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Let $P$ be a convex polyhedron in $\mathbb{R}^3$, and $d(P)$ its intrinsic diameter, i.e., the longest shortest surface path between two points. Say that $P$ is of class

  • $D_0$ if neither endpoint of $d(P)$ is a vertex.
  • $D_1$ if one endpoint is a vertex and the other not.
  • $D_2$ if both endpoints are vertices.

(I believe these classes are mutually exclusive but I haven't proven that.)

My question is:

Q. In the space $\mathcal{P}_n$ of all convex polyhedra on $n$ vertices, what proportion are of class $D_0, D_1, D_2$ ?

My intuition is that polyhedra of class $D_0$ are rare, whereas "most" polyhedra are of class $D_1$. Even approximate relative proportions would be useful.

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    $\begingroup$ What is the measure on the space of convex polyhedra with $n$ vertices? By default I assume that the $3n$ coordinates all have independent standard normal distributions, and we look at the subspace corresponding to convex polyhedra. $\endgroup$
    – user44143
    Commented Aug 12, 2022 at 13:54
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    $\begingroup$ Can you give an easy example in class $D_0$? $\endgroup$
    – user44143
    Commented Aug 12, 2022 at 13:58
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    $\begingroup$ @MattF. For a tall triangular prism, the centroids of the base and top determine the diameter. I have an image in one of my papers but I cannot access it at the moment. $\endgroup$
    – Joseph O'Rourke
    Commented Aug 12, 2022 at 14:25
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    $\begingroup$ That puts the polyhedra in a metric space, but how do we get the probability measure? The Hausdorff measure on its subspace of convex polyhedra with unit surface area is available, but that seems complicated. $\endgroup$
    – user44143
    Commented Aug 12, 2022 at 14:37
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    $\begingroup$ You could add the question: “Is it true that classes $D_0$ and $D_2$ are nowhere-dense?” That would imply proportions of 0, 1, 0 for any reasonable measure. $\endgroup$
    – user44143
    Commented Aug 14, 2022 at 20:59

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