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19 views

Ehrhart polynomial of lattice tetrahedrons in $\Bbb{R}^4$

Let $\lbrace v_1 , v_2, v_3 , v_4 \rbrace \subset \Bbb{Z}^4$ be linearly independent, and denote by $P$ the convex hull of this set. Now, $P$ is a 3-polytope residing in four-dimensional space. What's ...
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0answers
21 views

Faces of the Permutahedron

We define the Permutahedron as the convex hull of all permutations of the vector $(1,2,\dots,n)\in\mathbb R^n$. I am having trouble seeing why the number of $n-k$ dimensional faces of this polytope is ...
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1answer
47 views

Why is the 24-cell (also called Icositetrachoron or Hyperdiamond) the unique regular convex polychoron which has no direct three-dimensional analog?

The 24-cell is self-dual, and is the unique regular convex polychoron which has no direct three-dimensional analog. http://mathworld.wolfram.com/24-Cell.html I don't understand why that is ...
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1answer
37 views

If we inscribed all the 6 regular convex four-dimensional polytopes in a sphere, which one would have the highest volume?

When a dodecahedron is inscribed in a sphere, it occupies more of the sphere's volume (66.49%) than an icosahedron inscribed in the same sphere (60.54%). But what about for the 6 regular convex ...
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0answers
27 views

What are the formulas for the number of vertices, edges, faces, cells, 4-faces, …, $n$-faces, of convex regular polytopes in $n \geq 5$ dimensions?

I know that in dimension $n \geq 5$ there are only 3 kind of convex regular polytopes in each dimension: the $n$-simplex, the $n$-cube and the $n$-orthoplex. I would like to know if there are ...
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0answers
26 views

Number of 3-way contingency tables

Denote by $f(n)$ the number of $2\times 2 \times 2$ contingency tables with all 2-dim marginals equal to $n$. What is $f(n)$? It is easy to see that $f(n)$ is polynomial of degree 4, since it ...
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0answers
17 views

Volume of a polytope with relaxed constraints

Consider a polytope in $n$ dimensions defined by a set of linear constraints: $$P = \{x \in \mathbb{R}^n : Ax \leq b\}$$ where A is some $m \times n$ constraint matrix, and $b = (b_1,\ldots,b_m)$ is ...
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0answers
42 views

building a polytop from polytop and finding its volume

Let $P$ be a symmetric polytope with $M$ vertices. Suppose we subdivide this polytope into $M$ equal parts $A_i, i=1, \ldots, M$ such that each part $A_i$ correspond to one vertex, $v_i, i=1, \ldots, ...
1
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1answer
28 views

What is a vertex-transitive polytope?

What is a vertex-transitive polytope? I can not seem to find a proper definition. The one on Wikipedia is overly vague. It just says that roughly, all vertices are the same. What does this mean? ...
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0answers
19 views

Is the volume of a convex polytope efficiently computable from the vertices

Is there an efficient method to compute the exact volume of a bounded full-dimensional convex polytope, given the coordinates of its vertices (V-representation)?
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1answer
13 views

How to know if polytope in V-representation is full dimensional?

Let $v_i$, for $i=1,...,V$ be the vertices of a convex polytope in $R^n$. That is, each $v_i$ is a point in $n$-dimensions. How can I determine if the polytope is full dimensional?
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38 views

Proof that a Polytope has vertices

As part of my Discrete Optimization course, I have a homework where I have to prove that a Polytope has vertices. I seems to have all tools in hand (definition of a vertex, polytop, convex hull, etc.) ...
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0answers
10 views

Polytopes generated as the convex hull of the orbit of a finite reflection group acting on a given vector

Consider a finite reflection group acting on $\mathbb{R}^N$. Pick a vector $x \in \mathbb{R}^N$ and look at the convex hull of its orbit. This is a polytope. There are some interesting particular ...
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0answers
56 views

max and min values on symmetric polytope

Let $-N\leq t \leq N$. Let $A$ be regular $(N-1)$-dimensional simplex with vertices $(t,0, \ldots, 0)\ldots (0, 0,\ldots, t)$ and $B$ be regular $(N-1)$-dimensional simplex with vertices $(t-N+1,1, ...
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1answer
23 views

condition for having a positive solution to a linear equation.

