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](https://mathoverflow.net/questions/64810/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?](https://mathoverflow.net/questions/66173/what-is-known-about-the-number-of-permissible-simplicial-complexes-given-the-numb)


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.



Number of simplicial polytopes with a given f-vector
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](http://en.wikipedia.org/wiki/Cyclic_polytope) 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](http://page.mi.fu-berlin.de/gmziegler/ftp/archiv/099utah.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.


