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Let $A$ be an open set in $\mathbb{R}^n$, $C \subset A$ a compact set. Does there always exist an open set $B \subset A$ such that $C \subset B$ and $\partial B \subset A$ is a $C^1$ regular surface?

As possible partial results, first notice that by compactness there is a finite collection of balls contained in $A$ whose union contains $C$ (the border of course is not regular in the intersections). One can also try to regularize the characteristic function $\mathbb{1}_B$ of an open set $B$ that satisfies the inclusions, and consider a level set of the regularized function. However, it seems non trivial to show that one can choose the regularization and the level set in such a way that the gradient never vanish in the level set.

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Indeed the regularization is a useful partial result: it then suffices to apply Sard theorem to the regularized function to find a superlevel set with regular border.

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  • $\begingroup$ Can you provide the details, please? $\endgroup$
    – Gustavo
    Commented Jun 22, 2019 at 19:19

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