Questions tagged [polyhedra]

Questions related to polyhedra and their properties.

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28 views

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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20 views

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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2answers
21 views

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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12 views

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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1answer
23 views

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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1answer
13 views

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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1answer
22 views

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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17 views

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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1answer
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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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1answer
32 views

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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58 views

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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1answer
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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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1answer
48 views

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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1answer
10 views

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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30 views

Computing the set of integral points of a convex hull

Assume that we have integral points $x_1, \ldots, x_n \in \lbrace 0, \ldots, l - 1 \rbrace^3$ for some $l \in \mathbb{N}_{> 0}$, that the vertices of the associated convex hull are given by $v_1, \...
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1answer
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Making sense of the definition of Polyhedron

A polyhedron is defined as the solution set of a finite number of linear equalities and inequalities: $$P = \{x | a^T_j x ≤ b_j , j = 1, . . . , m, c^T_j x = d_j , j = 1, . . . , p\}$$ A polyhedron ...
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How can I define a Goldberg polyhedral with nearly-regular hexagons, given a hexagon width and sphere circumference?

I'm trying to create a "hexagonal grid" that covers the planet earth with nearly regular hexagons. I understand that, by including 12 pentagons, the remainder of a sphere can be covered by pseudo-...
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Algorithm for checking if a polyhedron is bounded. [closed]

Given a polyhedron in the form $Ax \leq b$, I need to know a way to determine if the polyhedron is bounded. I need an algorithm for which given a matrix $A$ and a vector $b$ as input, it should ...
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1answer
33 views

To what is $V-E+F$ transformed to when $n>3$?

I guess that convex polyhedra can be well-defined in $\mathbb R^n$ when $n>3$ and that they are well-studied so would like to know to what does the expression $V-E+F$ transforms to when $n>3$ ...
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1answer
78 views

Proof that there exists a 7-sided polyhedron dice

We know that we can build a polyhedron shaped dice with $2n$ faces using two regular pyramids with n-sided bases, but how can we build a fair polyhedron dice with 7 faces ? Can we generalize the ...
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1answer
34 views

integral polyhedron and projection

Let $$P=\{x\in R^m: Ax=b, Bx\leq d, x\geq 0 \}$$ and $$Q=\{(x,y)\in R^{m+n}: Ax=b, Bx+y=d, x\geq 0, y\geq 0 \}$$ be given systems of linear inequalities. Assume $Q$ is an integral polyhedron with ...
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1answer
21 views

non-degenerate vertices adjacent to exactly $n$ vertices

For $A \in \mathbb{R}^{m \times n}$, $b \in \mathbb{R}^{m}$ let $\mathcal{P}$ be a polytop given by $$ \mathcal{P} = \{x \in \mathbb{R}^{n} \mid Ax \leq b\}. $$ Let $x$ be a vertex of $\mathcal{P}\...
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25 views

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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1answer
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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 ...
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Expressions for the dual of a polyhedral cone

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,\...
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Mixed integer linear programs determined by their continuous variables

What is known about mixed integer linear programs of the form $$ \begin{array}{rll} \max & c^T \mathbf{x} + d^T \mathbf{y} & \\ \text{subject to:} & a_i^T \mathbf{x} + b_i^T \mathbf{y} \...
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Proving a candidate for the normal cone of a polyhedra is closed

I am trying to prove the formula for the normal cone of a polyhedron and in the process I want to show the following set is closed so I can use the strict convex separation theorem $\{ A^T \lambda | \...
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Is there terminology for the polyhedron in N-space where every vertex is equidistant to both its adjacent vertices and a circumcentre?

For example, in 2-space, a regular hexagon's edges all have the same magnitude, but also share a magnitude with the radius of the circumcentre (intuitive by sticking a six equilateral triangles ...
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1answer
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Equality of two polyhedral cones

I am reading a proof which is showing that two (polyhedral) cones $K$ and $K'$ are equal. These cones are constructed from matrices $A\in\mathbb{R}^{p\times k}$ and $B\in\mathbb{R}^{k\times q}$, where ...
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Polyhedron $f$-vector/$r$-boundary calculation?

For this example in Sommerville's book on calculating $r$-boundaries, i.e. a boundary of $r$ dimensions, and $N_r$ being defined as the number of $r$ boundaries. Similarly, defining $N_{pq}$ as the ...
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max - min of an LP

I am reading a proof which states without proof that: If a polyhedron $P$ in $\mathbb{R}^n$ has dimension dim($P$) $\geq 1$ then there exists $c\in\mathbb{R}^n$ such that max$\{c^Tx\text{ }|\text{ }x\...
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1answer
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Existence of planar graph whose faces correspond to the faces of a convex polyhedron

Wikipedia states that Steinitz's theorem says: "a given graph $G$ is the graph of a convex three-dimensional polyhedron, if and only if $G$ is planar and $3$-vertex-connected" So, given a convex ...
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1answer
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Duals of Convex Isohedra are Uniform?

All of the convex isohedra are either Platonic, Catalan, bipyramids, or trapezohedra. Their duals are Platonic, Archimedean, prisms, or antiprisms respectively, all of which are uniform, i.e. having ...
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Archimedean Solids Definition: Is Uniformity Necessary?

Is every convex and isogonal polyhedron uniform? That is, are the faces of a convex isogonal polyhedron all regular polygons? If so, then the Archimedean solids could be defined as convex and ...
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Helical “packing” of deltahedra?

I'm looking for help with terminology: I'm a glass artist / PhD researcher aiming to use a modular, geometric approach to making. I've been looking to use repeated polyhedra (with regular polygonal ...
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Packing/tessellating 3 dimensional space fully by polytopes? Give examples.

What is a shortlist of first few simplest (say 5~10 simplest) possible shapes of polyhedra/polytopes (with a minimum number of edges shared) to pack the 3-dimensional flat space (say $\mathbb{R}^3$) ...
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Questions relating to explanation of polyhedra and simplexes

I am told that the definition of a polyhedron is $$P = \{ x \vert a_j^T x \le b_j, j = 1, \dots, m, c_j^T x = d_j, j = 1, \dots, p \}.$$ I am then told that the compact notation is $$P = \{ x \vert ...

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