, Volume 10, Issue 3, pp 271-280

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

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Abstract

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 integerk≥1 there is an integer f(k) such that everyd-polytope,d≥f(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 ℝ5 with crosspolytopes.

Supported in part by a BSF Grant and by I.H.E.S, Bures-Sur-Yvette.