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# Questions tagged [convex-polytopes]

Convex polytopes are the convex hulls of a finite set of points in Euclidean spaces. They have rich combinatorial, arithmetic, and metrical theory, and are related to toric varieties and to linear programming

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### How can the same polytope have three different volumes?

I'm quite new to geometry and I came across the idea that the same convex polytope can have at least three different volumes. Consider the permutohedron, formed by the convex hull of the n! points ...
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### Edges of the contact polytope of the Leech lattice

Let $P\subset\Bbb R^{24}$ be the contact polytope of the Leech lattice, that is, $P$ is the convex hull of the 196,560 shortest vectors of $\Lambda_{24}$. Question: What are the edges of $P$? Let'...
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### Given $H_{N}=\{\vec{x} \in [-1,1]^N:\sum_{i=1}^N x_i = 0\}$, what is the smallest subset $S \subset H_{2N}$ such that $\mathrm{conv}(S)=H_{2N}$

Motivation: This is related to a different question I asked in April. It occurred to me while thinking about the sums of uniform random variables and it stuck in my mind because it's the special case ...

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