In elementary geometry, a polytope is a geometric object with flat sides, and may exist in any general number of dimensions n as an n-dimensional polytope or n-polytope. (Def: http://en.m.wikipedia.org/wiki/Polytope)

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Interesting cube subdivisions: what is going on here, and what are these polytopes?

I was messing around recently with a unit cube. If you draw vertices on the midpoint of each edge of the cube, then connect those points by new edges, you will form the wireframe of what I figured ...
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Triangulations of combinatorially equivalent polytopes

I am wondering which relation(s) there are between triangulations of combinatorially equivalent polytopes that use no new points: Let $P,Q$ be a $n$-polytopes such that their face lattices are ...
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is this sufficient to define a simplex?

I want to define a simplex based on the following properties A convex polytope All vertexes share an edge with all others For a given vertex $v_i$ the set of all facets that the vertex belongs to ...
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what is special about simplexes?

Is it true that the $d$-simplex is the only convex polytope such that All vertexes share an edge with all others, It is transitive under the action of a group?
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Projection of hyper-cubes via multiple variable elimination

I am not a mathematician but I do use some tools from geometry in robotics. So, I apologize if what I am writing here is not mathematically consistent but I really do need your help. I have a linear ...
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42 views

Polytopes defined by $x_i >=0, Ax = b$ are generic ? (Understanding simplex method)

Consider polytopes in $R^n$ defined by $x_i >= 0, Ax = b$, for $b > 0$. Assume $A$ is of full rank $r$ and $Ax=b$ has solutions. The following properties seems to be correct. I would be ...
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Shelling of a polytope

During line shelling of a convex polytope in d-dimension, it is easy to see that visible facets are shellable. In the same way non visible facets are also shellable. But while combining these two ...
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Property of pairs of disjoint convex polytopes in $\mathbb{R}^n$

I am having trouble proving the following property. Maybe the result already exists, but I could not find anything on the topic. Proposition: Let $P_1$ and $P_2$ be two disjoint polytopes of ...
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Most efficient way of transforming from V-representation to H-representation

What is an efficient way to transform from the v-representation of a convex hull (in terms of vertices) to its h-representation ($Ax \leq b$)?
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15 views

What are the combinatorial types of the facets of a CNTBF?

This question is about Composition-Named Tridiagonal Birkhoff Faces (or CNTBFs), defined in this question. (One goal of this question - to be addressed in a corollary question - is to establish that ...
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14 views

Convex hull of union of polytopes in halfspace representation

Suppose I have two polytopes in $\mathbb{R}^n$ given in H-representation as $P_1: \{x | H_1 x\leq b_1 \}$ $P_2: \{x | H_2 x\leq b_2 \}$ My question is, if it is possible to efficiently (i.e., avoid ...
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9 views

convex polytopes where every vertex pairwise shares a facet

for arbitrary dimension, what are the convex polytopes such that all vertices share a facet of some dimension, which is not the top facet (the entire polytope), with all other vertices? One example is ...
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Affine hull of a linearly constrained set (convex polytope)

For any $A \in \{0,1\}^{m \times n}$ and $r \in \mathbb{R}^m_{>0}$ consider the set $S := \{x \in \mathbb{R}^n_{\geq 0} \ | \ A x = r\}$ of non-negative solutions to the linear system $Ax - r = 0$. ...
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44 views

Rotations of 4-Cubes

I have recently learned the orbit stabilizer theorem, and have encountered unexpected results pertaining to the rotations of a tesseract; I am curious if there is any intuition for this. A $4$-Cube ...
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competing definitions for polytope dual

Along with a polytope P one has the notion of its dual which is officially defined via the inner product. However, in three dimensions at least, the dual is often pictured simply by placing a point ...
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30 views

On a convex polytope

Let $e_i$-s denote the standard unit vectors of $\mathbb{R}^n.$ Denote $$\mathcal{C}_k = \left\{ \sum_{i \in S} \pm e_i \colon S \subseteq \{ 1,2,..., n\} \mbox{ and } |S| \leq k \right\}$$ the set ...
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25 views

Simplicial polytope Dehn-Sommerville Equations

Let's suppose we have a polytope P with $dim(P)=d$ and the h vector $ h(P,x)=\sum\limits_{i = 0}^{n} h_ix^{d-i}$ i have to prove that if $h_{k}=h_{d-k}$(simplicial polytope) then $xh(P,x)=h(P,1/x)$ ...
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The polytope of ways to express an interior point of a polytope as an average of the vertices

