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Questions tagged [polytopes]

In elementary geometry, a polytope is a geometric object with flat sides, and may exist in any general number of dimensions n as an n-dimensional polytope or n-polytope. (Def: http://en.m.wikipedia.org/wiki/Polytope)

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The idea behind Delzant construction of a toric manifold from a convex polytope

I am trying to understand how to visualize a symplectic toric manifold from its moment polytope, following chapter 29.4 in "Lectures on Symplectic Geometry" by Ana Cannas da Silva: https://people.math....
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Why can't a vertex of a $d$-dimensional polytope be in fewer than $d$ edges?

This is motivated by the definition of simple polytopes: if all vertices of a $d$-dimensional convex polytope $P$ are in exactly $d$ edges (i.e. $1$-dimensional faces of $P$), then $P$ is simple. I ...
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Smooth function from $\mathbb R^n$ to a polytope

Given an open bounded convex polytope $C$ in $\mathbb{R}^n$, is there a way to construct a smooth function from $\mathbb{R}^m$ to $\mathbb{R}^n$ whose image is $C$? For instance, if $C$ is the unit ...
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How to compute the image of a polytope after a linear transformation (formal proof) ?

I have been looking for a result with (formal proof) of the following question (exactly the same question as posed in https://mathoverflow.net/questions/179282/mathcalh-polyhedron-under-a-linear-map?...
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Projection of convex polytope onto plane

A convex polytope is given by a system of linear inequalities in 4D space. How do I find its projection onto plane given by equation $x_1 + x_2 = 0$? If it was $x_1 = 0$, I'd use Fourier-Motzkin ...
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25 views

Prove that the polar set of a convex set contains the origin

Let the polytope $P = \text{conv}\{p_1,\ldots,p_N\} \subset \mathbb{R}^n$ and the polar set defined by $$Q = \{x\in \mathbb{R}^n : p_i^Tx \le 1, \forall i \in \{1,\ldots,N\}\}.$$ Prove that the origin ...
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How to compute the vertices of a hypercube given in H-form

I am working with a hypercube given in H-form: {x: Vx <= 1}. How would I compute the vertices of this hypercube?
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The meaning of the dimension of a Newton polytope

I am not very familar with algebraic geometry. Thus, maybe my question is a really basic one, but I have not find an answer in literature. I consider Newton polytopes of multivariate polynomials $f(x)...
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37 views

Number of vertices of a partiuclar polytope

Question: I would like to know the number (or a good approximation of it) of vertices of the polytope arising from the following constraints: $\{ (a_{1,1},\ldots,a_{1,n}, a_{2,1}, \ldots a_{2,n},\...
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computing polynomials whose roots are the vertices of 4-polytopes of circumradius one interpreted as quaternions

Suppose we have the vertices of 4-polytopes which have been scaled so that their are on the unit sphere. Interpret them as quaternions. If these have been computed before, I'm looking for a ...
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25 views

Consider a simple convex polyhedron, determine the vertices

Consider a simple convex polyhedron Δ in ℝ𝟛 with 2013 faces. How many vertices are there in Δ ? How many of the vertices have index one with respect to linear function?(not equal to a constant on ...
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Algorithm for Union of Polychora (4D Polytopes)

In the course of my research (radio engineering), I need to solve the following problem, for which I do not feel well equipped. I would like to calculate the union of polychora (4-polytopes, bounded ...
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22 views

Mean width distance and containment of ball

Let $B_n$ denote the Euclidean unit ball in $\Bbb R^n$, and let $P\subset B_n$ be a polytope such that $w(B_n)-w(P) \leq \epsilon$, where $w(K)=2\int_{\Bbb S^{n-1}} h_K(u)\,d\sigma(u)$ is the mean ...
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Procedure to generate 2-dimensional convex polytopes with a given number of vertices

I need to deal with the following. Let $[B]$ denote the set of integers $\{0,1,\ldots,B\}$. Consider an integer $n \geq 3$ and a grid $[B]\times [B]$, and let $P_{n,B}$ denote the set of convex ...
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Can one induce the Dual Subdivision of Tropical Signomials Similarly to Tropical Polynomials?

Preliminaries. Consider the semiring $\mathbb{T} = \{\mathbb{R} \cup \{-\infty\}, \oplus, \odot \}$, where the $a \oplus b = \max\{a,b\}$ and that $a \odot b = a + b$. The addition and scalar ...
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Why does a d-polytope being k-neighborly for 1 <= k <= d imply the polytope is neighborly for all k' such that 1 <= k' <= k?

