Questions tagged [polyhedra]

Questions related to polyhedra and their properties.

Filter by
Sorted by
Tagged with
0
votes
0answers
19 views

A geometric realization of a 4-polytope with 7 vertices

I am looking for a geometric realization of a 4-polytope with 7 vertices. A list can be found in "An Enumeration of Simplicial 4-Polytopes with 8 vertices" by Grünbaum and V. P. Sreedharan. ...
0
votes
0answers
14 views

Do a polytope $P$ and its polar dual $P^\circ$ have the same centroid?

One natural choice for a center point of a convex polytope $P\subset\Bbb R^d$ is the average of all its vertices $$c(P):=\frac{v_1+\cdots +v_n}{n}.$$ Call $P$ centered if $c(P)=0$ Question: If $P$ is ...
1
vote
1answer
11 views

Number of Deformation Parameters of Simple Polyhedron Equals Number of Edges

Suppose $P$ is a simple convex polyhedron with $n$ faces. Euler's formula and the handshaking lemma tell us that the number of edges $E=3n-6$. By a deformation of $P$, I mean a polyhedron, ...
0
votes
0answers
20 views

Finding the minimum volume of all possible regions made by a concave polyhedron and its convex hull

I'm working on a project creating 3d-printed joint models and I would like to minimize the amount of plastic needed in adding protrusions that would form the perfect makebed. A visual example of the ...
0
votes
1answer
22 views

Prove that this linear relaxation has half-integral extreme points

Given a graph $G=(V,E)$, here is a Linear Relaxation of the edge cover polytope: (1) For each $v \in V, \sum_{e \in \delta(v)} x_e \geq 1.$ (2) For each $e \in E$, $0 \leq x_e \leq 1.$ Here $\delta(S)$...
1
vote
2answers
68 views

How do I find or derive circumspherical radii, surface area, and volume for these 5 non-traditional concave polyhedra?

How do I find or derive circumspherical radii, surface area, and volume for these 5 non-traditional concave polyhedra? As much of a mathematics enthusiast I am, I'm stuck on a huge roadblock in ...
1
vote
1answer
16 views

Finding the Canonical Polyhedron associated with a 3-connected simple graphs.

I am not a professional mathematician but I am a reasonably competent programmer and I am also no stranger mathematics, though I must say that my usual domain is closer to calculus and functions ...
0
votes
0answers
28 views

Is there any new developments on the Barnette's conjecture?

When I searching for interesting math problems. I find there is a graph theory conjecture called the Barnette's conjecture. The statement is: Is every bipartite simple polyhedron Hamiltonian? A early ...
0
votes
1answer
19 views

Prove the solution set of a Linear programming problem is a polyhedron

Problem: Prove the solution set of a Linear programming problem is a polyhedron. I have proved the feasible set of an LPP is a polyhedron (as the constraints are inequations). Now I want to show the ...
2
votes
0answers
14 views

Every convex polyhedron has a stable face

Recently stumbled across the idea of monostatic polytopes, and I was reminded of an old book I'd read that gave a "proof" that every polyhedron has at least one stable face. Since such a ...
1
vote
1answer
67 views

How to understand the reduced cost in simplex method?

I’m reading a note on the simplex method. The author mentions a quantity called “reduced cost”, yet no interpretation of it is provided. Here’s the setup: $c \in \mathbb{R}^n$ is the cost vector, $A \...
14
votes
0answers
111 views

What is the largest volume of a polyhedron whose skeleton has total length 1? Is it the regular triangular prism?

Say that the perimeter of a polyhedron is the sum of its edge lengths. What is the maximum volume of a polyhedron with unit perimeter? A reasonable first guess would be the regular tetrahedron of side ...
10
votes
1answer
144 views

Do there exist uniform triangular prisms with all vertices in $\mathbb Z^3$?

