Questions tagged [abstract-polytopes]

An "abstract polytope" is a poset satisfying a list of properties shared by face lattices of polytopes.

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Can you perturb an inscribed polytope so all its edges grow?

Consider the family of convex simplicial polytopes with vertices in the unit sphere of $\mathbb{R}^n$ which have the origin as an interior point. My question is the following: Let $P, P'$ be two non-...
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3 votes
1 answer
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Existence of wild regular abstract polytopes

Is it possible for an edge to connect two non-adjacent vertices of a polygonal face in a regular abstract polytope? Here “adjacent” means that the two vertices are connected by an edge that is a facet ...
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Realizing spherical complexes as convex polytope

A spherical polytope is the intersection of some closed hemispheres which is non-empty and does not contain a pair of antipodal points. A spherical complex is a tiling of the whole (d−1)-dimensional ...
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Realizability of abstract polytopes

What are the conditions that allow an abstract polytope to have a non-skew (but not necessarily convex) realization in Euclidean space?
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Relation between C-groups and reflection groups

Take a set of reflections $\{r_1,\ldots,r_k\}$ of $\mathbb R^n$. Sometimes, the group presentation will turn out to be a C-group – this is where the regular planar polytopes in Euclidean space, ...
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Amalgamation problem for the 11-cell and 57-cell

Are there any finite regular abstract 5-polytopes whose facets are 11-cells and whose vertex figures are 57-cells?
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2 votes
1 answer
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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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29 votes
7 answers
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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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Gluing simplices through a common point/ realisation of a convex simplicial polytope

Given $m≥d+1$ a positive integer, is it always possible to find m d-dimensional simplices $\Delta_i=\mathrm{Conv}(M,V_{i,1},…,V_{i,d})$ such that 1) they all share the common vertex M 2) the ...
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9 votes
3 answers
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Minimal combinatorial data needed to define a polytope [duplicate]

Suppose I give a list of vertices $(v_1, v_2, ..., v_n)$, and a list of "adjacencies", i.e. pairs of vertices $(v_i,v_j)$. Does it exists a unique polytope that has this vertices and realises the ...
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1 vote
1 answer
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Is volume of abstract polytope realisation bounded by length of edges?

Suppose we have abstract polytope $F$ of dimension $d$ (that is the greatest rank facet has rank $n$). Such abstract object may have realisations in d-dimensional Euclidean space as polytopes $A_i(F)$,...
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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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2 votes
1 answer
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Is a finite abstract polytope of Euler characteristic 0 Eulerian?

An abstract polytope is a poset $X$ (here finite), whose elements are called faces, satisfying these 4 conditions: There is a least face and a greatest face. All flags (i.e. maximal chains) have the ...
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Question on abstract polytopes

Let $(P,\le)$ be an abstract $n$-polytope, with $n\ge 2$. Let $H,H',K$ be $m$-faces, with $0\le m \le n-2$. Is it true that there is a sequence $\{H_0=H,H_1,\ldots,H_{r-1},H_r=H' \} \subseteq P$ so ...
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3 votes
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Criteria for abstract polytopes to be convex polytope

Suppose I have an abstract polytope defined by a poset. Are there any methods for determining whether the abstract polytope can be geometrically realized as convex-hull on its set of vertices?
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Build a topological polytope with a specified CW-structure

I am a topologist and not quite familiar with the tools for building a polytope. I would like to build some topological polytope which is an somewhere in between permutohedron and associahedron which ...
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8 votes
1 answer
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Labeled polytopes

In a "problem of a week" contest in my school, I gave the following problem to students: we assign to each vertex of a cube a number $1$ or $-1$. And we associate to each face the product of the ...
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7 votes
1 answer
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What's the best way to test if a sphere is a polytope? (algorithms for the Simplicial Steinitz Problem)

The problem of recognizing whether a simplicial face lattice is polytopal is sometimes called the Steinitz problem. Sturmfels and Bokowski advanced a set of methods in the late 80s to test whether ...
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8 votes
2 answers
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Which finite groups are generated by n involutions?

One of the interesting problems in abstract polytope theory is to determine, for a given finite group, when that group is the automorphism group of a regular abstract polytope. This is equivalent to ...
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