If a convex polytope in $\mathbb R^n$ has more than $n+1$ vertices, then each point in its interior is a convex combination of the vertices in more than one way. For each interior point, the set of ...
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18 views

Facets shared by two points on a convex polytope

I have a convex polytope of arbitrary dimension. Let $\mathcal{F} (A)$ denote the set of facets that vertex $A$ belongs to. If two vertices share an edge, is it true that the disunion of $\mathcal{F} ...
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140 views

Projection of hybercube without Fourier-Motzkin Elimination

I am not a mathematician but I do use some tools from geometry in robotics. So, I apologize if what I am writing here is not mathematically consistent but I really do need your help. I have a linear ...
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28 views

“Regular polytopes” in Minkowski spacetime

Is there an analogue of regular "polytopes" (hyperbolic honeycombs?) in the 4D Minkowski spacetime of special relativity, just as there are six regular polytopes in Euclidean 4D space? If so, what is ...
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Delzant theorem for polyhedra

Delzant theorem says that there is a 1-1 correspondence between compact toric symplectic manifolds (modulo equivariant symplectomorphism) and the Delzant polytopes (modulo lattice isomorphism). The ...
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Volume of convex polytope the vertices of which are vertices of the unit hypercube

I have an infinite family $P_{a,b}$ of non-degenerate convex polytopes of dimension $ab$. Each polytope is given explicitely by a list of its vertices and all of these vertices are elements of ...
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30 views

Permutahedron edge

Let's suppose we have the permutahedron (1,2,3,4) and all the possible permutations . I try to find an edge of this polytope . I try to find a hyperplane a1*x1+a2*x2+a3*x3+a4*x4 = b, such that ...
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37 views

What are the exact critera for a CW-complex being a polytope?

Everybody talks about the fact that polyhedra are special CW complexes, and some of the higher dimensional abstract polytopes are too, but nobody tells the exact criteria for a CW complex being a ...
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19 views

Compute the simplex which projects to a polytope

Given a polytope $P=\{\vec{x}\mid A\vec{x}\leq b\}$, it is known that $P$ can be represented as an affine projection of a simplex. The question is, how do I come up with the affine projection and ...
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1answer
37 views

polytope vs simplex

It is stated that any polytope is an affine projection of a simplex. I do not quite understand this: on the plane, a simplex has exactly 3 vertices, but let's consider a polytope $P=\{(x,y)\mid ...
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22 views

Facets of binary polytopes

I have a problem that seems like it should have a slick, elegant solution but I'm having trouble finding one. I'm working with convex polytopes with vertices that are subsets of $\{-1,1\}^n$. When ...
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20 views

Polytope's H-representation

I have the $H$-representation of a polytope in $\Bbb R^3$: \begin{align*} z&\le 1 \\ x+y+z &\le 2\\ x-y+z &\le 2\\ -x+y+z &\le 2\\ -x-y+z &\le 2\\ -z &\le 0 \end{align*} But I ...
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Dimension of a polytope

I try to prove that the dimension of a polytope $P=\{x\in\mathbb R^n | Ax\leq b\}$ is equal to $n-d$ ($d\in\{1,2, \dots, n-1\}$). I already know $d$ linearly independent hyperplanes which includes ...
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Symmetry group of product polytope

The symmetry group of the interval $[-1,1]$ is $\mathbb Z_2$, since it consists only of the identity and the reflection at the origin. Consider now the square $[-1,1]^2$. Obviously, its symmetry ...
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All polygons satisfy the “normal” property.

A fancy explanation is below, but here's an edited simpler explanation because I think the jargon makes the problem seem inaccessible. In reality this problem is super accessible and I'm sure the ...
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Polynomial = 0 on edges of tesseract?

Can there exist a polynomial on $w,x,y,z$ whose value is zero on the edges of a tesseract – or rather, on the projection of those edges to the unit sphere – and nonzero everywhere else on the unit ...
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104 views

Faces of Polyhedra

I am starting to study convex geometry. Let $\mathcal{P}$ be a polyhedron in $\mathbb{R}^n$. I want to show that the face $\mathcal{F}'$ of a face $\mathcal{F}$ of $\mathcal{P}$ is still a face of ...
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Is difference of convex hulls a convex hull of differences?

