# Fat 4-polytopes

The *fatness* of a 4-polytope is defined to be where is the number of faces of of dimension .

**Question**Does there exist a fixed constant so that every convex 4-polytope has fatness at most ?

The -vector of a -dimensional polytope is the vector where is the number of faces of dimension . Let us denote by the collection of all -vectors of convex -dimensional polytopes. Steinitz proved that the set is completely characterized by the following three conditions:

- \item , \item , \item .

The first of these conditions is Euler's formula. The second and third are easy inequalities which are tight for simplicial (all faces triangles) and simple (all vertices of degree 3) polytopes, respectively.

In sharp contrast to this, the situation for seems to be quite complicated. For instance, it has been shown that does not contain all elements of which lie in the convex hull of ; i.e., has "holes" in it. For the extreme examples of simple and simplicial polytopes, the -theorem of Billera-Lee and Stanley gives a complete description of all possible -vectors, but in general very little is known.