1 research outputs found
How to take shortcuts in Euclidean space: making a given set into a short quasi-convex set
For a given connected set in dimensional Euclidean space, we
construct a connected set such that the two sets
have comparable Hausdorff length, and the set has the property
that it is quasiconvex, i.e. any two points and 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 and
. Thus, for any set in dimensional Euclidean space we have a set
as above such that has comparable Hausdorff
length to a shortest connected set containing . Constants appearing here
depend only on the ambient dimension . In the case where is
Reifenberg flat, our constants are also independent the dimension , and in
this case, our theorem holds for in an infinite dimensional Hilbert
space. This work closely related to spanners, which appear in computer
science. Keywords: chord-arc, quasiconvex, k-spanner, traveling salesman.Comment: Made referee edit