# Questions tagged [polyhedra]

Questions related to polyhedra and their properties.

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### The closure of a rational polyhedron is a rational polyhedron.

I'm reading the following proof, where the closure of the rational polyhedron $P$ is denoted $P'$. I don't get the line where $Y$ is defined. This is a set of linear expressions of the form $y^TA$...
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### Example: Projection of a Mixed-Integer Program is not a Mixed-Integer Program

one can show that the projcetion of a Mixed-Ineger Program (MIP) is not a MIP. Our professor told us to consider the following example: \begin{align*} -0.5z_1+z_2 \geq 0\\ 0.5z_1-z_2 \geq 0\\ z_1 \geq ...
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### For a polyhedron $P$ associated with an LP, find a polyhedral description of the integer polyhedron $P_{IP}$ using Gomory-Chvátal cuts.

Consider the polyhedron $P=\left\{\left(x_{1}, x_{2}\right) \in \mathbb{R}_{+}^{2}: x_{1}+4 x_{2} \leq 8, x_{1}+x_{2} \leq 4\right\} .$ Find a polyhedral description of $P_{I P}$ using GC-cuts. What ...
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### Euler's Formula and Existence of Solids

Euler's formula tells us that the number of vertices, edges and faces of a 3D solid have to satisfy the relationship $V+F=E+2$. How about the converse, if I have a triple of numbers that fulfill this ...
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### Gluing of $20$ copies of a specified tetrahedron can be appeared as an icosahedron?

Consider a $2$-dimensional triangle $[ ABC]$. When we glue six copies in $\mathbb{R}^2$ along boundaries, then we have a convex disc whose boundary is $6$-gon. Question : Fix a $3$-dimensional ...
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### What is the vertex figure of the cyclic polytope $C(7,4)$?

The cyclic polytope $C(n,d)\subset\Bbb R^d$ is a certain $d$-dimensional convex polytope on $n$ vertices, in which any two vertices are joined by an edge (a so-called neighborly polytope). I wonder: ...
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### Check whether a polyhedron is empty or not

I have a polyhedron of the form $$P=\{\mathbf{x}\in\mathbb{R}^n\ |\ \mathbf{Ax}\leq\mathbf{b}\}.$$ This polyhedron is an intersection of two other polyhedra, and I want to know if the intersection ...
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### Proof for sum of vertex polygons in polyhedra

What's the rigirous proof for the statement: When the internal angles meeting at a vertex are added, if the sum $<360$ then it's the polyhedra is convex, if the sum $= 360$ it's flat and $>360$ ...
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### Compute H-representation of a polytope after projection onto lower-dimensional space

For a convex polytope described by many equalities and inequalities: $P = \{(x,y): Ax + By <= c, Dx+Ey = f\}$, can we get H-representation of the polytope $Q$ after projecting $P$ to x-space. In ...
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### Find description of convex set by equations

I have given the polyhedron $P$ which is given by the convex hull of the points $v_1 = (0,0,0), v_2 = (1,0,0), v_3 = (0,1,0), v_4 = (0,1,1)$. I want to find the representation of $P$ as the ...
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### Criteria for checking if points are the vertices of a hypercube

I asked a question over at Code Golf Stack Exchange which essentially asked folks to write a program to determine if a collection of $2^n$ points in $\mathbb{Z}^m$ is the vertex set of some $n$-...
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### Comparing interior angles and dihedral angles in tetrahedra

Let $S\subset\Bbb R^3$ be a tetrahedron (not necessarily regular, just the convex hull of any four points in general position). Let $v,e,\sigma\subset S$ be a vertex, an edge and a face of $S$, so ...
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### Why is the euler characteristic of a sphere 2?

When calculating the Euler Characteristic of any regular polyhedron the value is 2. Since a sphere is homoeomorphic to all regular polyhedrons, the sphere ought to have a Euler Characteristic of 2 as ...
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### Rotating a 4d Polytope in a higher-dimensional vector space to make it 'full dimensional'

I have a 4-dimensional polytope centered at the origin of a higher dimensional, $n>4$, space. I have vertices for this polytope in $n$ dimensions, but I would like to write the vertices with 4 ...
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### How to go from a skeleton of a polyhedron to a pretty good drawing?

