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Consider the space of non-empty, compact, and convex subsets of $\mathbb{R}^d$ equipped with the Hausdorff metric.

Are simplicial polytopes a dense subset of that space?

Probably this is just a reference request, since I am currently traveling far from my area of expertise.

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  • $\begingroup$ I'd say the answer is yes for any $d\ge0$. Why $d=2$ is special? Aren't then simplicial polytopes just the convex polygons? $\endgroup$
    – Pietro Majer
    Commented Sep 15, 2022 at 7:47
  • $\begingroup$ No, only triangles: en.wikipedia.org/wiki/Simplicial_polytope If I had to prove the statement, I would try to prove it for polytopes and then show that simplicial polytopes lie dense in the polytopes, but again that cannot work for $d=2$. I rather not, however, since I assume this should be known. $\endgroup$
    – user467453
    Commented Sep 15, 2022 at 7:50
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    $\begingroup$ Sorry, I still don't get the definition you are referring to, even checking the wikipedia link. Isn't for you 1) A 2D convex polytope just a convex polygon? 2) A facet of a polygon just an edge, which is a 1D simplex? $\endgroup$
    – Pietro Majer
    Commented Sep 15, 2022 at 8:13
  • $\begingroup$ The approximation you want is very easy then: take the convex hull of a finite $\epsilon$-net $S$ of the compact convex set $K\subset \mathbb R^d$, with the additional property that no $(d+1)$-subset of $S$ is affinely dependent (which is true for a dense open set in $(\mathbb R^d)^{|S|}$ ). Then $\text{co}(S)$ is a simplicial polytope whose Hausdorff distance from $K$ is not larger than $\epsilon$ . $\endgroup$
    – Pietro Majer
    Commented Sep 15, 2022 at 8:24
  • $\begingroup$ I am sorry for my earlier mistake. I was confused. Thanks a ton! $\endgroup$
    – user467453
    Commented Sep 15, 2022 at 9:15

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An almost algorithmical construction: Consider a polytope $P$ with barycenter $B$. Add an additional vertex outside $P$ at infinitesimal distance of $P$ on all rays starting at $B$ and running through the barycenter of a $2$-face which is not a triangle. This leads to a polytope having only triangles as $2$-faces. Iterate the construction with $3$-faces which are not simplices etc. The final result is a simplicial polytope.

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