Let $Y$ be a member of $\mathbb{R}^m$. I need a necessary and sufficient condition on a $n\times m$ binary matrix $A$ for having a solution to the linear equation: $$AX=Y$$ Such that $X_i\geq 0$, ...
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1answer
31 views

Finding volume of convex polyhedron

I am trying to compute the volume of the convex polyhedron with vertices (0,0,0), (1,0,0),(0,2,0),(0,0,3) and (10,10,10). I am supposed to use a triple integral but am struggling with how to set it ...
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0answers
22 views

Generalized permutahedron and random polytopes

The Birkhoff polytope $B_n$ is defined as the convex hull of the set of permutation matrices, which gives us the set of doubly stochastic matrices. A concept which is intimately related is that of the ...
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0answers
36 views

Three theorems for Polyhedra, Polytopes, and Cones

Is there anybody have readable proofs of the following theorem? A polytope (bounded polyhedron) is the convex hull of a finite set of points. A polyhedral cone is generated by a finite set of ...
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0answers
38 views

showing that all convex polehedron graphs are 3-connected

I'm trying to figure out how to show that two nonadjacent vertices in the graph of a convex polyhedron can be disconnected from one another by the removal of at least three vertices. I know what a ...
1
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1answer
89 views

Splitting polytope into equal parts

Let $-N\leq t \leq N$. Let $A$ be regular $(N-1)$-dimensional simplex with vertices $(t,0, \ldots, 0)\ldots (0, 0,\ldots, t)$ and $B$ be regular $(N-1)$-dimensional simplex with vertices $(t-N+1,1, ...
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2answers
61 views

What's the fewest number of sides required to make a polytope in n dimensions?

In 2 dimensions it takes at least 3 sides to make a polygon, the triangle, and in 3 dimensions it takes at least 4 faces (so far as I'm aware) to make a polyhedron. Can this rule be generalized to ...
4
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1answer
47 views

Understanding McMullen's Upper Bound Theorem

I'm a computer science student working on a paper regarding constrained delaunay triangulations. I have been searching for a proof regarding the upper bound for the number of triangles in a ...
5
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2answers
188 views

Vertex-transitive polytope with large facet

Consider a vertex-transitive convex polytope with a facet containing more than the half of all vertices. Does it already have to be a simplex or are there other examples? I am particularly interested ...
4
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1answer
70 views

Existence and unicity of a complete bounded cell in a generic hyperplane arrangement.

Let $n>d$ be integers and $H_1,\ldots,H_n$ be hyperplanes in $\mathbb{R}^d$ in generic position. By generic position I mean that if we change slightly their position, then the configuration does ...
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1answer
52 views

Why are all symmetry groups of regular polytopes are finite Coxeter groups.

Why are all symmetry groups of regular polytopes are finite Coxeter groups?
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2answers
66 views

Polytope parametrization

How one could parametrize a convex polytope? By parametrization I mean something like in multiple integrals, when to integrate over an area one can integrate over one variable in an interval $[l,r]$ ...
1
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1answer
42 views

What do you call a convex polyhedron whose symmetry group is transitive on the facets?

I'd like to know a name/source for the following concept: Let $P$ be a convex polyhedron in $\mathbb{R}^3$. Let $G$ be its symmetry group, and let $F$ be the collection of (top-dimensional) faces of ...
3
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1answer
26 views

Cyclic polytope of dimension 4

I don't quite know how to count the number of $k$-dimensional faces of a $4$-dimensional cyclic polytope (http://en.wikipedia.org/wiki/Cyclic_polytope) without using the standard formula. Any advice? ...
3
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1answer
26 views

Graph of polytope and hyperplane

Suppose that $P$ is a compact and convex polytope in $R^d$ and let $G$ be the graph of $P$ ($V(G)$ are the vertices of $P$ and $E(G)$ are the $1$-dimensional faces - for example polyedral graphs are ...
3
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1answer
219 views

Intersection of hypercube and hyperplane - features of resulting polytope?

Consider a set in $R^n$ defined as the intersection of the unit hypercube $[0,1]^n$ with a hyperplane defined by $\sum x_i = k$. Assume $k \in (0,n)$ so the intersection has positive volume. Can we ...
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1answer
64 views

Linear isoparametrics with dual finite elements

The subject presented here is some content of the Wikipedia page about Platonic solids combined with my own experience on Finite Elements.To start with the latter, there is a standard piece of Finite ...
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1answer
52 views

Is there a lower bound on the number of facets of a full-dimensional convex polytope

