Lecture 22: More on counting faces of polytopes

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 f_k of k-faces (or the greatest h_k) that a polytope can have? The “cyclic polytope” C_d(n) 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 g_k = h_k-h_{k-1} for k \leq \frac{n}2 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 h_1 new facets other than P. Then at a certain you will be able to see exactly h_2 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”.

 

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