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Hamilton Path Polytope

How can someone prove that the $dim(H)=\frac{n(n-1)}{2}$ where $H$ is the Hamilton path polytope of a complete graph $K_n$, that consists of vertices $x\in\{0,1\}^{\mid E\mid}$
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30 views

Uniform Sampling over Convex Polytope (not full-dimensional)

I want to simulate a uniform distribution on a convex polytope that is not full-dimensional for optimization purposes (to generate random points on the set I want to minimize over). The polytope is ...
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0answers
38 views

What are higher degree cantellations?

Compare higher degree rectifications. What are higher degree cantellations? I didn't find any definition on internet, at least no operational description. You can find constructions by ...
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1answer
23 views

Prove that a convex $d$-polytope has at least $d+1$ facets

This seems trivial but I can't come up with a formal proof. I think there should be a way to do this inductively but I can't figure out how$\ldots$ Any help much appreciated
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19 views

Discrete Geometry (Polytopes)

I have to try to prove the following: Let $V = {v_1,...,v_n} \subset R^d$ be a point configuration affinely spanning $R^d$ (i.e., $aff(V) = R^d)$. Let H be the collection of hyperplanes spanned by ...
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2answers
33 views

Number of Half-Unit Pentachora in a Unit Hexacosichoron

As the title summarizes, I am unable to find out how many 5-cells {3,3,3} (pentachora) with a circumscribed diameter (d) of 1/2 can fit into a 600-cell {3,3,5} (hexacosichoron) with d=1, {3,3,3,6}. ...
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26 views

How to show that the vertices of a convex hull are given by these specific subsets…

We work over $\mathbb{R}^N$. Let $V$ be the corners of the unit cube $[0,1]^N$, or equivalently the set of vectors whose coordinates take values $0$ or $1$. Let $d:\{0, \ldots, N\} \to \mathbb{R}_+$ ...
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1answer
17 views

Proof: Minkowski sum polytope implies A and B polytopes

Suppose $A$ and $B$ are convex sets and their Minkowski sum $A+B$ is a polytope. How do you prove that $A$ and $B$ are polytopes as well?
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33 views

On the Name of the Amplituhedron

Shouldn't the 'amplituhedron' really be called an 'amplitutope' since it's really a polytope and not strictly a polyhedron?
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1answer
8 views

Count and description of vertices of certain faces of the Tridiagonal Birkhoff polytope $\Omega^t_{d+k}$

For $k \ge 1$, $d \ge 2$ and $k \le d - 1$, let ${}^f_d\Omega^t_{d+k} (d;c_k(d - 1))$ be the intersection of $k - 1$ facets of the Tridiagonal Birkhoff polytope $\Omega^t_{d+k}$ with equations: ...
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15 views

dimension of Weber set and selectope (as a operator)

Let $\Omega$ be a finite set of players. For a selector $\alpha:(2^{\Omega}-\{\emptyset\})\rightarrow\Omega$, we define a marginal value operator as a linear operator $m^{\alpha}$ ...
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0answers
13 views

Extra information needed to distinguish combinatorially isomorphic polytopes

The title pretty much sums up my question: what extra information do we need (or what is an example of sufficient information) on top of the face lattice in order to completely characterize a convex ...
2
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0answers
28 views

Discrete Analogue of the Poincaré Conjecture and Simple Connectedness

I apologize if this question is badly worded or obvious, but I have no formal topology background. I have put some effort into trying to find something, but nothing turned up, perhaps due to my lack ...
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0answers
38 views

linear algebra question

Consider $n$ convex polytopes $S_1, \cdots, S_n$ and a set of matrices $W$ such that each matrix $A\in W$, we have that the $i$-th row of $A$ is a member of $S_i$. (In general $W$ is infinite.) ...
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1answer
23 views

What are the facets of the Tridiagonal Birkhoff $d$-polytope $\Omega^t_{d+1}$?

The Birkhoff $d$-polytope $\Omega_{d+1}$ is the convex polytope in $\mathbb{R}^{d+1}$ $\times$ $\mathbb{R}^{d+1}$ of doubly stochastic matrices: • All matrices contained in $\Omega_{d+1}$ have ...
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1answer
39 views

Projection onto hypercube [0,1]^n

Given a positive n-dimensional vector $\mathbf{z}$ (all its elements are positive), my goal is to project it to a unit hyperplane $[0,1]^n$. However my projection is defined with respect to ...
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0answers
33 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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1answer
70 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
77 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
49 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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1answer
48 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
30 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
20 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
45 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
34 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
20 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
15 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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40 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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12 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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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
28 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
40 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
31 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
56 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
40 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 ...
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1answer
91 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
70 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 ...
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1answer
64 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 ...
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2answers
199 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 ...
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1answer
73 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
61 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
84 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 $[l,r]$ ...
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1answer
45 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 ...
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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? ...
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1answer
32 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 ...
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1answer
292 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
90 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
59 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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162 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
25 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 ...