Some reflections on a simple geometric problem (Q2786924)

The article deals with the extension of the well-known problem in Euclidean 3D space. The extension is stated in the Theorem:NEWLINENEWLINE If \(A=A_1, A_2,\dots, A_n\) and \(B_1,B_2,\dots, B_m\) is a split of the \(n+m=d+1\) vertices of a \(d\)-dimensional simplex in \(E^d\) into two non-empty disjoint sets, then there exist two parallel hyperplanes \(\pi\) and \(\sigma\), such that the set \(A\) is in the hyperplane \(\pi\) and the set \(B\) is in the hyperplane \(\sigma\) and the intersection \(\pi \cap \sigma\) is included in the hyperplane at infinity.NEWLINENEWLINE One can split the vertex set of a \(d\)-simplex in \(E^d\) into more than two non-empty subsets. We get the following generalization: If the \(d + 1\) vertices of a \(d\)-simplex in \(E^d\) are split into \(k\) mutually-disjoint non-empty subsets \(A^1=A_1,1, A_1,2,\dots\), \(A^2= A_2,1, A_2,2,\dots\), \dots, \(A^k= A_k,1, A_k,2,\dots\) then there exist \(k\) hyperplanes \(\pi_1\), \(\pi_2\), \dots, \(\pi_k\), such that \(A^i\) is in \(\pi_i\) for all \(i\), and the intersection of all the hyperplanes is an affine flat at infinity of the suitable dimension.