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.
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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. Does there exist a Lipschitz injection of ℤd into the open set of a site percolation process on ℤD, if the percolation parameter p is sufficiently close to 1? We prove a negative answer when d = D and also when d ≥ 2 if the Lipschitz constant M is required to be 1. Earlier work of Dirr, Dondl, Grimmett, Holroyd and Scheutzow yields a positive answer for d < D and M = 2. As a result, the above question is answered for all d, D and M. Our proof in the case d = D uses Tucker’s lemma from topological combinatorics, together with the aforementioned result for d < D. One application is an affirmative answer to a question of Peled concerning embeddings of random patterns in two and more dimensions. We introduce a new class of multifunctions whose graphs under certain "kernel inverting" matrices, are locally equal to the graphs of Lipschitzian (single-valued) mappings. We characterize the existence of Lipschitzian localizations of these multifunctions in terms of a natural condition on a generalized Jacobian mapping. One corollary to our main result is a Lipschitzian inverse mapping theorem for the broad class of "max hypomonotone" multifunctions. We apply our theoretical results to the sensitivity analysis of solution mappings associated with parameterized optimization problems. In particular, we obtain new characterizations of the Lipschitzian stability of stationary points and Karush-Kuhn-Tucker pairs associated with parameterized nonlinear programs. We show that the Goemans-Linial semidefinite relaxation of the Sparsest Cut problem with general demands has integrality gap @math . This is achieved by exhibiting @math -point metric spaces of negative type whose @math distortion is @math . Our result is based on quantitative bounds on the rate of degeneration of Lipschitz maps from the Heisenberg group to @math when restricted to cosets of the center.