A theorem on polyhedra which is related to Alexander's separation theorem (Q1263837)

For a subset \(M\subset\mathbb{R}^ n\) and \(\alpha\in(0,\pi]\) the \(\alpha\)- angle-zone of \(M\) is defined as the union of \(M\) and all points \(x\in\mathbb{R}^ n\setminus M\) for which there exist points \(y,z\) in M such that the angle at x in the triangle \(\Delta(x,y,z)\) has arc measure \(\geq\alpha\). Let \(\alpha_ n\) be that number of the interval (0,\(\pi)\) for which \(\cos \alpha_ n=-1/(n+1).\) A lattice cube of \({\mathbb{R}}^ n\) is an n-dimensional interval \([a_ 1,a_ 1+1]\times...\times [a_ n,a_ n+1],\) where the \(a_ i\) are integers. Then the following is proved: Let \(C_ 1,...,C_ k\) be finitely many lattice cubes of \({\mathbb{R}}^ n\) and \(F_ 1,...,F_ k\) non-empty compact subsets of \({\mathbb{R}}^ n\) such that the following holds: \(C_ i\cap C_ j=\emptyset \Rightarrow F_ i\cap F_ j=\emptyset\). For i,j\(\leq k\) the cubes \(C_ i\), \(C_ j\) are called neighboring if \(C_ i\cap C_ j\neq \emptyset\) or if there exists a bounded region of \({\mathbb{R}}^ n\setminus (C_ 1\cup...\cup C_ k)\) whose boundary intersects \(C_ i\) and \(C_ j\). Then every bounded region of \({\mathbb{R}}^ n\setminus (F_ 1\cup...\cup F_ k)\) is a subset of \(\sqcup \{\alpha_{n-1}(F_ i\cup F_ j)|\) \(i,j\in \{1,...,k\}\) and \(C_ i\), \(C_ j\) are neighboring\(\}\). This theorem entrains the theorem of Jordan/Brouwer/Alexander on the invariance of the separating set. If the \(F_ i\) are polyhedra, the above theorem can be interpreted as a discrete combinatorial version of this invariance theorem. [Corr.: On p. 241, line 2, add: (0) For every \(i\in \{1,...,k\}\) \(F_ i\) is not a subset of \(U_ i:=\sqcup \{F_ j|\) \(j\neq i\}\). And in line 5 replace {\(\overset\circ F_ i\) by \(\overset\circ F_ i\setminus U_ i\).].}