## 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.

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### 3 Responses to Faces of Simple 4 Polytopes

1. Kristal Cantwell says:

If you started with the standard packing of spheres of unit radius in three dimensions and then wrapped it around a hypersphere of very large radius(there is going to be an error term from this since this doesn’t wrap exactly and then somehow corrected it through adjustments so it was simple. It seems to me that this idea ought to produce a class of simple polytopes that can have as large a number of facets as desired that have a a ratio of two dimensional faces as triangles to facets that as number of facets increases tends to a positive number. I am looking at this as a possible counterexample.

2. Kristal Cantwell says:

I am not sure the ideas in my first post work. I have thought of a simpler way to show the conjecture is false. Assume the conjecture is true start with a simple 4-polytope. Then simply use hyperplanes two cut of small simplices at all the vertices. Then all the original faces will now have twice as man sides so they will have at least 6 sides. The Newly formed faces will all be triangles. And so we now have simple four dimensional polytope all of whose faces do not have four sides.

3. Gil Kalai says:

Dear Kristal, the problem in your construction is that you add a facet for every vertex of the original polytope. Then in the new polytope the number of 2-faces will become just linear in the number of facets and the conjecture holds trivially.