On low-dimensional faces that high-dimensional polytopes must have

We prove that every five-dimensional polytope has a two-dimensional face which is a triangle or a quadrilateral. We state and discuss the following conjecture: For every integerk1 there is an integer f(k) such that everyd-polytope,df(k), has ak-dimensional face which is either a simplex or combinatorially isomorphic to thek-dimensional cube.We give some related results concerning facet-forming polytopes and tilings. For example, sharpening a result of Schulte 25] we prove that there is no face to face tiling of ⁵ with crosspolytopes.Supported in part by a BSF Grant and by I.H.E.S, Bures-Sur-Yvette.