every connected set of finite length is quasiconvex, and this note is about the problem of adding shortcuts to make it so.
2021-05-22
7 min read