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 $n$, $\lim_{d\to\infty} \frac{c_s(n,d)}{c(n,d)}$.

Is this known? Any help would be much appreciated.

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

I 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 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 interesting 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 $lim_{d\to\infty} \frac{c_s(n,d)}{c(n,d)}$.

Is this known? Any help would be much appreciated.

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

I 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 $n$, $\lim_{d\to\infty} \frac{c_s(n,d)}{c(n,d)}$.

Is this known? Any help would be much appreciated.

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