# Faces of Simple 4 Polytopes

In the conference celebrating Klee and Grünbaum’s mathematics at Seattle Günter Ziegler proposed the following bold conjecture about 4 polytopes.

Conjecture: A simple 4-polytope with $n$ facets has at most a linear number (in $n$)  two dimensional faces which are not 4-gons!

If the polytope is dual-to-neighborly then the number of 2-faces is quadratic in $n$. For the dual-to-cyclic polytope the assertion of the conjecture is true.