Reifenberg Theorem for Locally Finitely Almost Splitting Sets

Abstract

The well-known Reifenberg theorem states that if a subset of ${\mathbb{R}}^n$ can be well approximated by $k$-planes at every point and every scale, then it is biHölder homeomorphic to a $k$-disk. This article concerns a subset $S$ of ${\mathbb{R}}^n$ which can be approximated by at most $N$ parallel $k$ planes at each point and scale. As a subset of ${\mathbb{R}}^n$ such an $S$ may be quite degenerate; $S$ may clearly not be homeomorphic to a disk, and indeed we will see may not be homeomorphic to a union of disks. However, we prove that $S$ is still the image of a multivalued map on ${\mathbb{R}}^k$, which is itself a biHölder homeomorphism of the disk into the set of subsets of ${\mathbb{R}}^n$.