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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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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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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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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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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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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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 ...
1 vote
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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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