# Questions tagged [polytopes]

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### 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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### 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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### 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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### 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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### 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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### Set system of supports of vertices of a polytope written in equational form (reference request)

Let $P \subseteq \mathbb{R}^n$ be a polytope defined by $Ax = b, x \geq 0$ (that is, $P$ is in equational form). For any $v \in \mathbb{R}^n$, let $\text{supp}(v) = \{ i : v_i \not = 0\}$. Let $V(P)$ ...
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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### What separates a cyclic polytope from a projective polytope?

I am having trouble understanding the difference between a cyclic polytope and a convex projective polytope as positive geometries. The link https://arxiv.org/pdf/1703.04541.pdf is the source of ...
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### Efficient $H$ representation of matrices with distinct cyclic shift permuted entries

Given points $v_1,\dots,v_n\in\mathbb Z^n$ in codimesion $1$ hyperplane $x_1+\dots+x_n=t$ with $0\leq x_{i}$ and a cyclic shift permutation $\sigma$ where $v_1,\dots,v_n$ when written as columns of ...
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### What does it mean polynomials share Newton polytope?

I have trouble understanding the connection between polynomials and Newton polytopes. I will try to make a short introduction to my problem and hope you will catch on. In the end I will ask questions. ...
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### “Totally transitive” polytopes which are not regular

Is it possible to have an abstract polytope which is vertex-transitive, edge-transitive, face-transitive, etc. (individually transitive on faces of each particular dimension) and yet not flag-...
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### Reference for this fact about perturbed polytopes?

Let $K \subset \mathbb{R}^n$ be a polytope (i.e., an intersection of finitely many halfspaces that has finite volume) and consider $F(K) := \int_K \|x\|^2\, {\rm d}x$, where $\|\cdot\|$ is the ...
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### Approximation of convex hull in high dimension

What are efficient methods (polytime) to compute an approximation of the convex hull in high dimension (say, $30000$) for a given set of points? Edit: I am looking for an algorithm for getting the ...
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### Digital topology, animal problem, 2-sphere and torus

I have the following question relating digital topology, surfaces, particularly $S^2$ and torus. Can a body $B$ constructed with cubes (without cavities or tunnels) and with frontier homeomorphic to ...
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### Nonperiodic points of piecewise-linear homeomorphisms

Suppose $K$ is a compact polytope and $T$ is a piecewise-linear homeomorphism from $K$ to itself. Suppose also that $T$ is not of finite order (that is, for no $n \geq 1$ is it the case that $T^n(x)=x$...
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