Linear relations in dissections into \(n\)-dimensional parallelepipeds (Q1078813)

The author generalizes his earlier result in Math. Notes 30, 942-946; translation from Mat. Zametki 30, No.6, 881-887 (1981; Zbl 0481.52006). Let \({\mathbb{P}}\) be the set of all parallelepipeds \(X\subset {\mathbb{R}}^ n_+\) of the form \(X=[c_ 1,c_ 1+d_ 1]\times...\times [c_ n,c_ n+d_ n],\) \(c_ i\geq 0\), \(d_ i\geq 0\). The equality \(X_ 0=\lambda_ 1X_ 1+...+\lambda_ mX_ m\), \(\lambda_ i\in {\mathbb{R}}\) means that for the characteristic functions \(\chi_ X\) the equality \(\chi_{X_ 0}=\lambda_ 1\chi_{X_ 1}+...+\lambda_ m\chi_{X_ m}\) holds except the set of boundary points of \(X_ i\) (having zero hypervolume). Particularly \(X_ 0=X_ 1+...+X_ m\) means that \(X_ 0\) is dissected into \(X_ 1,...,X_ m\). The main theorem (Theorem 1) asserts that if \(X_ 0=\lambda_ 1X_ 1+...+\lambda_ mX_ m\) then for each covering \(\{M_ 1,...,M_ n\}\) of the set \(M=\{1,..,m\}\) by its arbitrary (possibly empty) subsets \(M_ i\subseteq M\), there exists an index i such that the equality \(X^ i_ 0=\sum_{j\in M_ i}\epsilon_ jX^ i_ j\) holds, where \(X^ i\) is the projection of X onto the i-th coordinate axis and \(\epsilon_ j\in \{0,\pm 1\}.\) As a consequence it is shown (Theorem 3) that if \(X_ 0=\lambda_ 1X_ 1+...+\lambda_ mX_ m\) with \(X_ 0\) a unit cube and all \(X_ i\) of equal hypervolume v, then 1/v is an integer.