How to take shortcuts in Euclidean space: making a given set into a short quasi-convex set
Jonas Azzam, Raanan Schul · Proceedings of the London Mathematical Society · 2012
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.