  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.
