Questions tagged [polyhedra]

Questions related to polyhedra and their properties.

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Approximate a convex body with a domain with smooth boundaries?

Let $K$ be a convex body in $\mathbb R^n$ (i.e. a set that is compact and convex). Is it possible to construct a domain with smooth boundaries that contains $K$ and is close enough in the sense of ...
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Naming a polyhedron with 6 faces and 7 vertices

As somebody who has to interpret different types of polyhedra in his research it is sometimes difficult to find a good name for a polyhedron that describes the coordination in a crystal structure. ...
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Polyhedra intersection

If $A$ and $B$ are polyhedra, how do we show that the intersection $A ∩ B$ is a polyhedron. Does the same apply if they are both polytopes, will the intersection $A ∩ B$ also be a polytope? The ...
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On abelian spaces

By Allen Hatcher's book, a space is called abelian if it has trivial action of $\pi_1$ on all homotopy groups $\pi_n$, since when $n=1$ this is the condition that $\pi_1$ be abelian. My quesion is ...
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If a polyhedron's faces and vertex figures are convex, is the polyhedron convex?

Suppose a polyhedron's faces are convex polygons, and its vertex figures are convex spherical polygons (or convex cones, depending on definitions). Must the polyhedron be convex? As a counter-example ...
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Tiling curved 3D space

A flat plane can be tiled, for example, with regular hexagons. If you try to tile a sphere with hexagons, however, it doesn't work--you have to introduce 12 pentagons to complete the tiling. Try to ...
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Confusion in one statement related to the Birkhoff polytope

I know that the set of doubly stochastic matrices $(\Omega(n))$ form a polyhedron. I only know about polyhedron is that it is a $3$-dimensional shape with flat polygonal faces, straight edges and ...
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MATLAB Plot $P = \{x|Ax\leq b\}$

So I am having this trouble with plotting my Polyhedral Set defined as $$P = \{x|Ax\leq b\}$$ The thing is I want to plot it in MATLAB. Let's say my matrix $A$ is $A = [1\ 1 \ ;\ 1\ -1;-5\ 1]$, and ...
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How to find extreme rays of a polyhedron?

I am trying to solve this example, and I don't have the knowledge on how to find the extreme rays of a polyhedron. I will appreciate any suggestions on how to attempt this. Problem $$U=\{u\in\mathbb{...
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Is there widely accepted notion of tangent plane(or similar) for a vertex of a polyhedron?

For a vertex v of a polygon, I think it is reasonable to define its tangent line as ...
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polyhedrons congruent if faces are all congruent + same connection status?

In 2 dimension, a unit square and a unit rhombus(of certain angle) has the same list of edges and connection status. But the unit square and the unit rhombus are not generally congruent. Can we find ...
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Euler characteristic of Kepler–Poinsot polyhedra

A lot of "well-behaved" polyhedra have Euler characteristic 2. By "well-behaved" I mean those that usually come up also in elementary courses e.g. convex polyhedra. Among the 4 ...
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Amenable groups, measure of volumes, and Paradoxical decomposition

I would like to kindly ask for your expertise on the following questions: It is known that the (additive) free commutative group Z^d is a discrete amenable group. Therefore, a finitely additive, left-...
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How do I show that this polyhedron is bounded? [closed]

How do I show that the Set with vectors $(a_1,a_2,a_3,a_4)$, where $a_i$ are real numbers and with $a_1+a_2=1, a_3+a_4=1, a_i\ge0$, is bounded?
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Separation of polyhedra via Farkas' lemma

Let $A\in\mathbb{R}^{m_1\times n}$, $B\in\mathbb{R}^{m_2\times n}$, $a\in\mathbb{R}^{m_1}$, and $b\in\mathbb{R}^{m_2}$. Consider the intersection of an "open" polyhedron and another closed ...
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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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Intersection of convex hulls

I have two polyhedral sets $\mathscr{P}_1, \mathscr{P}_2,$ defined as convex hulls $$\mathscr{P}_1 = \mbox{conv} \left\{ v_{1},\dots, v_{N} \right\}, \qquad \mathscr{P}_2 = \mbox{conv} \left\{ w_{1},\...
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Build an explicit polyhedral representation of $\operatorname{Epi}(f)$

Let $f : \mathbb{R}^n \rightarrow \mathbb{R}$ be defined by $f(x) = \max\limits_{1\leq i < j \leq n} \{ |x_i| + |x_j| \}$. Furthermore, let $\operatorname{Epi}(f) = \{ [x; t] \; : \; f(x)\leq t\}$. ...
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How to build an explicit polyhedral representation of $P_n$

