# Open problems in polyhedral combinatorics

Gil Kalai lists five open questions about polyhedra. Two of them are ones I've worked on in a paper with Kuperberg and Ziegler:

How big can the numbers of \( k \)-faces be in a polytope with n vertices and \( n \) facets? Gil asks the question for four-dimensional polytopes (where the issue is whether the numbers of edges and ridges can be larger than the numbers of vertices and facets by more than a constant factor) but the same question can be asked in any higher dimension. A simple construction forms polytopes with \( \Omega(n^{\lceil (d-1)/3 \rceil}) \) intermediate-dimension faces and it's reasonable to hope that's the right answer. My paper with Kuperberg and Ziegler showed that there are cell complexes that look like 4-polytopes combinatorially and topologically but where the number of edges and vertices is large, but it's not obvious whether these complexes can be realized as polytopes.

Are there polytopes that look simplicial in their low-dimensional faces and simple in their high dimensional faces? That is, all 2-faces should be triangles, all 3-faces tetrahedra, ..., and the same for the dual polytope. In the same paper, we showed that there are infinitely many 2-simple 2-simplicial 4-polytopes. Gil is asking for 5-simple 5-simplicial 10-polytopes.

Another of his questions, the one about whether centrally symmetric polytopes always have \( 3^d \) or more faces, came up earlier in the comment threads of this post by Terry Tao.