# Questions tagged [polyhedra]

Questions related to polyhedra and their properties.

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0answers
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### 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. ...
0answers
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### 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 ...
1answer
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### 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, ...
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### 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 ...
1answer
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### 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 $\delta(S)$...
2answers
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### 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 ...
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 ...
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 ...
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 ...
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 ...
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### 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 ...
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### 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 $S =\{{x\in{R^n}} \mid Ax\ge b\}$,...
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-...
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 ...
1answer
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### 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 ...
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, ...
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### 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 ...
0answers
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### 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 ...
1answer
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### 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 ...
1answer
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### 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, ...
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 $U_i$ ...
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 &...
1answer
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### 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 $\Gamma v$, ...
1answer
26 views