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

For a given connected set $\mathrm{\Gamma }$ in $d-$dimensional Euclidean space, we construct a connected set $\tilde{\mathrm{\Gamma }}\supset \mathrm{\Gamma }$ such that the two sets have comparable Hausdorff length, and the set $\tilde{\mathrm{\Gamma }}$ has the property that it is quasiconvex, i.e. any two points $x$ and $y$ in $\tilde{\mathrm{\Gamma }}$ can be connected via a path, all of which is in $\tilde{\mathrm{\Gamma }}$, 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 $\tilde{\mathrm{\Gamma }}$ as above such that $\tilde{\mathrm{\Gamma }}$ 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 $\mathrm{\Gamma }$ is Reifenberg flat, our constants are also independent the dimension $d$, and in this case, our theorem holds for $\mathrm{\Gamma }$ 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.