# On the proportion of simplicial $d$-polytopes on $n$-vertices

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

• I assume that $d$-polytope means 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. – M. Winter Feb 9 at 9:45
• Good point M. Winter. I was thinking of n=g(d), say n=d+3. I have made that clear. – Guillermo Pineda-Villavicencio Feb 9 at 13:01