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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 automorphism $\iota_P\in\mathrm{Aut}(G)$ of the edge-graph, and likewise $\iota_Q\in\mathrm{Aut}(G)$ for $Q$.

Question: do we have $\iota_P=\iota_Q$?

It was pointed out by David E Speyer in the comments that this question can be interpreted in two different (yet still very related) ways. I admit, I don't know which version is the "right one" to ask for my application, and a solution to either would be welcome. I suspect that one of the versions is stronger, and this would then be the favorable one. But I also suspect that they are equivalent, in the sense that the answer is No to both.

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  • $\begingroup$ This is not true even when P=Q. A polytope can have two different involutions. $\endgroup$
    – Alexandre Eremenko
    Commented Jan 31, 2023 at 14:22
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    $\begingroup$ @AlexandreEremenko: maybe I'm misunderstanding what you're saying, but to say that $P$ is centrally symmetric means that the linear map $x \mapsto -x$ in the Euclidean space it lives in is maps $P$ to itself, and this map (restricted to $P$) is then called the central symmetry. So I don't understand in what sense a polytope could have two different central symmetries. Maybe you are saying an isometric copy of $P$ has a different one (though that seems hard to imagine...)? $\endgroup$
    – Sam Hopkins
    Commented Jan 31, 2023 at 14:47
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    $\begingroup$ It seems to me that writing $i_Q = i_P$ is confusing, as these are involutions on two different sets. What one wants to write is $i_Q = \sigma \circ i_P \circ \sigma^{-1}$, where $\sigma$ is a bijection from the vertices of $P$ to those of $Q$. Once you note this issue, there are two versions of the question: $\endgroup$
    – David E Speyer
    Commented Jan 31, 2023 at 15:11
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    $\begingroup$ (1) "Given centrally symmetric polytopes $P$ and $Q$ and a bijection $\sigma : v(P) \to v(Q)$ inducing an isomorphism of edge graphs, do we have $i_Q = \sigma \circ i_P \circ \sigma^{-1}$?" (2) "Given centrally symmetric polytopes $P$ and $Q$ with isomorphic edge graphs, does there exist a isomorphism $\sigma$ of the edge graphs with $i_Q = \sigma \circ i_P \circ \sigma^{-1}$?" $\endgroup$
    – David E Speyer
    Commented Jan 31, 2023 at 15:11
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    $\begingroup$ Indeed, I have since learned of a theorem of Grunbaum: A centrally symmetric polytope with $2n$ vertices in $\mathbb{R}^{2d}$ can have at most $(1-2^{-d}) \tfrac{(2n)^2}{2}$ edges. So the graph $K_{2n} \setminus C_{2n}$ can't be achieved in any fixed dimension. See Prop 2.1 for a quick proof arxiv.org/abs/math/0611893 . $\endgroup$
    – David E Speyer
    Commented Feb 2, 2023 at 14:09

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