Bi-Lipschitz Expansion of Measurable Sets

We show that for $0<γ, γ' <1$ and for measurable subsets of the unit square with Lebesgue measure $γ$ 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 $1-γ'$.

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