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For a given connected set Γ in  d ‐dimensional Euclidean space, we construct a connected set    Γ  ˜   ⊃ Γ such that the two sets have comparable Hausdorff length, and the set    Γ  ˜    has the property that it is quasiconvex, that is, any two points  x  and  y  in    Γ  ˜    can be connected via a path, all of which is in    Γ  ˜   , 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    Γ  ˜    as above such that    Γ  ˜    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 of the dimension  d , and in this case, our theorem holds for Γ in an infinite‐dimensional Hilbert space. This work is closely related to  k ‐spanners, which appears in computer science.

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