Factorization and piecewise affine approximation of bi-Lipschitz mappings on large sets

A well-known open problem asks whether every bi-Lipschitz homeomorphism of ${\mathbb{R}}^d$ factors as a composition of mappings of small distortion. We show that every bi-Lipschitz embedding of the unit cube $[0,1{]}^d$ into ${\mathbb{R}}^d$ factors into finitely many global bi-Lipschitz mappings of small distortion, outside of an exceptional set of arbitrarily small Lebesgue measure, which cannot in general be removed. Our main tool is a corona-type decomposition theorem for bi-Lipschitz mappings. As corollaries, we obtain a related factorization result for bi-Lipschitz homeomorphisms of the $d$-sphere, and we show that bi-Lipschitz embeddings of the unit $d$-cube in ${\mathbb{R}}^d$ can be approximated by global piecewise affine homeomorphisms outside of a small set.