It's quite easy to find a regular square prism (cube) or a regular triangular antiprism (octahedron) with vertices in $\mathbb Z^3$. Take for instance, take the convex hulls $$ \begin{align*} &\...
0
votes
0answers
9 views

Hyperplane definitions from set of half-spaces

A $d$-dimensional polyhedron in $\mathbb{Q}^d$ can be described by a set of (closed) half-spaces $AX \ge B$, where $A \in \mathbb{Q}^{n \times d}$, $X = (x_1, \dots, x_d)$, $B \in \mathbb{Q}^n$. If ...
1
vote
1answer
37 views

Existence of an inner point in a nonempty polyhedron

I was reading some notes on polyhedral analysis and encountered a proof that confused me. The proof is in the book named "Integer and Combinatorial Optimization" by Wolsey and Nemhauser. Let ...
0
votes
0answers
16 views

Criterion for polytope being full-dimensional using bounding hyperplanes

Suppose $P$ is a polytope of $\mathbb{R}^n$ defined by a (bounded) finite family of half-spaces $t_i \cdot x \leq q_i$ with every $t_i \in \mathbb{R}^n$ non-zero and $q \in \mathbb{R}$. Let $V \...
9
votes
0answers
154 views

Are there “close” solutions to Hilbert's third problem?

Hilbert's third problem (or a modern formulation thereof) asks whether two polyhedra $P,Q$ of equal volume are equidecomposable by cutting $P$ into finitely many polyhedral pieces and rearranging them ...
1
vote
2answers
47 views

Having trouble understanding difference between polyhedron and polytope

Hi i´m reading a pdf about linear programming and i´m having trouble understanding the difference between a polyhedron and polytope between those two definitions A polyhedron P ⊆ $R^{n}$ is the set ...
3
votes
0answers
65 views

Finding the best (yet suboptimal) extreme point in an unbounded LP / open polyhedron?

Given the following open example polyhedron: \begin{equation} \begin{aligned} x_1 & & & \ge 1 \\ x_1 & -x_2 & -s & \le 0 \\ x_1 & +x_2 & & \le 3 \\ x_1 ...
1
vote
0answers
35 views

An example of polytope with exponential number of vertices?

I am looking for an example of polytope with exponential number of vertices? (Like $2^n$ vertices) I guess that dual of cyclic polytopes has exponential number of vertices. Are there simpler & ...
0
votes
1answer
44 views

Does this tetrahedron have a name?

Consider the tetrahedron with vertices at: $$(0,0,0)$$ $$(2,-\sqrt2,0)$$ $$(2,\sqrt2,0)$$ $$(2,0,2)$$ This tetrahedron is not regular but does it have any notable properties? It appears to have some ...
0
votes
0answers
5 views

Computation of maximal angle of polyhedral cone with respect to a subspace

Let $\mathbb R^n_{\geq 0} = \{x\in\mathbb R^n:x\geq0\text{ entrywise}\}$ as usual be the non-negative orthant, and suppose that $U$ is a $k$-dimensional subspace of $\mathbb R^n$ that intersects $\...
1
vote
1answer
36 views

Proof of dimension of Polyhedron

Before proving the theorem the book introduces a new definition called implicit equality. What does exactly the below equality mean? $$ A^{=}x=b^{=} $$ What is inside this set? $$ \{x \in \mathbb{R}^...
0
votes
1answer
23 views

Is the following set polyhedral at all of its points?

I'm trying to understand the following definition: Let $x \in A \subset \mathbb R^n$. We say that $A$ is polyhedral at $x$ iff there is a neighborhood $U$ of $x$ and a polyhedron $B$ such that $A \...
1
vote
0answers
30 views

Voronoi Cell of the Dual Lattice?

A lattice is a discrete (we may assume full-rank) subgroup of $\mathbb{R}^n$, often written as the image of $\mathbb{Z}^n$ under a particular matrix $\mathbf{B}\in\mathbb{R}^{n\times n}$ (a basis of ...
0
votes
0answers
33 views

Determine whether $S_1=\{{x\in{R^2}}\mid |x_1|+|x_2|\le 1\},S_2=\{{x\in{R^2}}\mid |x_1|+|x_2|\ge 1\}$ are polyhedral.

