# Questions tagged [polytopes]

The polytopes tag has no usage guidance.

84
questions

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### Inverting “codimension matrix” for polytopes?

Let $P$ be an abstract polytope. Let's construct its square matrix $A$ as follows. Its lines and columns are labelled by all faces of $P$, of all dimensions. Put $A(F_1,F_2)=t^m$ if $F_1$ is a subface ...

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28 views

### Preserving the Holomorphicity of a Complex Differentiable Form on a Polytope

I had originally intended the following to be a secondary question to my original post but then realized that it merited a separate entry entirely.
Question: Could it be possible to approximate a ...

**2**

votes

**1**answer

96 views

### Quasiconformal map from a subset of $\mathbb{C}$ to a polytope

Question. Does a quasiconformal map exist between a subset of $\mathbb{C}$ (such
as a unit disc or rectangle) and a polytope?
Here, I take a polytope to be a two-dimensional surface that could be ...

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22 views

### Sectioning facets from Coxeter-Dynkin diagram

Terminology: a sectioning facet $Q$ of a $D$-dimensional polytope $P$ is another polytope which uses a $(D-1)$-dimensional subset of $P$’s vertices and edges and is maximal (it is not a diminishing of ...

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40 views

### An easy way to recognize the edges of an orbit polytope?

Given a finite (orthogonal) matrix group $\Gamma\subseteq\mathrm O(\Bbb R^d)$ and a point $x\in\Bbb R^d$. The corresponding orbit polytope is
$$\mathrm{Orb}(\Gamma,x):=\mathrm{conv}\{Tx\mid T\in \...

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64 views

### Number of vertices in a polyhedron

Consider polytopes
$$A_1[x_{1,1},\dots,x_{1,m_1},z_{1}]'\leq b_1$$
$$A_2[x_{2,1},\dots,x_{2,m_2},z_{2}]'\leq b_2$$
$$B[z_{1},z_{2},z]'\leq c$$
having vertex count $v_1,v_2$ and $v$ respectively.
We ...

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27 views

### Intrinsic definition of a cone in a normal fan

Let $P\subseteq \mathbb{R}^n$ be a full dimensional polytope. Let us assume that $P$ has a facet description with the following inequalities:
$$ \left<x,u_F\right> \geq -a_F$$
where $u_F\in \...

**3**

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45 views

### Classifying/enumerating vertex-transitive simplicial polytopes

I'm interested in understanding the class of simplicial polytopes in $\mathbb R^n$ whose Euclidean isometry group $G$ acts transitively on the vertices. These are examples that I know of:
simplicial ...

**5**

votes

**3**answers

431 views

### Convex lattice polygons with equal area and perimeter

A convex polygon all of whose vertices have integer coordinates is a convex lattice polygon.
Do there exist mutually non-congruent convex lattice polygons which have the same area and same perimeter?
...

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102 views

### A combinatorial characterization of the central inversion of a polytope?

Given a convex full-dimensional polytope $P\subset\Bbb R^d$ (convex hull of finitely many points and not contained in any proper affine subspace) and a symmetry thereof (a linear map $\smash{T\in\...

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43 views

### Classification of pseudoregular polyhedra

In contrast to a regular polyhedron, which has one orbit of flags, I’ve been studying what I call pseudoregular polyhedra, which have two orbits of flags interchanged by conjugation (explained here). ...

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31 views

### Are there uniform compounds of 135 $BC_8$ polytopes?

The Coxeter group $D_8$ is an index-135 subgroup of $E_8$. One of the consequences of this is that the rectified 8-orthoplex, whose coordinates can be given as all permutations and sign changes of $\{...

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42 views

### When does blending occur?

First, a definition. Blending is the operation of taking two or more polytopes, arranging them in a compound so that some elements coincide completely, and removing those coincident pairs.
Last year, ...

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93 views

### Are there any other regular compounds?

Ever since I first read Coxeter’s definition of a regular compound (which seems to be the most commonly used), I didn’t like it on account of it being completely different than for properly connected ...

**5**

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**1**answer

349 views

### What are the “simplest” polytopes with an automorphism group of $\mathrm M_{11} \hspace {-1.25pt} $?

Do any polytopes have an automorphism group of the smallest of the sporadic groups, the Matthieu group $\mathrm M_{11} \hspace {-1pt} $? Indeed, they must exist. What are the simplest such polytopes ...

**5**

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**1**answer

143 views

### Is there a polytope with an essentially unique shape?

More percisely:
Question: Is there a (convex) polytope that has a unique realization up to, say, projective transformations?
I suppose I have to assume that it has more than $d+2$ vertices/facets if ...

**9**

votes

**2**answers

314 views

### Do Bernoulli polynomials know about face vectors?

This question is grounded firmly in numerology. It originates in an observation about some Bernoulli polynomials and the regular icosahedron. Let $F_{k+1}(n)=\sum_{i=1}^n i^k$ be the sum of the ...

