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

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

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