I am having difficulty with the following question. Let $P_n = \{x\in\mathbb{R}^n \; : \; |x_i| \leq 1, i\leq n, \sum_i |x_i| \leq 2 \}$. Build an explicit polyhedral representation of $P_n$ that is ...
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Optimal basic feasible solution in which reduced cost vector has a negative component

I am stuck on the following question. I am mainly having a hard time coming up with an illustrative example. The question is as follows: Consider the linear program $(P): \min\limits_x \{ c^T x \; : \;...
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How to find an LP for the robust counterpart of the uncertain LP problem

I am trying to answer the following problem but with no success: Let $c\in \mathbb{R}^n, b\in \mathbb{R}^m, \bar{A}\in \mathbb{R}^{m\times n}$ and $\Delta\in \mathbb{R}^{m\times n}_+$ be given. ...
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How to prove that $\min_{x\in X} \max_{y\in Y} \{a^T x + b^T y + y^T Ax \} = \max_{y\in Y} \min_{x\in X} \{a^T x + b^T y + y^T Ax \}$

If $X := \left\{x \in \mathbb{R}^n \mid Px \leq p \right\}$ and $Y := \left\{ y \in \mathbb{R}^m \mid Qy \leq q \right\}$ are nonempty and bounded polyhedral sets, and if $a\in \mathbb{R}^n, b\in\...
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System of linear inequalities which is feasible if and only if $P^1 = P^2$

I am completely stuck on the following problem. Let $P^i = \{x \mid A^i x \leq b^i \}$ for $i=1,2$. Derive a system of linear inequalities which is feasible if and only if $P^1 = P^2$. It is easy to ...
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Generalizing regular polyhedra by repelling points on a sphere

Find the arrangement of $N$ identical point charges on a sphere. For uniqueness, assume one charge sits on the north pole and another one lies on a fixed latitude of the sphere Given a circumference ...
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Construct $(P^*)^*$ for a polyhedron

I am trying to solve the following problem: Consider the polyhedron $P = \{x\in \mathbb{R}^3 \mid x_1 + x_2 + x_3 \geq 1, x_j \geq 0 \; j \in [3] \}$. The polar of a set $S$ is defined as $S^* := \{y\...
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Let P be a tetrahedron, and let vector (V, E, F) = (4, 6, 4) represent the number of vertices, edges, and faces of P.

What are the possible (V, E, F) vectors that can be obtained by stacking pyramids over faces of P? Justify this in depth.
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Given a simplex $S$ with vertex set V, show that any the convex hull of any subset $V' \subseteq V$ is also a face of $S$.

Not looking for an outright solution but some back-and-forth. My thoughts so far: A face of any polytope $S$ can be defined as $F = H \cap S,$ where $H$ is some supporting hyperplane of $S$. We know ...
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On the nature of mosaic specified by Schlafli symbol $\{p,q\}$

I was reading a paper on hyperbolic pascal triangle and the author stated that for Schlafli symbol $\{p,q\}$, if $(p-2)\;(q-2)=4$, it determines the Euclidean mosaic. For $(p-2)\;(q-2) <4$ a sphere ...
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Proving a relationship between vertices, faces, and the face lattice.

Let $P$ be a polytope with vertex set $V$ such that $v_1, v_2, ..., v_k \in V.$ Let $F \in \Phi(P)$ be a face of $P$, where $\Phi(P)$ is the face lattice of $P$. I'd like to show that if $\frac{1}{k}(...
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Counting the faces of a polytope and its dual

I have developed the way to count the faces of a $d-$dimensional polytope $P$ and show that the face numbers satisfy the Euler-Poincare relation, and we'll say $C(k,P)$ is the number of $k-$...
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how to find the intersection between a vertex and an edge in a Polyhedra?

So I'm having this kind of problem, I'm doing a functionality in a "CAD like" software where I need to split the following polyhedra in the point A = (5,0,0) and B = (5, -25.84, 0), so my ...
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4 votes
1 answer
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Geometry Question: A property of a convex polyhedron.

I'm trying to interpret a verified solution for the following problem. Show that $v_3+f_3>0$. Here, $v_n$ denotes the number of vertices of a convex polyhedron that meet with $n$ edges, and $f_n$ ...
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Describe the cone generated by vectors $(1,0,0,1), (0,2,3,4), (0,0,3,1), (1,3,2,4), (2,4,6,4)$ by its irredundant linear inequalities.

I'm really not sure on how to tackle this. I have tried considering all matrices comprised of three of the five vectors -- as rows -- and row reducing them, but that hasn't really lead anywhere (so ...
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Consider a convex polyhedra such that Euler's Polyhedra Formula applies...