**3**

votes

**1**answer

81 views

### Solid angles at points in an orthosimplex

Given a point ${\bf x} = (x_1,x_2,\dots,x_n)$ in the orthosimplex $K = \{(x_1,x_2,\dots,x_n)\ : \ 0 \leq x_1 \leq x_2 \leq \dots \leq x_n \leq 1\}$, what proportion of a ball of radius $\epsilon$ ...

**16**

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**1**answer

606 views

### Can every simple polytope be inscribed in a sphere?

It is known that not every convex polytope (even polyhedron, e.g. this one) can be made inscribed, that is, we cannot always move its vertices so that
all vertices end up on a common sphere, and
the ...

**4**

votes

**3**answers

250 views

### Minimal data required to determine a convex polytope

Let $P\subset \Bbb R^d$ be a convex polytope.
Suppose that I know
its combinatorial type (aka. the face-lattice),
the length $\ell_i$ of each edge, and
the distance $r_i$ of each vertex from the ...

**1**

vote

**0**answers

45 views

### What are the corners of this polytope?

Let $f$ be a non-negative function on the positive integers such that $f(s+t)\geq f(s) + f(t)$ for all $s,t\in\mathbb{Z}^+$. Consider the polytope consisting of all $x\in \mathbb{R}^n$ such that $$\...

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94 views

### Can we combine the symmetries of two polytopes to create a more symmetric polytope?

Suppose that there are two combinatorially equivalent (convex) polytopes $P_1,P_2\subset\Bbb R^d$, that is, both with the same face lattice $\mathcal L$.
The symmetry group $\mathrm{Aut}(P_i)\subset\...

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**1**answer

105 views

### A polytope with congruent facets and an insphere that is not facet-transitive?

Is there a $d$-dimensional convex polytope (convex hull of finitely many points, not contained in a proper subspace), with $d\ge 4$ and the following properties?
All facets are congruent,
it has an ...

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votes

**2**answers

241 views

### Is a polytope that has in-spheres for faces of all dimensions already regular?

Let $P\subset\Bbb R^d$ be a convex polytope (convex hull of finitely many points).
A $k$-in-sphere of $P$ is a sphere centered at the origin to which each $k$-face of $P$ is tangent. So a 0-in-sphere ...

**5**

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**1**answer

132 views

### What is known about the duals of cyclic polytopes?

What is known about the duals of cyclic polytopes, in particular, their facets (or equivalently, the vertex-figures of cyclic polytopes)?
In even dimensions, all facets of the dual are ...

**3**

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**1**answer

94 views

### Reference for “every 5-dimensional polytope has a 3-gonal or 4-gonal face”

It seems to be folklore that every 5-dimensional convex polytope has a 3-gonal or 4-gonal face of dimension two. I was not able to track down a source for that claim.
Alternatively, I would be ...

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244 views

### Is there a 4-polytope without 3-gonal and 4-gonal faces, other than the 120-cell?

The question is in the title:
Question: Is there any 4-dimensional polytope without 3-gonal and 4-gonal faces (of dimension two), other than the 120-cell?
I consider only convex polytopes (convex ...

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71 views

### Polytopes with large dihedral angles

The regular $d$-simplex has dihedral angle $\arccos(1/d)<90^\circ$, and the $d$-cube has dihedral angle exactly $90^\circ$.
The maximal dihedral angle of a prism over a $(d-1)$-simplex is also $90^\...

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56 views

### If all 2-faces of a polytope are $2n$-gons, is the edge-graph bipartite?

This question on MSE has not received a satisfying answer. It can be summarized as follows:
Question: Is is true that the edge-graph of a (convex) polytope is bipartite if and only if all 2-faces ...

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votes

**1**answer

69 views

### Uniquely describing a polytopal complex by prescribing the local structure around its vertices

Let $C$ be a $d$-dimensional (abstract) polytopal complex.
Most of what I say below could be asked in this general setting, but for a start, let's further restrict to simple polytopal spheres, that is,...

**26**

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2k views

### Why are we interested in permutahedra, associahedra, cyclohedra, …?

The following families of polytopes have received a lot of attention:
permutahedra,
associahedra,
cyclohedra,
...
My question is simple: Why?
As I understand, at least the latter two were ...

**1**

vote

**1**answer

121 views

### The Fano plane, stericated 6-simplex, and pentallated 6-simplex

According to this link:
https://en.wikipedia.org/wiki/Stericated_6-simplexes
the stericated 5-simplex "scal" has 105 vertices defined as permutations of (0,0,1,1,1,1,2).
In the course of my team's ...

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55 views

### A matrix that commutes with all symmetries of a vertex-transitive polytope

Let $P\subset\Bbb R^d$ be a vertex-transitive polytope aka. an orbit polytope.
Can there be a matrix $T\in\mathrm{SO}(\Bbb R^d)$ that commutes with all symmetries in $\mathrm{Aut}(P)\subset\mathrm O(\...

