# Questions tagged [polytopes]

In elementary geometry, a polytope is a geometric object with flat sides, which may exist in any general number of dimensions $n$ as an $n$-dimensional polytope or $n$-polytope.

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### Polar Set of a Polyhedra is a Polytope

I am having trouble to verify Proposition 2 from the following MIT OCW document: Is there a way for us to see the equivalence of $C_1$ and $C_2$ simply through the definition? I do notice there was a ...
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### Number of vertices of product of polytopes

If one takes the product of a polytope $P_1$ with another polytope $P_2$, where $P_1$ has $n$ vertices and $P_2$ has $m$ vertices, will the product of the two polytopes have $mn$ vertices? It is not ...
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### A question about compactness and polytopes

I saw a statement as follow: Let $A$ be a compact set in $\mathbb{R}^n$, and $P\subset A$ a closed subset of $A$. By compactness, to prove that $P$ is a polytope, it sufffices to work locally about a ...
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### 3-dimensional convex polytope with adjacent vertices is a 3-simlpex

Let $P=\rm{conv} ( \textit{V} )$ be an convex $3$-dimensional polytope with vertices $V$ in which every two vertices $x,y \in V$ are adjacent. Show that $P$ is a $3$-Simplex. I think we can use radon'...
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### Is there a standard construction for Geodesic Polytopes in high dimensions?

Geodesic polytopes in $\mathbb R^3$ can be used to construct "simple" triangulations of $\mathbb S^{2}$, the 2-sphere. They can be constructed, for example, by taking a regular octahedron ...
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