Projected products of polygons

Abstract

It is an open problem to characterize the cone of f-vectors of 4-dimensional convex polytopes. The question whether the “fatness” of the f-vector of a 4-polytope can be arbitrarily large is a key problem in this context. Here we construct a 2-parameter family of 4-dimensional polytopes (Formula Presented) with extreme combinatorial structure. In this family, the “fatness” of the fvector gets arbitrarily close to 9; an analogous invariant of the flag vector, the “complexity,” gets arbitrarily close to 16. The polytopes are obtained from suitable deformed products of even polygons by a projection to ℝ4. © 2004 American Mathematical Society.