Questions tagged [polytopes]

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51 views

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
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1answer
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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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 \...
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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 ...
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3answers
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? ...
5
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1answer
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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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 ...
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1answer
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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1answer
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 ...
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2answers
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
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1answer
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 $\epsilon$ ...
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1answer
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 ...
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3answers
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 ...
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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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1answer
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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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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1answer
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 ...
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1answer
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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2answers
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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1answer
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,...
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7answers
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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 ...
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1answer
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 ...
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1answer
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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2answers
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 ...
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1answer
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 ...
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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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3answers
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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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 ...
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1answer
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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2answers
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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1answer
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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1answer
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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1answer
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 ...