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.

simplifythe situation a bit": unlikely. $\endgroup$