Let $K \subset \mathbb{R}^n$ be an n-dimensional simplex. Let $e_i$ denote the $i^{th}$ standard unit vector. Define $K-K$ as follows: $$ K-K=\{x-y: x,y \in K\} $$ We know that $K=$ convex hull ...
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32 views

About the interior of a polyhedron

Let us consider a polyhedron in $\mathbb{R}^n$ (in this context it must NOT be bounded) $\mathcal{P} = \{ x: A \cdot x \leq c\}$ for some matrix $A$. Let $\mathcal{I} = \{ x: A \cdot x < c\}$ be a ...
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What is number of faces in a k-ary n-dim cube?

What is the number of $(n-r)$ dim faces for a $k$-ary $n$-dim cube ? Definition of k-ary cube: In a $k$- ary $n$- cube , each node is identified by an $n$-bit base-$k$ address $b_{n − ...
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How can a nonconvex polytope be defined (not by an LMI)?

A convex polytope can be defined by an LMI (linear matrix inequality) or a list of points. How can a nonconvex polytope be defined?
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Finding a vector normal to a face of a polytope

Suppose I have an $n-1$ dimensional facet of an $n$ dimensional polytope, where the facet is expressed by a set of points. For a given $n-2$ face of the facet, how can I find a vector that is both: ...
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What is this specific convex polytope called?

I have the intersection of two spaces: 1) The complete solution to underdetermined matrix equation $Ax=b$ (more rows that columns) 2) The N-Simplex (N is the number of columns/variables in $Ax=b$) ...
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Intuition of the Hessian of the Log Barrier Function

I have a convex polytope defined by $\mathbf{Ax \leq b}$ (row-wise) The log-barrier function is defined as: $$\phi(x) =-\sum{\log(b_i - a_ix)}$$ The Hessian of the log-barrier is : ...
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77 views

Optimizing over intersection of polytopes inside polytope

I have the following situation, as shown in the following diagram: $W=\{w_i\}_{i=1..|W|}$ is a set of vertices (show in diagram in blue) located within a regular simplex and having coordinates $\in ...
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General formula or at least check of existence of some given n-1 polytope cross section obtain by one cut on a given n polytope?

This is a curious observation inspired after beating this game So for these two levels in question, after some prior experience in common cross sections and loads of trial and error, I found it is ...
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Why does the surface area of the hypersphere go to zero as the number of dimensions goes to infinity?

Why does the surface area of the hypersphere go to zero as the number of dimensions goes to infinity? Is there an intuitive reason?
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How do I demonstrate Jordan measurability of a compact convex polytope?

Ex 1.1.9 in Tao's An introduction to measure theory asks us to show that any compact convex polytope in $\mathbb{R}^d$ is Jordan measurable. Is the following an efficient (or even valid) approach to ...
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Analytic Center of Convex Polytope

I have a convex polytope defined by $Ax \leq b$. I want to know how to find the "analytic center" of my convex polytope, because my goal is to sample from the polytope using Monte-Carlo Markov ...
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Smooth Fano Polytopes and Hypersurfaces

This is a rather extended question, so I will try to make it as compact and readable as possible. I am trying to practice with the Macaulay2 software, in particular the polyhedra and ...
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1answer
25 views

Question on relation between convex sets

The book I am referring regarding bell inequalites has the following two things it mentions about two convex sets which I am unable to understand. There are two convex sets $C$ ($C$ is a polytope ) ...
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28 views

How to see transformations on polytopes?

I have a polytope in six dimension with extreme points $(1,0,0,0,0,0)$ $(0,1,0,0,0,0)$ $(0,0,1,0,0,0)$ $(1,1,0,1,0,0)$ $(1,0,1,0,1,0)$ $(0,1,1,0,0,1)$ $(1,1,1,1,1,1)$ $(0,0,0,0,0,0)$ Each of the ...
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How to find the facet inequalities for Bell-Wigner polytope?

The Bell-Wigner polytope has the following extreme points $(1,0,0,0,0,0)$ $(0,1,0,0,0,0)$ $(0,0,1,0,0,0)$ $(1,1,0,1,0,0)$ $(1,0,1,0,1,0)$ $(0,1,1,0,0,1)$ $(1,1,1,1,1,1)$ $(0,0,0,0,0,0)$ I checked ...