Cube-Simplex conjecture

Importance: High ✭✭✭
Author(s): Kalai, Gil
Subject: Geometry
» Polytopes
Recomm. for undergrads: no
Posted by: mdevos
on: May 9th, 2008

\begin{conjecture} For every positive integer $k$, there exists an integer $d$ so that every polytope of dimension $\ge d$ has a $k$-dimensional face which is either a simplex or is combinatorially isomorphic to a $k$-dimensional cube. \end{conjecture}

It is an easy consequence of Euler's formula that every 3-polytope has a face which is either a triangle, a quadrilateral, or a pentagon. The \Def{120-cell} is a 4-polytope in which every 2-face is a pentagon (in fact every 3-face is a regular dodecahedron). Perles and Shephard asked whether there exist higher dimensional polytopes in which all 2-faces have at least 5 vertices. This question was answered in the negative by Kalai [K] who showed that every 5-polytope has a 2-face with at most 4 vertices. So, if we define $f(k)$ to be the smallest integer $d$ satisfying the above conjecture for $k$, or $\infty$ if none exists, then $f(2) = 5$.

This conjecture is still open for simple polytopes. However, it is known that for every positive integer $k$, there exists an integer $d$ so that every simple polytope of dimension $\ge d$ either has a 2-dimensional face which is a triangle, or a $k$-dimensional face which is combinatorially isomorphic to a cube. This was proved by Kalai [K] using some earlier results of Nikulin and of Blind and Blind. Actually, something much stronger holds here: simple polytopes of sufficiently high dimension without 2-faces which are triangles must have most $k$-dimensional faces combinatorially isomorphic to the $k$-cube.

The following is an interesting weakening of the above conjecture.

\begin{conjecture} For every positive integer $k$, there exists an integer $d$ and a finite list $L$ of $k$-dimensional polytopes, so that every polytope of dimension $\ge d$ has a $k$-dimensional face which appears in $L$. \end{conjecture}

Defining $h(k)$ to be the smallest integer $d$ satisfying this conjecture for $k$, or $\infty$ if none exists, we find that $h(2) = 3$ (by the consequence of Euler's formula in the first paragraph). Meisinger, Kleinschmidt, and Kalai [MKK] proved that $h(3) \le 9$ with the help of FLAGTOOL, a computer program which can compute linear relations for $f$-vectors. This weaker conjecture is known to be true for simple polytopes.

Bibliography

[MKK] G. Meisinger, P. Kleinschmidt, and G. Kalai, Three theorems, with computer-aided proofs, on three-dimensional faces and quotients of polytopes. The Branko Grünbaum birthday issue. Discrete Comput. Geom. 24 (2000), no. 2-3, 413--420. \MRhref{MR1758060}

*[K] G. Kalai, On low-dimensional faces that high-dimensional polytopes must have. Combinatorica 10 (1990), no. 3, 271--280. \MRhref{MR1092544}


* indicates original appearance(s) of problem.