Some comments on the f-vectors of polytopes.
1. (Upper bound theorem.) Given a dimension d and a number of vertices n, what is the greatest number of k-faces (or the greatest ) that a polytope can have? The “cyclic polytope” maximizes all of these numbers simultaneously. (Those numbers can be computed explicitly)
This was conjectured by Motzkin in 1957 and proved by McMullen in 1970.
2. (g-theorem) We could ask (and Zach did ask) what are the possible h-vectors of polytopes. I said that this seems to be a very hard question for arbitrary polytopes. However, for *simplicial* polytopes, Billera-Lee and Stanley gave a remarkable answer, which is one of the greatest accomplishments in the theory of polytopes.
By the Dehn-Somerville relations, the f-vector of such a polytope only depends on the numbers for which form the “g-vector”. The “g-theorem” gives a set of necessary and sufficient inequalities that a vector must satisfy to be the g-vector of a simplicial polytope. Stanley proved the necessity while Billera-Lee proved the sufficiency – they are both very interesting.
3. (h-vectors and “line shellings”) Let me describe more precisely what the h-vector counts. Suppose that you stand very close to a facet F of P but outside of the polytope – you can only see F where you are. Now “fly away” from the polytope on a straight line in a fixed direction. At a certain point, you will (simultaneously) be able to see exactly new facets other than P. Then at a certain you will be able to see exactly new facets, etc. (You need to wrap around R^n until you see all the facets.) This interprets the h-vector of P as a counting problem, which gives the same answer regardless of the starting facet and the direction of flight!
Can you see why this proves the Dehn-Somerville relations?
4. I forgot to tell you that the Dehn-Somerville relations are *the only* relations satisfied by the f-vector of a simplicial polytope. Any other relation is a linear combination of these.
To read more about these topics, my favorite sources are Ziegler’s beautiful book “Lectures on Polytopes”, and Grunbaum’s “Convex Polytopes”.