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2
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0answers
10 views

Nice embedding of the permutohedron of order $n$ in ${\mathbb R}^{n-1}$

The permutohedron $P_n$ of order $n$ ($n\geqslant 2$) is the convex hull of the points $P_\pi=(\pi(1),\dots,\pi(n))$ where $\pi$ ranges over all permutations of $\{1,2,\dots,n\}$. Obviously, since ...
1
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0answers
17 views

Intersection numbers on blow ups of toric varieties

Suppose we have a smooth, complete toric varietiy $X_{\Sigma}$ of dimension $n$. Let $\sigma_k \in \Sigma(k)$ a smooth $k$-dimensional cone in $\Sigma$ and suppose we make the toric blow up at the ...
2
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0answers
34 views

From a set of vertices, find smallest polytope enclosing another point

Out of a set of vertices $V=\{\vec v_i\in \mathbb R^D\}$, I am constructing a piecewise linear interpolating function $f:\mathbf{conv}(V)\rightarrow R$ as follows: given a point $\vec d\in ...
1
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1answer
17 views

How to find the center of mass (not vertex average) of a convex hull?

I have found results that say that computing the average of vertices of a polytope presented by inequalities is a #$P$-hard problem. However what if we want the true center of mass (determined by ...
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0answers
16 views

What is the definition of a polytope in terms of set theory?

I know that the set definition of a polytope $P^+_f$ associated with a polymatroid function f is: $$ P^+_f = \{ y \in \mathbb{R}^V_+ : y(A) \leq f(A), \forall A \subseteq V\}$$ where $y(A) = \sum_{i ...
1
vote
1answer
60 views

Uniform lattice sample inside a particular convex polytope

[update]: hardmath suggests using tools from linear programming. This looks like a good idea indeed. I can now tell that my feasible set is described by: $Set = \{d \in \mathbb{N}^c, -B.d\leqslant ...
0
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0answers
5 views

How to generate a ploytope with a finite number of simplices

Well to solve this problem I wanted to show that a polytope with r- lineraly independent vertices is the finite union of r-simplices, but I am stuck in that because I dont know how to proceed and ...
1
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1answer
31 views

The number of vertices in a polytope is finite [duplicate]

I want to prove the following: Let $K$ be a convex polytope. Show that $K$ has a finite number of extreme points. I have seen the bound for the cardinality of the set of extreme points: $|E| \leq ...
0
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0answers
25 views

Prove that a convex polytope has a finite number of extreme points

I want to solve this problem: Let $K$ a convex polytope.Show that $K$ has a finite number of extreme points. My attempt: Is clear that the extreme points of a polytope are the vertices, now, these ...
2
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0answers
25 views

3D models of the unfoldings of the hypercube?

There are (apparently) 261 distinct unfoldings of the 4D hypercube, a.k.a., the tesseract, into 3D.1 These unfoldings (or "nets") are analogous to the 11 unfoldings of the 3D cube into the plane.2 ...
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0answers
6 views

Does a totally normalized version of a sub modular function form the same polymatroid?

Consider any sub modular function $f$: $$f(A) + f(B) \geq f(A \cup B) + f(A \cap B)$$ and a totally normalized version of f $\bar{f}(A) = f(A) - m(A)$ where $m(A)$ is a modular. Consider the ...
1
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0answers
13 views

How many top-$k$ orderings can be induced by linear functionals on integer points in a square?

Let $X = ([1,n] \times [1,n]) \cap {\mathbb N}\times {\mathbb N}$ for positive integer $n$. For $k \leq n^2,$ a top-$k$ ordering $(x_{a_1},x_{a_2}, \ldots, x_{a_k})$ induced by a vector $c$ is a ...
8
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3answers
115 views

Exposed point of a compact convex set

I'm trying to show that given a compact convex set $K$ in $R^d$, there must be at least one exposed point (where $v$ is exposed if there exists a hyperplane H such that $H \cap K = \{v\}$ . This is a ...
0
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0answers
27 views

Vertex Figure of Leech Lattice

I'm doing some research on the newton number (kissing #) problem and it would be extremely useful to know the type and number of elements, particularly the facets, of the vertex figure formed from the ...
1
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0answers
19 views

Is there any study on noncrossing partitions for cyclic polytopes?

I am from another area of maths, and has never learnt combinatorics and polytope theory properly before, so please feel free to point out any mistake/misuse of terminology. The starting point is the ...
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1answer
48 views

Sparse & Dense Polynomials

I've been reading up on Bernstein's theorem for an algebraic geometry assignment and I've come across the terms "dense" and "sparse" in relation to the polynomials. However, I have been unable to find ...
4
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0answers
64 views

Edges of a permutohedron

Consider a permutohedron $P_n$ (this is a polytope which is a convex hull of $n!$ points, which are obtained from $(1,2,...,n)$ by all possible permutations of coordinates). I have to prove the ...
2
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1answer
52 views

Hamilton Path Polytope

How can someone prove that the $\{dim(H)=\frac{n(n-1)}{2} uncorrect\ see\ update\}$ where $H$ is the Hamilton path polytope of a complete graph $K_n$, that consists of vertices $x\in\{0,1\}^{\mid ...
0
votes
0answers
48 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 ...
2
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0answers
43 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 ...
0
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2answers
33 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
1
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1answer
39 views

The number of $(d-1)$-faces in a $d$-polytope is at least $(d+1)$

I have to try to prove the following: Let $V = {v_1,...,v_n} \subset \mathbb{R}^d$ be a point configuration affinely spanning $\mathbb{R}$ (i.e., $\operatorname{aff}(V) = \mathbb{R}^d)$. Let $H$ be ...
0
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2answers
38 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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0answers
34 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 $d:\{0, \ldots, N\} \to \mathbb{R}_+$ ...
0
votes
1answer
43 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?
2
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0answers
40 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?
1
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1answer
43 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: ...
2
votes
0answers
18 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 $m^{\alpha}$ ...
1
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0answers
16 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
32 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
43 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.) ...
1
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1answer
27 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 ...
1
vote
1answer
60 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 ...
2
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0answers
37 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 ...
2
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1answer
141 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 ...
1
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1answer
124 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 ...
0
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1answer
60 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 ...
0
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1answer
92 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
35 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
26 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 ...
1
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0answers
49 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
38 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
26 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
21 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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0answers
60 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
14 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
57 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, ...
0
votes
1answer
38 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 $X_i\geq 0$, ...
0
votes
1answer
46 views

Finding volume of convex polyhedron given vertices

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 ...
1
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0answers
41 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 ...