Approximation of a Reifenberg-flat set by a smooth surface

TL;DR

This paper demonstrates that Reifenberg-flat sets can be approximated by smooth surfaces, enabling the application of existing results to establish bi-Hölder homeomorphisms and properties of the complement for certain sets.

Contribution

It provides a method to approximate Reifenberg-flat sets with smooth surfaces, extending the applicability of geometric measure theory results.

Findings

Existence of smooth surface approximation for Reifenberg-flat sets

Construction of bi-Hölder homeomorphisms mapping the surface to the set

Approximation of the complement of the set by smooth domains in special cases

Abstract

We show that if $E \i \R^n$ is a Reifenberg flat set $E$ of dimension $d$ at scale $r_0$, we can find a smooth surface $\Sigma_0$ of dimension $d$ which is close to $E$ at the scale $r_0$. When $E$ is a Reifenberg flat set, this allows us to apply a result of G. David and T. Toro [Memoirs of the AMS 215 (2012), 1012], and get a bi-H\"older homeomorphism of $\R^n$ that sends $\Sigma_0$ to $E$. If in addition $d=n-1$ and $E$ is compact and connected, then $\Sigma_0$ is orientable, and $\R^n \sm E$ has exactly two connected components, which we can approximate from the inside by smooth domains.