I'm reading Grunbaum's Convex Polytopes where he cites the following theorem in a proof by contradiction for a larger theorem: "If $P$ is a $k$-neighborly $d$-polytope, and if $1 \le k' \le k$, ...
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Sampling extreme points from Minkowski Sum

I recently stumbled upon the following subproblem: we are given zonotopes $P_1, \dots, P_m$ in $\mathcal{V}$ representation (i.e. we are given the extreme points of each $P_i$). Denote the Minkowski ...
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38 views

Proving full rank of a special type of a (0,1,2)-integer matrices

My question arise at the consideration of Newton polytopes. In that context I consider integer matrices $ A =(a_{ij})\in \mathbb{Z}^{(n+1) \times N} $ with $n+1 \leq N$ having the following ...
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Is there a specific name for polytopes that only have 0s and 1s as coordinates?

I only know of the name 0/1-polytope, but whenever I search for it on Google, I get articles written by the same author. So I think there is another name for it.
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Are the corner hypercubera polytopes self-dual?

Motivation: The polyhedron whose vertices are seven of the vertices of a cube (four on the bottom and three on top) - called a cubera - is self-dual. Does an analogous construction produce a self-dual ...
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How to algorithmically find only the edges of a high dimensional convex hull?

Given to me is a set of points $p_1,...,p_n\in\Bbb R^d$ in general position. I want to determine only the edges of the convex hull $C:=\mathrm{conv}\{p_1,...,p_n\}$, and this as efficient as possible. ...
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In what sense are “projecting” and “taking sections” of polytopes dual operations?

It seems to be folklore that projecting and taking sections of polytopes are somehow "dual operations" (e.g. explicitly noted in the abstract of this paper, or suggested by this answer to an MO ...
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Is this construction of the “edge polytope” known?

Given a convex polytope $P\subseteq\Bbb R^d$. I am going to construct a new polytope from its edges (I call it the edge polytope) with the following steps: Take the 1-skeleton of $P$. Extract the ...
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What is known about these “transparent” polytopes?

I am looking for the name (if there is one), simple properties and possible literature for the following class of polytopes (by polytope I mean the convex hull of finitely man points in $\Bbb R^n,n\ge ...
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Number of facets intersecting a face of a non-simple polytope

Let $P$ be a non-simple polytope and $E$ a face of codimension $k$ with $\dim E \geq 1$. Is it always true that $E$ is an intersection of $k$ facets of $P$? When $P$ is simple, this is always true. (...
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Techniques for finding the edges of a 4-dimensional convex hull

I am looking for some techniques in general, but let's do it on a (for me relevent) example. I have a set of $n^2$ points $$p_{i,j}:=\frac1{\sqrt 2} \begin{pmatrix} \cos(2\pi i/n) \\ \sin(2\pi i/n) \\...
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Convex Hulls and maximizing volume

I thought of a function (recreational mathematics) and wonder if there is any existing math about it. Google searching did not turn anything up. Let $n\in \mathbb{N}$ be the dimension, and $x\in \...
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Are (quasi-)regular polytopes uniquely determined by their edge graph?

I consider polytopes $P\subset\Bbb R^n,n\ge 2$ of arbitrary dimension (intersection of finitely many halfspaces, therefore convex), which are vertex- and edge-transitive (also called quasi-regular). ...
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63 views

Reformulation of inequality constraint

Version 1 Let $\pi$ be a given permutation of the integers $\{1,\ldots,n\}$ and let $$\mathcal{X}=\{x\in\mathbb{R}_{+}^{n} \mid x_{\pi(1)}\geq\cdots\geq x_{\pi(n)}\}$$ Suppose we seek some $a \in \...
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Polytope in Minkowski sum

Is the following statement true? Suppose that $P$ is a polytope contained in the Minkowski sum $A+B:=\{a+b: a\in A, b\in B\}$ of two convex compact sets $A$ and $B$. Then there exist polytopes $Q\...
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$n$-sphere enclosing the Birkhoff polytope

I am not a mathematician by training, so please feel free to correct my logic or descriptions when necessary: Let $P$ denote a $\textit{permutation matrix}$: $$ \begin{equation} P := \{X \in \{0,1\}^{...
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What are the facets of the Birkhoff Polytope when $n=2$?

I've read in several sources that the number of facets of the Birkhoff polytope $\mathcal{B}(n)$ is $n^2$. Is this supposed to hold when $n=2$? Since $\mathcal{B}(2)$ has dimension $1$, the facets ...
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Lower bound for the number of edges in a Minkowski sum

I was wondering whether the following statement about a Minkowski sum of two polytopes is true: Let $P=P_1 + P_2$ be the Minkowski sum of two polytopes $P_1$ and $P_2$ in $\Bbb R^n$, i.e. $P=\{p_1+...
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Check if convex polytope contained inside a union of convex polytopes?

