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4 votes
2 answers
254 views

Does the edge-graph of a centrally symmetric polytope determine which vertices are antipodal?

Given two origin symmetric convex polytopes $P_1$ and $P_2$ (that is $P_i=-P_i$) with the same edge-graph, but potentially of different dimensions and combinatorial types. Let $\phi: G_{P_1}\to G_{P_2}...
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4 votes
0 answers
132 views

Can a polytopal graph be "centrally symmetric" in more than one way?

Let $P,Q$ be two centrally symmetric convex polytopes, potentially of different dimensions and combinatorial type, but with the same edge-graph $G$. The central symmetry of $P$ induces an involutory ...
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2 votes
0 answers
93 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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2 votes
0 answers
138 views

Is the projective symmetry group of a polytope more general than its linear symmetry group?

Give a (convex) polytope $P\subset\Bbb R^d$ (the convex hull of finitely many points). Consider its linear and projective symmetry groups: \begin{align} \DeclareMathOperator{\Aut}{Aut} \...
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4 votes
0 answers
114 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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6 votes
1 answer
212 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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2 votes
1 answer
96 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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3 votes
2 answers
344 views

Is a vertex- and edge-transitive polytope already a uniform polytope?

I want to consider (convex) polytopes $P=\mathrm{conv}\{p_1,...,p_n\}\subset\Bbb R^d$ which are both, vertex- and edge-transitive (or maybe stronger: 1-flag-transitive). Question: Is every such ...
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3 votes
2 answers
172 views

Maximal edge length of symmetric polytopes

For me, a polytope is the convex hull of finitely many points. It is said to be vertex-transitive / edge-transitive if its symmetry group acts transitively on its vertices / edges. Let's call a ...
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6 votes
1 answer
133 views

How rich is the class of vertex- and edge-transitive polytopes?

There are only a few regular polytopes (five in 3D, six in 4D, three in any dimension above). In contrast, the class of uniform polytopes becomes very rich with higher dimensions. The class of vertex-...
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