# Five Open Problems Regarding Convex Polytopes

## The problems

1. The $3^d$ conjecture

A centrally symmetric d-polytope has at least $3^d$ non empty faces.

2. The cube-simplex conjecture

For every k there is f(k) so that every d-polytope with $d \ge f(k)$ has a k-dimensional face which is either a simplex or combinatorially isomorphic to a k-dimsnional cube.

3. Barany’s question

For every d-dimensional polytope P and every k, $0 \le k \le d-1$,  is it true that $f_k(P) \ge \min (f_0(P),f_{d-1}(P))$?

(In words: the number of k-dimensional faces of P is at least the minimum between the number of vertices of P and the number of facets of P. )

4.  Fat 4-polytopes

For 4-polytopes P, is the quantity $(f_1(P)+f_2(P))/(f_0(P)+f_3(P))$ bounded from above by some absolute constant?

5.  five-simplicial five-simple polytopes

Are there 5-simplicial 5-simple 10-polytopes? Or at least 5-simplicial 5-simple d-polytope for some d?

(A polytope P is k-simplicial if all its faces of dimension at most k, are simplices. A polytope P is k-simple if its dual P* is k-simplicial.)

And on a personal note: My beloved, beautiful,  and troubled country celebrates 60 today: happy birthday!

Update (May 12): David Eppstein raised in a followup a sort of a dual question to Barany’s. For a d-polytope with n vertices and n facets what is the maximal number of k-faces. For a fixed d and large n the free join of pentagons is conjectured to give asymptotically the best upper bound.

Update (July 29) Gunter Ziegler reminded me of the following additional problem of Barany: Is the number of saturated chains in a d-polytope bounded by some constant (depending on d) times the total number of faces (of all dimensions) of the polytope. A saturated flag is a 0-face inside a 1-face inside a 2-face … inside a (d-1)-face.