A map is -bilipschitz if for all . This definition makes sense if X and Y are general metric spaces, but let’s suppose they are subsets on the plane .
Definition 1. A set has the BL extension property if any bilipschitz map can be extended to a bilipschitz map . (Extension means that is required to agree with on .)
Lines and circles have the BL extension property. This was proved in early 1980s independently by Tukia, Jerison and Kenig, and Latfullin.
Definition 2. A set has the controlled BL extension property if there exists a constant such that any -bilipschitz map can be extended to a -bilipschitz map .
Clearly, Definition 2 asks for more than Definition 1. I can prove that a line has the controlled BL extension property, even with a modest constant such as . (Incidentally, one cannot take .) I still can’t prove the controlled BL extension property for a circle.