Projected products of polygons

Abstract

It is an open problem to characterize the cone of f f -vectors of 4 4 -dimensional convex polytopes. The question whether the “fatness” of the f f -vector of a 4 4 -polytope can be arbitrarily large is a key problem in this context. Here we construct a 2 2 -parameter family of 4 4 -dimensional polytopes π ( P n 2 r ) \pi (P^{2r}_n) with extreme combinatorial structure. In this family, the “fatness” of the f f -vector gets arbitrarily close to 9 9 ; an analogous invariant of the flag vector, the “complexity,” gets arbitrarily close to 16 16 . The polytopes are obtained from suitable deformed products of even polygons by a projection to R 4 \mathbb {R}^4 .