Determine whether $$S_1=\{{x\in{R^2}} \mid |x_1|+|x_2|\le 1\} $$ $$S_2=\{{x\in{R^2}} \mid |x_1|+|x_2|\ge 1\} $$ are polyhedral. The definition of polyhedron is that $S =\{{x\in{R^n}} \mid Ax\ge b\} $,...
3
votes
2answers
153 views

How to construct a chamfered dodecahedron with equilateral or coplanar faces?

According to wikipedia, "The chamfered dodecahedron is a convex polyhedron with 80 vertices, 120 edges, and 42 faces: 30 hexagons and 12 pentagons. It is constructed as a chamfer (geometry) (edge-...
2
votes
2answers
60 views

Can each edge of a Wythoffian polytope be flipped by a reflection?

A Wythoffian polytope $P\subset\Bbb R^d$ is an orbit polytope of a finite reflection group, that is, $$P:=\mathrm{Orb}(\Gamma,x):=\mathrm{conv}\{Tx\mid T\in\Gamma\},$$ where $\Gamma$ is a finite ...
1
vote
1answer
69 views

Proving that a $3$-dimensional compact polytope such that every two vertices are adjacent is a tetrahedron

So I have to prove that given a 3-dimensional compact polytope such that every two vertices are adjacent, then it is a tetrahedron. Somehow I'm able to visualise from the fact that every two vertices ...
3
votes
1answer
108 views

Given dihedral angles, find a set of edges

In the paper Space Vectors Forming Rational Angles a special set of tetrahedra is mentioned. "The remaining three are in the R-orbit of the tetrahedron with dihedral angles (π/7, 3π/7, π/3, π/3, ...
2
votes
0answers
31 views

Extreme points of a “boundary” of a polyhedral set are the extreme points of the polyhedral set

Consider a matrix $A_{m\times n}$ and $b_{n\times 1}$. Let the space be $\mathbb R^n$. Let $S:=\{ x\in\mathbb R^n:Ax\le b, x\ge 0\}$ and $T:=\{x\in\mathbb R^n:Ax=b, x\ge 0\}$. I want to show that if $\...
8
votes
1answer
236 views

Given isosceles triangles $\triangle ABC$ and $\triangle DBF$ (all spherical chords), identify the chord $\overline {DF}$ so that $|AD| = |DF| = |FC|$

tl;dr: As shown in the image below, find the chord $\overline {DF}$ so that $|\overline {AD}| = |\overline {DF}|$, and have the answer be in the form of the ratio between $|\overline {AC}|$ and $|\...
0
votes
0answers
24 views

farkas-minkowski theorem specific exmaple

I was reading about farkas-minkowski theorem, which basically said a convex cone is polyhedral iff it's finitely generated. The theorem make sense, but when I played with some example I met the ...
2
votes
0answers
55 views

If $Q$ is a polyhedron with lineality space $L$, then dim$(Q)$ = dim$(Q/L)$ + dim$(L)$, where $Q/L$ is $Q$ with $L$ “modded out”.

The definition of $Q/L$ from my course is a bit vague, but I assume it is as follows: if $Q = L + S$ (where “$+$” is the Minkowski sum), then $Q/L = S$. I realize that if we define $W$ to be the ...
4
votes
1answer
59 views

Polyhedra which can be perfectly split into self-similar pieces

A cube can be perfectly split into smaller equally sized cubes. Similarly, a triangular prism can be perfectly split into smaller equally sized triangular prisms. Is there a name for or list of the ...
1
vote
1answer
35 views

Intersection of a pointed cone and a hyperplane is a polytope?

