# 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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### Adjacent vertices of a permutohedron which is the orbit of a point $x$ with repeated coordinates

Question: Let $x=(x_1, ..., x_n) \in \mathbb{R}^n$, and let $P$ be the convex hull of the points formed by permuting the coordinates of $x$. Given a vertex $y$ of $P$, what is a general rule for ...
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### Can you perturb an inscribed polytope so all its edges grow?

Consider the family of convex simplicial polytopes with vertices in the unit sphere of $\mathbb{R}^n$ which have the origin as an interior point. My question is the following: Let $P, P'$ be two non-...
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### Secondary polytope

Given a polytope $P$, what do the points of the secondary polytope correspond to? I know that the vertices of the secondary polytope correspond to regular triangulations of $P$. But what do the ...
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### Roundest polyhedra: how well can we bound the edge count of their faces?

By "roundest" I mean having the lowest surface area for the highest volume, given a fixed number of faces $n$. There've been a few questions about them on here (including from me), but I'm ...
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### Graduate level convexity - Intersection of an r-polytope with a hyperplane is an r-1 polytope

I am trying to follow Roger Webster's Convexity 's proof of Euler's celebrated result on the relationship between the number of faces of a polytope. An image of the proof is here. In the course of the ...
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### Is my open set a ball?

Let $P$ be a polytope of dimension $n$, and let $\mathcal{C}$ be a polyhedral subdivision of $P$. That means that $\mathcal{C}$ is a finite collection of polytopes whose union is $P$, such that the ...
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### Tiling space with supertile of hypercube unfoldings

Two students in my class asked and answered what might be a novel question. It is well known that the cube has exactly $11$ edge-unfoldings (or "nets"), as shown below:         (Image from ...
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