Suppose I have a polytope $\mathcal P$ and a set of other polytopes $\{\mathcal S_0,\dots,\mathcal S_N\}$. Is there a computationally efficient way to check that $\mathcal P\subseteq\bigcup_{i=0}^{N}\...
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(LP problem) Find an optimal plane that contains a set of points in its positive half-space, such that it does not coincide with another plane

Given is a set of points $S = \{s_1,...,s_N\in\mathbb{R}^n\}$ and a plane $H_1$ with $H_1^+ = \{ x \in \mathbb{R}^n ~\vert~ n_1^Tx - b_1 \geq 0\}$ being its positive half-space, such that $\forall s\...
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Finite overgroups of finite 4D point groups

In 3D, every finite point group is either prismatic, a subgroup of [4, 3] or a subgroup of [5,3]. This Wikipedia diagram, along with the rest of the article, seem to imply that in 4D, every finite ...
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What separates a cyclic polytope from a projective polytope?

I am having trouble understanding the difference between a cyclic polytope and a convex projective polytope as positive geometries. The link https://arxiv.org/pdf/1703.04541.pdf is the source of ...
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What's the collective term for the concepts of truncation, gyroelongation, snubification, etc., for polyhedra?

I recently watched a video on a youtube channel "DONG" led by Michael from Vsauce titled "Making every strictly convex deltahedron" Some concepts like elongation, gyroelongation, expansion, ...
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Combinatorial property of cross polytopes [duplicate]

I apologize in advance; I know that this site is for research-level mathematics, not for elementary learning ground. But I tried to understand the following question on my own but it is still ...
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tangent cone and simplex

Let $X \subset \mathbb R^d$ be a simplex and $x \notin X$ Prove that there exists a unique face $F \subseteq X $ with minimal dimension $\dim(F)=\dim(aff(F))$ such that $x \in K_F$ where $K_F:=\{x+...
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How to compute volume of polytope?

I have a polytope $P$ described as the convex hull of finite points $u_1,..., u_m\in \mathbb R^n$. Is there an easy way to compute the volume of $P$ in $\mathbb R^n$? So far I have it written as $$ ...
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describing a Simplicial complex of a regular polytope through the action of the automorphism group

This is from page 41 of "Abstract regular polytopes" Suppose that we have a group $\Gamma = \langle\sigma_1,\cdots,\sigma_n\rangle$ generated by involutions and denote $\Gamma_J = \langle\sigma_j\ \...
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Intuition: 5 regular polyhedra, 6 regular 4-polytopes, and then 3 regular d-polytopes

This question was posted on MathEducators a few days ago. Users there suggested I post on MSE. I am seeking an intuitive explanation (that would make sense to U.S. college students) why the number ...
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1answer
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Facets of the Birkhoff polytope

I recently became interested in the geometric structure of the Birkhoff polytope since it's connected to a problem that I'm working on. The Wikipedia article states that the Birkhoff polytope has $n^2$...
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simplex summation formula

Let $D$ be a simplex and $F$ be any face of $D$ and $G$ be any other face that contains $F$ Prove $\sum_{F\subseteq G}(-1)^{\dim(G)} =0$ I know that $\sum_{F\subseteq D}(-1)^{\dim(F)} =1$ but how ...
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Number of faces in a simplex

Let $D \subset \mathbb R^d$ be a $d$-simplex with vertices $V=\{ v_1,v_2, \ldots, v_{d+1} \}$ Prove that for every $W \subset V$ $\operatorname{conv}(W)$ is a face of $D$ My definition of a face ...
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abstract chiral polytopes

An abstract chiral polytope $P$ is an abstract polytope such that the automorphism group $\Gamma(P)$ has two flag orbits and such that adjacent flags are always in distinct orbits. Now we are ...
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Obtaining a planar graph from a non-planar graph through vertex addition

Wondering if it is possible to construct a planar graph from any non-planar graph by adding vertices (or splitting edges arbitrarily), and if so, if there is any technique to it. Wondering if there is ...
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How to find vertices, edges and faces of a polyhedron analytically?

I have a polyhedron which is defined by the following system of inequalities: $$ \left\{ \begin{array}{c} x \geq 0 \\ y \geq 0 \\ x + y + z \leq 2\\ x + y + 2z \geq 2 \end{array} \right. $$ The ...
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Meaning of the letter 𝕂

I've seen $\mathbb{K}$ in a few places defining affine function without a definition, wondering what it means. For example: A function $f : \mathbb{K}^m \to \mathbb{K}^n$ is affine if there exists a ...