How to take shortcuts in Euclidean space: making a given set into a short quasi-convex set

Abstract

For a given connected set $\Gamma$ in $d-$dimensional Euclidean space, we construct a connected set $\tilde\Gamma\supset \Gamma$ such that the two sets have comparable Hausdorff length, and the set $\tilde\Gamma$ has the property that it is quasiconvex, i.e. any two points $x$ and $y$ in $\tilde\Gamma$ can be connected via a path, all of which is in $\tilde\Gamma$, which has length bounded by a fixed constant multiple of the Euclidean distance between $x$ and $y$. Thus, for any set $K$ in $d-$dimensional Euclidean space we have a set $\tilde\Gamma$ as above such that $\tilde\Gamma$ has comparable Hausdorff length to a shortest connected set containing $K$. Constants appearing here depend only on the ambient dimension $d$. In the case where $\Gamma$ is Reifenberg flat, our constants are also independent the dimension $d$, and in this case, our theorem holds for $\Gamma$ in an infinite dimensional Hilbert space. This work closely related to $k-$spanners, which appear in computer science. Keywords: chord-arc, quasiconvex, k-spanner, traveling salesman.