We show that for @math and for measurable subsets of the unit square with Lebesgue measure @math there exist bi-Lipschitz maps with bounded Lipschitz constant (uniformly over all such sets) which are identity on the boundary and increases the Lebesgue measure of the set to at least @math .
The question of existence of bi-Lipschitz homeomorphisms between different subsets of @math , satisfying certain conditions, has been studied classically, notably by McMullen @cite_0 . It is shown in that paper that there exists a positive function @math on @math , @math , such that there does not exist a bi-Lipschitz homeomorphism @math with @math . It is also proved that there exists separated nets in @math ( @math ) which cannot be mapped into @math in a bi-Lipschitz way.
@cite_0
2122101130
We prove two stability results for Lipschitz functions on triangulable, compact metric spaces and consider applications of both to problems in systems biology. Given two functions, the first result is formulated in terms of the Wasserstein distance between their persistence diagrams and the second in terms of their total persistence.