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

G. Kalai*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

13 Scopus citations

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 integer k≥1 there is an integer f(k) such that every d-polytope, d≥f(k), has a k-dimensional face which is either a simplex or combinatorially isomorphic to the k-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.

Original languageEnglish
Pages (from-to)271-280
Number of pages10
JournalCombinatorica
Volume10
Issue number3
DOIs
StatePublished - Sep 1990

Keywords

  • AMS subject classification (1980): 52A25, 52A20

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