Bi-Lipschitz Expansion of Measurable Sets

Abstract

We show that for $0<\gamma ,{\gamma }^{\prime }<1$ and for measurable subsets of the unit square with Lebesgue measure $\gamma$ 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 $1-{\gamma }^{\prime }$.