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

TL;DR

This paper demonstrates that bi-Lipschitz embeddings of the unit cube can be factorized into small-distortion mappings outside a negligible set, using a corona-type decomposition, and can be approximated by piecewise affine homeomorphisms.

Contribution

It introduces a corona-type decomposition theorem for bi-Lipschitz mappings and shows how to factorize and approximate such embeddings with small distortion outside small sets.

Findings

Bi-Lipschitz embeddings of the cube factor into small-distortion mappings outside small measure sets.

Bi-Lipschitz homeomorphisms of the sphere can be similarly factorized.

Embeddings can be approximated by piecewise affine homeomorphisms outside negligible sets.

Abstract

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.