Reifenberg Theorem for Locally Finitely Almost Splitting Sets

TL;DR

This paper extends the Reifenberg theorem to sets approximated by multiple parallel planes, showing such sets are images of multivalued biHölder maps, even if they are degenerate or not homeomorphic to disks.

Contribution

It introduces a generalized Reifenberg-type result for sets approximated by multiple planes, establishing their structure as images of multivalued biHölder maps.

Findings

Sets approximated by multiple parallel planes are images of multivalued biHölder maps.

Such sets may be degenerate and not homeomorphic to disks or unions of disks.

The result generalizes classical Reifenberg theorem to more complex geometric structures.

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 biH\"older 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 biH\"older homeomorphism of the disk into the set of subsets of $\mathbb{R}^n$.