On low-dimensional faces that high-dimensional polytopes must have (Q2640884)

The author proves that every five-dimensional polytope has a two- dimensional face which is a triangle or a quadrilateral. He also states and discusses the conjecture: For every integer \(k\geq 1\) there is an integer f(k) such that every d-polytope, \(d\geq f(k)\), has a k-dimensional face which is either a simplex or combinatorially isomorphic to the k- dimensional cube. In addition, the author gives some related results concerning facet-forming polytopes and tilings.