Number of simplicial polytopes with a given f-vector

Question

Plenty of very nice literature is available on the characterization of f-vectors of simplicial complexes of diverse sorts (results by Billera, Bjoerner, Kalai, Stanley, among others). I mention, as an example, the Dehn-Sommerville equations, the Upper- and Lower Bound Theorems, for Simplicial Polytopes.

Are there any results on the enumeration of simplicial, convex d-polytopes with a given f-vector?

A slightly simpler question is: How does information about an f-vector (say, specifying the number of vertices, edges and triangles) determine the amount of (simplicial, convex d-) polytopes having this numbers fixed. Are there some cases where the f-vector specifies completely the polytope?

This is related to these posts:

Number of graphs with a given number of nodes, edges and triangles

What is known about the number of permissible simplicial complexes given the number of k-cells?

And the reason I am concerned about this is that in the first of the posts, it has been commented that the problem may be way too difficult, so I was wondering whether imposing the condition that the simplicial complex be a convex polytope may simplify the situation a bit.

Answer 1

These are just some random remarks, with one hopefully useful reference.

"Are there some cases where the $f$-vector specifies completely the polytope?"

This is hardly what you are seeking, but for 3-polytopes, $f_2=2f_0-4$ is achieved exactly for the $f$-vectors of simplicial polytopes. And of course the stacked and cyclic polytopes achieve the lower and upper bounds respectively.

I don't know if you have seen Günter M. Ziegler's "Convex Polytopes: Extremal Constructions and $f$-Vector Shapes" (IAS/Park City Mathematics Series Volume 14, 2004), which seems to directly address your questions, albeit as of several years ago. Here is the PDF.

Here is one tidbit. He mentions, as a measure of our ignorance, that not even this "suspiciously innocuous conjecture" of Imre Bárány is settled:

For any $d$-polytope, $f_k \ge \min \{f_0, f_{d−1}\}$.

It is (or was in 2004) only proven for $d \le 6$.

Günter has a particularly careful description of what's known about the $f$-vectors of 4-polytopes, a specialty of his. In particular, the set of these $f$-vectors "is not the set of all integral points in a polyhedral cone, or even in a convex set." It has concavities and holes.