Reifenberg Parameterizations for Sets with Holes

2009/10/26 by David, Guy, Toro, Tatiana · 4 citations
#28A75 #49K99 #49Q05 #49Q20 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.0910.4869

Abstract

We extend the proof of Reifenberg's Topological Disk Theorem to allow the case of sets with holes, and give sufficient conditions on a set E for the existence of a bi-Lipschitz parameterization of E by a d-dimensional plane or smooth manifold. Such a condition is expressed in terms of square summability for the P. Jones numbers β₁(x,r). In particular, it applies in the locally Ahlfors-regular case to provide very big pieces of bi-Lipschitz images of \R^d.

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