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 Γ ⊃ Γ such that the two sets have comparable Hausdorff length, and the set has the property that it is quasiconvex, that is, any two points x and y in can be connected via a path, all of which is in, 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 as above such that 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.