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40 views

### Every point in a regular polytope has its own antipodal point or antipodal face

I apologize for using non-common language. When this problem comes to my mind, it seems quite easy but It's not.
Maybe It can be rewritten as,
There exists a unique facet containing the most far ...

**5**

votes

**1**answer

230 views

### Classification of vertex-transitive zonotopes

Zonotopes are convex polytopes that can be defined in several equivalent ways:
parallel projections of cubes,
Minkowsi sums of line segments,
only centrally symmetric faces,
...
I wonder whether ...

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**2**answers

126 views

### 4-polytopes with only one kind of regular facet

Is there a neat way to show (or a reference that already proves) that
the 4-cube is the only convex 4-polytope in which all facets are regular 3-cubes?
the 24-cell is the only convex 4-polytope in ...

**3**

votes

**1**answer

137 views

### Are there any more polytopes whose 2-faces are identical 4-gons?

What are examples for convex polytope $P\subset \Bbb R^d,d\ge 3$ for which holds
$P$ is 2-face transitive (that is, all 2-faces are equivalent under the symmetries of $P$), and
all 2-faces of $P$ are ...

**2**

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**0**answers

125 views

### The topological complexity of polytopes

Polytopes arise naturally when modelling fundamental structures in Biology such as RNA and proteins [1,2]. Recently, it occurred to me that a complexity measure on the topology of polytopes might be ...

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280 views

### Is there any edge- but not vertex-transitive polytope in $d\ge 4$ dimensions?

I consider convex polytopes $P\subset\Bbb R^d$. The polytope is called vertex- resp. edge-transitive, if any vertex resp. edge can be mapped to any other by a symmetry of the polytope.
I am looking ...

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45 views

### Realizing 0/1-polytopes with shortest possible edge lengths

Has there been something written about the following question?
Question: Given a 0/1-polytope, what is the shortest edge lengths with which this polytope can be realized as a 0/1-polytope.
The ...

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**0**answers

48 views

### Matroids which are transitive on minimal basis exchanges

I am looking for matroids in which all minimal basis exchanges look the same, that is, the matroid is transitive on these. Let me explain what I mean by that.
Consider a finite matroid $M$. Define a ...

**6**

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**1**answer

162 views

### Edges of the contact polytope of the Leech lattice

Let $P\subset\Bbb R^{24}$ be the contact polytope of the Leech lattice, that is, $P$ is the convex hull of the 196,560 shortest vectors of $\Lambda_{24}$.
Question: What are the edges of $P$?
Let'...

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**2**answers

95 views

### Products of polytopes and the normals of their facets

I need to compute the normals of the facets of certain polytopes that can be represented as products of polytopes in smaller dimensions.
Searching the bibliography I found that the facets of the ...

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**1**answer

126 views

### Number of bitangents to convex polytopes

Let me state my question prior to defining terms:
Q. Let $P_1$ and $P_2$ be disjoint convex polytopes
in $\mathbb{R}^d$ of $n$ vertices each.
What is the maximum number of distinct bitangent
...

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28 views

### When do projection maps of polyhedra factor?

Given three polyhedra $P$, $Q$, and $R$ in dimensions $a$, $b$, and $c$ respectively, with $a\leq b\leq c$, with the additional condition that: $P=\pi_1(Q)=\pi_2(R)$, where $\pi_1$ and $\pi_2$ are ...

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**1**answer

168 views

### On decomposition of polytopes

Given $m$ number of convex polytopes each with $v$ vertices and described by $h$ hyperplane inequalities in $\mathbb R^t$ are there operations on these polytopes that combine then to give an $v^{\...

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41 views

### Covering a simplex efficiently by efficiently describable polytopes?

Take a standard simplex or cube in $\mathbb R^n$.
Is there a way to cover it with $O(poly(\log n))$ convex polytopes each describable by only $O(poly(\log n))$ half-plane inequalities?
If not what ...

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123 views

### Is there a method to cut a hypercube into disjoint cubes [closed]

Since Borsuk conjecture hold for centrally symmetric convex sets in $\mathbb{R}^n$
so we can cut a hypercube into at least $n+1$ disjoint parts.
Is there a method how can one do that?

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vote

**1**answer

95 views

### Projections of particular simplex yielding boundary of a regular polygon?

What is the maximum $m$ such that there is a simplex with $n$ vertex points in $n-1$ dimensions whose projection yields boundary of a regular $m$-gon on $2D$ plane?

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### Tiling with Horn's polytopes

Let $n\ge2$ be an integer. Consider the hyperplane $H_n$ of ${\mathbb R}^n$ defined by the equation $x_1+\cdots+x_n=0$ and then the sector $P_n\subset H_n$ defined by the inequalities $x_1\le\cdots\le ...