1. The conjecture
A centrally symmetric d-polytope has at least non empty faces.
2. The cube-simplex conjecture
For every k there is f(k) so that every d-polytope with 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, , is it true that ?
(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 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.