Consider the proposition of Euler that V - E + F = 2 for polyhedra. It is famously known that Euler's original proof failed to preserve convexity when considering the removal of vertices from ...
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Proving that the free sum of two Polyhedra is a Polyhedra

For definitions and reference, we are working from Guenter Ziegler's Lectures on Polytopes. We define the free sum as follows. Given two polyhedra $P, Q \in \mathbb{R}^d$, the free sum $P \bigoplus Q =...
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Proof regarding Euler's Polyhedra Formula when stacking onto simplicial polyhedra.

Prove that if we either stack a pyramid onto a face of a simplicial polyhedron P, or if we truncate a vertex of a simple P in order to obtain a new polyhedron P-hat, then the formula V −E + F for P ...
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Dividing a sphere as evenly as possible

I want to put $n$ points on a sphere such that they are as far apart as possible. I know how to do this for certain particular values of $n$. For example, $n=2$ would just be 2 points on opposite ...
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A convex polyhedron has $f_n$ faces with $n$ edges and $v_n$ vertices at which $n$ edges meet.

This problem is from "Notes on Geometry" by E. Rees. A convex polyhedron has $f_n$ faces with $n$ edges and $v_n$ vertices at which $n$ edges meet. Show that $\sum nf_n=\sum nv_n$ $\sum f_{...
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Request detailed explanation about Stephen Boyd cvxbook-solutions-manual exercise 2.8(a) expressing a set S in the form S = {x | Ax<=b, Fx = g}

cvxbook-solutions exercise page-5 exercise 2.8(a) 2.8(a) $S = \{y_1a_1 + y_2a_2 | − 1 ≤ y_1 ≤ 1, − 1 ≤ y_2 ≤ 1\}, \text{where }a_1, a_2 ∈ R^n$. The following is the solution mixed with my question. ...
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Is the 2D projection of the maximum volume inscribed ellipsoid still inside the 2D projection of the polyhedron?

I have a set of points for which I computed the convex hull (which is a Polyhedron). I then computed the maximum volume inscribed ellipsoid of it. Because this problem is six-dimensional, plotting for ...
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How to proof the intersections of halfspaces?

I have two sets in R3 given by P1 = {(x, y, z) ∈ R3 : |x| ≤ 1, |y| ≤ 1, |z| ≤ 1} , and P2 = {(x, y, z) ∈ R3 : |x| + |y| + |z| ≤ 1} . How do I show that the halfspaces of each set intersect in order to ...
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Consider the set in $\Bbb{R}^3$ given by: $P1 = \{(x,y,z) \in \Bbb{R}^3 : |x| \le 1\, |y| \le 1\, |z| \le 1\}$. Show that the set is a polyhedron

I am encountering trouble with this mathematical proof question for an optimisation context. Question : Consider the set in $\Bbb{R}^3$ given by: $P1 = \{(x,y,z) \in \Bbb{R}^3 : |x| \le 1\, |y| \le 1\,...
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What are some face and vertex transitive polyhedra that are not edge transitive?

A way I know to define the Platonic solids is that they are the only convex polyhedra that are edge, face, and vertex transitive. If we retain only the vertex transitivity, one finds a new family of ...
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Faces for Polytopes.

We are using the definition for faces, as an intersection with a supporting hyperplane. I have to show, $F$ is a face of polyhedron $Q$, if and only if $F$ is convex and for every $0 < x < 1$, $...
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2 votes
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Self-dual polyhedral and self-complementary graph

I proved that if $ G$ is a self-dual polyhedral graph that is also self-complementary then we have that the order (number of vertices) is $8$ and the size (number of edges) is $14$. Moreover, I proved ...
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Finding all vertices of a polyhedron in $\Bbb{R}^n$ defined by $x_i+x_j\geq1$ and $0\leq x_i\leq 1$ [closed]

Assume that we have a polyhedron in $\mathbb{R}^n$ defined by the following inequalities: $$ \begin{cases} x_i + x_j \geq 1, \forall i, j \in [1 ,n] : i \neq j \\ 0 \leq x_i \leq 1, \forall i \...
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Why this proof doesn't work for non convex polyhedra?

Here is a proof that for any convex polyhedron there exist 2 faces with equal number of edges. I was told that it doesn't work for non convex polyhedra but I can't find out why. Could someone show a ...
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How can we exclude vertices of a compact polyhedron and represent it as a convex hull?

It is known that every compact (closed and bounded) polyhedron $P$ can be written as a convex hull of finitely many points, i.e., $\text{conv}\{x_1, \dots, x_m\}$. Questions How can I make sure that, ...
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Simplicial polyhedral cones

I would appreciate a reply to my question: let' s consider the Euclidean d-dimension real vector space. A polyhedral cone K is said to be simplicial if K is generated by linearly independent vectors (...
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Weighted volume of simplices making up a cube not equal to volume of cube?

I am trying to implement an algorithm to calculate the volume of a polyhedron by dividing it into simplices with their apex as some arbitrary vertex of the polyhedron and then summing up the volume of ...
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