Cutting a polytope (Q1900020)

Theorem 1. There exist two tetrahedra which can be positioned such that: (1) the origin is in the interior of both the tetrahedra, and (2) no closed half-space whose boundary plane passes through the origin contains more than 5 of the 8 vertices of the two tetrahedra. Theorem 2. (The main theorem) There exists a convex 4-dimensional polytope \(P\) with vertices \(v\) and \(w\) for which no hyperplane containing \(v\) and \(w\) has more than one facet in either closed half-space. This paper, in spite of its brevity, gives a fine survey of conjectures and results concerning how hyperplanes intersect the boundary of polytopes. There is a substantial bibliography.