Suppose you have a graph which represents the skeleton (vertices and edges) of a polyhedron. I know I can easily construct a planar embedding Tutte embedding, but how do you convert this to a figure ...
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### How do I prove that the intersection of two convex polyhedra is a convex polyhedron?

I'm studying about convex geometry, and that is my problem. for more details: A polyhedron is a convex hull of finite points. P is a polyhedron then P := conv{x1,..,xn}
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### Showing that a polyhedron doesn't contain an integral point

I have the following question: I have to decide if a polytope $P = \{x\in\mathbb{R}_{\geq 0}^{\ell}\mid Ax=0\}$ contains an integral point except $x=0$, for hundreds or thousands of different matrices ...
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### Polyhedron with Euler characteristic equal to $3$ [duplicate]

Basically as the title says. I'm not able to find a polyhedron with Euler characteristic equal to $3$.
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### Proving a set has integer extreme points

The hint says to use TU Properties, but I don't know how to express P as a matrix to use the properties Any help is appreciated
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### A difficulty in polyhedron chamfering

I am trying to write a program to chamfer a given polyhedron, but I got stuck, and would like some help. My understanding of the process of chamfering is like so: shrink the given polyhedron's faces ...
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### How does a TDI System change if you add non-negativity constraints

I know that if the system $Ax \leq b$ is TDI the TDI property does not change if you add redundant inequalities. My question is how does a TDI system change if you add non-negativity constraints, i.e. ...
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### Does any regular surface have a sequence of inscribed polyhedra whose surface area arbitrarily exceeds the area of the surface?

The Schwartz lantern is a family of sequences of noble polyhedra inscribed within a cylinder that have the counterintuitive property that, as the number of vertices grows larger, the total surface ...
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### Find the maximum spherical convex polyhedron area given great circles

Consider a unit sphere centered at 0 and $n$ hyper-planes containing the point 0. The intersection between the sphere and the hyper-planes are great circles. These great circles partition the surface ...
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### Chromatic polynomial of the $1$-skeleton of the $24$-cell

I'm interested in computing the chromatic polynomial of the $24$-cell. Trying to compute this in Mathematica in a naïve way (...
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### How can the equation $Ax=b$ represent a polyhedron?

I can understand how inequalities can be used to define a polyhedron, for example, each plane in a 3d setting would be one face and putting all the planes together we would get a closed body with the ...
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### dimension of proper face of a polyhedron

Let $\mathcal{P} = \mathcal{P}(A,b) = \{x \in \mathbb{R}^{n} \mid Ax \leq b\}$ be a polyhedron with $A \in \mathbb{R}^{m \times n}, b \in \mathbb{R}^{m}$. Let $\mathcal{F} \subset \mathcal{P}$ be a ...
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### Every convex polyhedron has only finitely many vertices

Let $A \in \mathbb{R}^{m \times n}$, $b \in \mathbb{R}^{m}$. Consider the Polyhedron $\mathcal{P}$ given by $$\mathcal{P} = \{x \in \mathbb{R}^{n} \mid Ax \leq b\}.$$ Is it true that $\mathcal{P}$...
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### Proving a vector is in a subspace via LP-duality

Let $P = \{x\in\mathbb{R}^n|Wx\leq b\}$ be a polytope with more than one point (strictly) such that $W\in\mathbb{R}^{m\times n}$ and $b\in\mathbb{R}^m$, and let $c\in\mathbb{R}^n$ be a vector such ...
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### How to generate similar polyhedron?

I'm not a matematician and I hope my question is clear enough. I've found some super interesting (at least to me) old images of polyhedron and I want to reproduce them. I found them in this website ...
Let $K$ be a polyhedral cone generated by the rows of a matrix $A\in\mathbb{R}^{p\times n}$, and constrained by the columns of a matrix $B\in\mathbb{R}^{n\times q}$, such that \$K = \text{cone}(A_1^T,\...