A $\mathcal{C}^{1}$ differentiable domain and Hausdorff dimension estimates

Question

Let us consider an open connected domain $E\subset \mathbb{R}^N$ and fix a point $x_0\in\partial E$ on its boundary. Suppose now, that there exists $R>0$ such that the set $\partial E \cap B_{R}(x_0)$ is $\mathcal{C}^1$, i.e. there exists a unique continuous normal vector \begin{eqnarray*} \nu \,\colon \,\,& B_{R}(x_0)\cap \partial E&\to &\mathbb{R}^N\\ & x&\mapsto & \nu_{x} \end{eqnarray*} to the set $\partial E\cap B_{R}(x_0)$. The question is: is it true that the set $\partial E$ is $F_\sigma$ and there exists some estimate on the Hausdorff dimension of $\partial E$?

Answer 1

Being a codimension 1 smooth submanifold, your set $\partial E\cap B_R(x_0)$ will have Hausdorff dimension $N-1$. The same goes for $\partial E$ since you can take compact sets $K$ and obtain Hausdorff dimension $N-1$ for each intersection $\partial E\cap K$ (by taking a finite subcover by balls $B_{R_i}(x_i)$). Finally, take an increasing sequence of compact sets $K_n$ with $\bigcup_{n=1}^\infty K_n=\mathbb R^N$ and note that $$\dim_H \partial E=\lim_{n\to\infty}\dim_H(\partial E\cap K_n)=N-1.$$