# 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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### Can a point be closer to all the vertices of a convex polytope than another point inside that polytope?

Consider a set $X = \{x_i \in \mathbb{R}^n\}$ and denote its convex hull $$C \equiv \bigg\{ \sum_i \lambda_i x_i : \lambda_i \geq 0 \text{ for all } i \text{ and } \sum_i \lambda_i = 1 \bigg\}.$$ I ...
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### What is special about the 11-cell and 57-cell?

Reading about the 11-cell and 57-cell I find two facts implied often: They are particularly notable among the abstract regular 4-polytopes. They are related to each other. I'll establish why I think ...
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### Is there such a thing as a convex polytope defined over a mixed-variable space: $\mathbb{Z}^m \times \mathbb{R}^n$

I am currently learning about polytopes and I have reached the stage of convex polytopes. Wiki says that a convex polytope is a special case of a polytope, having the additional property that it is ...
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### Counting lattice points in cross-sections of a polytope

Let $P$ be polytope in $\mathbb{R}^d$ with vertices in $\mathbb{Z}^d$ and let $h(z_1,\ldots,z_d)\in\mathbb{Z}[z_1,\ldots,z_d]$ be a degree one polynomial with integer coefficients, thought of as a &...
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### Ant walking on a cube riddle generalization to higher dimensions

There's a pretty common riddle floating around which goes something like this: An ant starts at one corner of a unit cube. They wish to reach the opposite corner. If they can traverse along any face, ...
1 vote
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### Knowing if the finite intersection of half spaces forms a closed polytope?

Say you have $N$ half-spaces $P$ in some dimension. We want to know if the intersection of all such half spaces results in a closed or open polytope, and if open, we want to introduce the minimum ...
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### Which manifolds admit symmetric tilings?

Let $M$ be a connected, boundaryless manifold. A tiling of $M$ is a regular cellulation of $M$, and a tiling is regular (i.e. symmetric) if the group of self-homeomorphisms of $M$ that map cells to ...
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### To cut a square pyramid in half by a plane parallel to one of its lateral faces

The image below depicts a pyramid $ABCDE$ with a square base. It is to be cut by the plane $FGHI$ which is parallel to the face $ADE$, such that this plane cuts the pyramid in two halves. Find the ...
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### Simplicial polyhedral fans and shellability

A simplicial polyhedral fan $\Sigma \subseteq \mathbb{R}^n$ is a collection of simplicial polyhedral cones $\{\alpha_1 v_1 + \dots + \alpha_r v_r : \alpha_r \geq 0\} \subseteq \mathbb{R}^n$, where the ...
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### When does a smooth projective toric variety admit a symplectic structure?

I'm trying to understand the relationship between toric varieties and their associated polytopes. There is a very crisp result in the case of symplectic toric varieties, namely that there is a 1:1 ...
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### Birkhoff polytope vs permutation polyhedron

I cannot understand a few things about the Birkhoff polytope. The Birkhoff polytope is defined as a polyhedron of all $n \times n$ doubly stochastic matrices. How is is it polytope then? To transform ...
1 vote
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### Simplification and "average" of polytopes

I have multiple (convex and bounded) polytopes, all of them describing the feasible area of a given LP (the rest of the LP is always the same), for a number of scenarios. Since those polytopes are too ...
1 vote
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### Polytopes: affine isomorphism implies combinatorial isomorphism

Let $P \subseteq \mathbb{R}^d$ and $Q \subseteq \mathbb{R}^e$ be (convex) polytopes. Define them to be affinely isomorphic if there is an affine map $f \colon \mathbb{R}^d \to \mathbb{R}^e$ whose ...
My understanding is that, for an affine transformation with square matrix $T$, the volume of a set transformed through this affine transformation gets scaled by |det($T$)|. However, if $T$ is ...