There is a statement about the partition of polyhedral boundary. It seems right for me, but I am not sure how to prove the existence precisely and whether there is a common statement as lemma or theorem.
If $E$ is a bounded set with polyhedral boundary, and that the outer unit normal to E (that is element aril you defined at $\mathcal{H}^{n-1}$-a.e. point of $\partial E$) is never orthogonal to $e_n$.
[Is the statement about outer unit normal a result of polyhedral boundary or a assumption? I think it’s a result, since you can partition the boundary into finite affine functions, the outer normals are finite directions. And except the points between two different affine functions, it’s well defined, hence $\mathcal{H}^{n-1}$-a.e. ]
By this assumption, and by the implicit function theorem, there exist a partition of the set $G = \{ z \in \mathbb{R}^{n-1} : \mathcal{L}^1(E_z)>0\}$ ($E_z$ is the slice.) into finitely many $(n-1)$-dimensional polyhedral sets $\{G_h\}_{h=1}M$ in $\mathbb{R}^{n-1}$, $G= \bigcup_{h=1}^M G_h$, and affine functions $v_h^k,u_h^k: G_h \to \mathbb{R}, 1 \leq h \leq M, 1 \leq k \leq N(h)$, with $$\partial E = \bigcup_{h=1}^M \bigcup_{k=1}^{N(h)} \Gamma(u_h^k, G_h) \cup \Gamma(v_h^k,G_h) \qquad \Gamma\text{ is the graph}$$ $$E = \bigcup_{h=1}^M \{ (z,t) \in G_h \times \mathbb{R}: t \in \bigcup_{k=1}^{N(h)}(v_h^k(z), u_h^k(z)) \}$$
[I know that since the boundary is compact, it can be partition into finite part and each part can be describe as finite affine functions. My question is the existence of this partition stated above.]
Further question: For $C^k$ boundary, can we get the same result ($v_h^k, u_h^k$ are $C^k$ functions) as above?
