# 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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### What is an upper bound on the diameter of a convex polytope?

Given a convex polytope defined by $Ax \le b$, with $V = \{ x_1, \ldots, x_n \}$ vertices, I would like to find the maximal distance $\max_{i,j} || x_i - y_i||_2$ as a function of $A$ and $b$ (some ...
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### Find in which face lie a point on a hyper polytope, inscribed in an hyper unit-sphere (and how to generate this hyper polytope)

I am working on clustering normalized point in a k dimension space. At first i used Spherical LSH strategy where i used random planes to cut the hyper-sphere but it induces a lot of random (because i ...
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### 120 cell generated from quaternions

The two quaternions $\omega={1\over 2}(-1,1,1,1)$ and $q={1\over 4}(0,2,\sqrt{5}+1,\sqrt{5}-1)$ generate a finite group under multiplication with 120 elements that form the vertices of a 600 cell, ...
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### Point in Polytope?

Context: This question is somewhat identical to this on MathOverflow, it’s different in that it only focuses on the formula of the solution to the underlying problem. Suppose I have a convex hull $H$ ...
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### How to calculate the centroid of a Polytope?

Given a polytope is divided into simplexes, is it correct to calculate the centroid of the polytope as the average sum of its simplex centroid coordinates
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### What's the combination of vectors with all coeffiencts between 0 to 1？

Assume a combination of the form: $\vec{S}=\sum_ip_i\vec{v}_i$ with $0\le p_i\le 1$, what is this combination called? And considering the set: $$\mathcal{S}=\{\sum_ip_i\vec{v}_i \mid 0\le p_i\le 1\}$$ ...
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### Ray through Polytope boundary(vertex/surface)?

This is a follow up problem to the question I asked sometime earlier also a similar problem to this, please what's the best way to quickly determine if a ray originating from a point intersects a n-...
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### Fastest way to solve vertex enumeration problem in python

I have a set of 73 linear non-strict inequalities that describe a convex polytope in the 36-dimensional space. All but one of the inequalities are of the form $x>=b$ or $x<=b$. In every but one ...
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### Rewriting an equation as a set of inequalities

I have a set $\\{x_1,x_2 \dots , x_d \in \mathbb{R} : |x_1| + |x_2| + \dots + |x_d| \leq 1 \\}$ and i would like to rewrite it as $\\{ u \in \mathbb{R}^d : Au \leq \textbf{1} \\}$ , where $\textbf{1}$ ...
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### Two definitions about reflexive polytopes

I am working through Computing the Continuous Discretely and they give the definition of a reflexive polytope as $$P=\{x \in \mathbb{R}^d : Ax \leq 1\}$$ where all entries from $A$ are integers. It's ...
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### Generalization of Caratheodory's theorem

Conjecture: Let $S\subseteq \mathbb{R}^d$ and let $x, y \in \text{conv}(S)$, the convex hull of $S$. Then, $x, y \in \text{conv}(S')$ for some $S' \subseteq S$ such that $|S'| \leq d+2$. Basically, ...
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### What does mean $\mathrm{conv}\{\{+1, -1\}^d\}$

What does mean $\mathrm{conv}\{e_1 , -e_l , \ldots , e_d , -e_d\}$ and $\mathrm{conv}\{\{+1, -1\}^d\}.$ I could not understand, need a simple explanation.
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### Are any higher-dimensional analogues of triangles flexible?

I don't know if my terminology is correct, so bear with. Triangles are 'inflexible'. By that, I mean that if you consider a triangle to be a graph of 3 edges and 3 vertices in a closed loop then you ...
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### The space of all convex combinations including given set.

It is possibly a simple question with me grossly overcomplicating things, but I'm not exactly familiar with the topic. The question is long and consists of three parts. A prerequisite construction I ...
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Can someone help me to expand the below equation for N=2 and N=3? $f_i=\sum \{e_i:j\leq i \leq k\}$ for some $1\leq j\leq k\leq N$. Here $e_i$ is the unit vector of coordinate i in an N-dimensional ...