I have a question regarding estimates for the proportion of simplicial $d$-polytopes on $n$-vertices.

Let $c_s(n,d)$ denote the number of combinatorial types of simplicial $d$-polytopes on $n$ labelled vertices and let $c(n,d)$ denote the number of combinatorial types of (general) $d$-polytopes on $n$ labelled vertices. I am interested in estimates for the following limits:

- For a fixed $d$, $\lim_{n\to\infty} \frac{c_s(n,d)}{c(n,d)}$, and
- For a fixed $n=g(d)$, say $n=d+3$ for instance, $\lim_{d\to\infty} \frac{c_s(n,d)}{c(n,d)}$.

Is this known? Any help would be much appreciated.

Thank you very much in advance, and best regards, Guillermo

full-dimensional$d$-dimensional convex polytope. Then $c_s(n,d) = c(n,d)=0$ if $d>n-1$, and so the second limit is not meaningful. $\endgroup$ – M. Winter Feb 9 at 9:45