APPROXIMATION OF A REIFENBERG-FLAT SET BY A SMOOTH SURFACE
Résumé
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 $r_0$. Then we can apply a result of G. David and T. Toro [Memoirs of the AMS 215 (2012), 1012], and get a bi-Hölder homeomorphism of $\R^n$ that sends $\Sigma_0$ to $E$. When $d=n-1$ and $E$ is compact and connected, $\Sigma_0$ is automatically orientable, and $\R^n \sm E$ has exactly two connected components.
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