Let $C = \text{cone}(u_1,\dots,u_m)$ for some $u_1,\dots,u_m \in \mathbb{R}^d \setminus \{\textbf{0}\}$ be a finitely generated pointed cone. Let $H_0 := \{x \in \mathbb{R}^d: \langle a,x \rangle = 0\}...
0
votes
2answers
51 views

Symmetric distribution of points on a sphere

Background: The VSEPR theory provides a useful guide to predicting molecular geometries. One implication of the theory is that the orientation of ligands in three-dimensional space is such that the ...
12
votes
3answers
188 views

Why does a convex polyhedron being vertex-, edge-, and face-transitive imply that it is a Platonic solid?

Suppose that we have a convex polyhedron $P$, such that the symmetry group of $P$ acts transitively on its vertices, edges, and faces (that is, it is isogonal, isotoxal, and isohedral). It then ...
0
votes
0answers
14 views

Can a 3D polyhedron have both v-representation and h-representation(constraints) in its definition?

I plan to use this polyhedra class which i am writing for spatial optimization, ...
2
votes
2answers
54 views

Let $P$ be a polytope of dimension $d$. Then for some $v \notin$ aff($P$), the pyramid $v \ast P$ has dimension $d+1$.

Let $P$ be a polytope of dimension $d$. Then for some $v \notin$ aff($P$), the pyramid $v \ast P$ has dimension $d+1$. I’m looking for a straightforward argument to prove this statement. All the ...
1
vote
3answers
43 views

Calculation of angles in regular polyhedrons

For any regular polyhedron (tetrahedron, cube, octahedron, dodecahedron, icosahedron, ...), how do I calculate an angle from the center of the regular polyhedron to the center of each face? Obviously, ...
0
votes
0answers
37 views

How to I get the polyhedron projection?

I have some problems to understand the polyhedron projection based on Fourier-Motzkin. The projection is defined like $Q := \pi_k(\{x \in R^n:Ax \geq b \}) = \{(x_1,...,x_k)^T \in R : \exists x_{k+1},....
0
votes
0answers
26 views

For a given set $S$, is there a nice way to compute a polyhedral set $P$ such that $S\subseteq P$ (ie, the “polyhedral envelope”)?

I wonder if there's a way to compute the polyhedral envelope of a set. That is, for a given set $S$, find the polyhedral set $P$ such that $S\subseteq P$. This polyhedral set $P$ will not be unique, ...
1
vote
4answers
42 views

Number of Zonotope Edges Parallel to Generator

Suppose we have a zonotope $Z$ that is the Minkowski sum of line segments $U_1+\dots +U_n$. All the edges of $Z$ are parallel to some $U_i$. Is it also true that the number of edges parallel to $U_i$ ...
2
votes
0answers
40 views

minimal description of polyhedron

I have to find minimal description of a polyhedron $P$ described as follows: \begin{align*}x_1 - x_2 &\leq 0\\-x_1 + x_2 &\leq 1\\2x_2 & \leq 5 \\ 4x_1 - x_2 &\leq 8 \\ x_1 + x_2 &...
1
vote
1answer
66 views

How to generate vertex-transitive tilings?

It is trivial to construct vertex-transitive polytopes: choose any finite matrix group $\Gamma\subseteq\mathrm O(\Bbb R^d)$ and some point $v\in\Bbb R^d$. Take the convex hull of the orbit $\Gamma v$, ...
1
vote
1answer
26 views

Problem to understand a solution about polyhedron.

in a tutorial on of our "teachers" written down this exercise and I do not understand the solution: Given is an polyhedron in standard form $K=\{ x \in R^n | Ax =b,x \geq 0 \}$ with matrix $...
4
votes
2answers
331 views

Why isn't the Euler characteristic equal to 2 for this polyhedron?

I've managed to confuse myself on a topic I thought I understood. To set up, if a surface $M$ is a compact Riemannian manifold with boundary $\partial M$, then the Gauss-Bonnet theorem states $$\int_{...
0
votes
0answers
33 views

Intersection of a Convex Polyhedron and a Plane

Let $$X= \{x|a_j^Tx\leq b_j, j=1,...,m \}$$ be a polyhedron, and p is a point inside a polyhedron as shown below. I want to derive an expression for calculating an orthogonal plane from point p which ...

1
2 3 4 5
19