As the title say's: Imagine a full-dimensional convex polytope. Is there a lower bound (or even exact formula) for the minimum number of facets the polytope has?
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0answers
156 views

categorical description of the Minkowski sum of polytopes

Consider the category $\textbf{Poly}$ of polytopes, where the objects are convex hulls of finite subsets of $\mathbb{R}^d$ for arbitrary $d \in \mathbb{N}$ and where the morphisms are affine maps ...
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1answer
24 views

Curious about Symmetric graphs of regular simplices

Found this picture on wikipedia: http://en.wikipedia.org/wiki/Simplex#Symmetric_graphs_of_regular_simplices Can anyone explain why some of these graphs contain their (geometric) center, and others ...
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1answer
116 views

3-d representation of $CQHRL_5$

Mathematically-oriented sculptors have created three dimensional representations of higher dimensional polytopes; for example George Hart. The 5-dimensional polytope $CQHRL_5$ is described in this ...
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7 views

Counts of simplex faces of the $CQHRL_D$ polytope family

This question relates to Counts of certain types of faces of the $CQHRL_D$ polytope family, where the polytope $CQHRL_D$ is described. How many simplex faces of each dimension are in $CQHRL_D$?
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1answer
59 views

Counts of certain types of faces of the $CQHRL_d$ polytope family

This is essentially a counting question, with motivation being supplied by a particular polytope family called crenellated quad hyper-rope loops of dimension $d$ ($CQHRL_d$) ($d$ $\ge$ $5$) and ...
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0answers
106 views

Polytopes inside polytopes

I would like to share some facts about polytopes and how to check whether a polytope is a subset of some other polytope. I will provide some methods and I would like to know whether there exist other ...
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0answers
62 views

Area of a 2D convex polytope made of halfspaces

For a computer program I am attempting to solve the area of a convex polytope defined by a finite number of halfspaces. I understand that this forms a polygon and given the vertices of a polygon I am ...
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0answers
41 views

Confusion related to k neighborly polytope

I was reading this paper related to neighborly polytope where they mentioned: Consider a $d \times n$ matrix $A$, with $d < n$. The problem of solving for $x$ in $y = Ax$ is underdetermined, ...
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0answers
61 views

Proof that the general polytope is measurable

So I have proved that a compact, convex polytope is Jordan measurable at the heed of Terrance Tao's "Introduction to measure theory" Exercise 1.1.9. My current question is how to expand this to the ...
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0answers
23 views

Given only dimensions for N vectors, is it possible to construct a polytope?

We are given a set of magnitude of vectors. These vectors are free to move and rotate in space. How can we verify that whether this set of given vectors form a polytope?
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2answers
64 views

Need a way of computing the vertices of intersection of two simplices

I have two simplices $\Delta_1, \Delta_2$ defined as: The first simplex, $\Delta_1$, is the set of points defined as follows: $$\Delta_1 = \left\{\sum\theta_iu_i, \theta_i >= 0, ...
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1answer
33 views

Polytopes characterization in $\mathbb R^n$

Given $P = \{x \in\mathbb R^n \mid a_1x_1 + \ldots + a_nx_n = \text{constant}\}$, $(a_1, \ldots , a_n) \ne 0$. Can $P$ be a polytope? I think that with $N = 1$, $P$ is a point. Can a point in ...
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1answer
103 views

Strong equidistribution of points on the n-sphere

The vertices of a Platonic solid are equally distributed on its circumscribing sphere in a very strong sense: each of them has the same number of nearest neighbours and all distances between nearest ...
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2answers
56 views

Very symmetric convex polytope

Let $C_n$ be the convex polytope in ${\mathbb R}^n$ defined by the inequalities (in $n$ variables $x_1,x_2, \ldots ,x_n$) : $$ x_i \geq 0, x_i+x_j \leq 1 $$ (for any indices $i<j$). Denote by ...
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0answers
47 views

Convex cone as sum of simplices?

In 3D a pyramid with a square base can be decomposed into the sum of two tetrahedra, i.e. two 3-simplices. I am dealing with a homogeneous N-dimensional system of inequalities and my solution is a ...
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1answer
165 views

Groups acting on polytopes

I am currently reading the paper "Polytopal Resolutions for Finite Groups" [1] by Graham Ellis, James Harris and Emil Skoeldberg and have a question regarding an early remark of theirs. Their basic ...
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1answer
108 views

Is the polar set of convex Polytope also Polytope

Let $P$ be a convex polytope. How can I prove that the polar set of $P$ (lets call it $P^*$) is polytope? where $P^*=\{x\in\mathcal R^n:\forall v\in P, |\langle x,v\rangle|\le1\}$ . $Thanks$