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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

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  • $\begingroup$ 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. $\endgroup$
    – M. Winter
    Commented Feb 9, 2020 at 9:45
  • $\begingroup$ Good point M. Winter. I was thinking of n=g(d), say n=d+3. I have made that clear. $\endgroup$
    – Guillermo Pineda-Villavicencio
    Commented Feb 9, 2020 at 13:01

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