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A zonotope is a polytope whose 2-faces are centrally symmetric.

Question: If a polytope $P$ is centrally symmetric and combinatorially equivalent to a zonotope, is it itself a zonotope?

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There are polytopes whose normal fans are central hyperplane arrangements which are not zonotopes. These are called "belt polytopes." I think they would give a counterexample to your question.

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  • $\begingroup$ Thank you for your answer. A quick google gave not much, but I will look deeper into that. If you have a more concrete example I would be very happy. $\endgroup$
    – M. Winter
    Commented Nov 25, 2020 at 17:50
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    $\begingroup$ Maybe see Exercise 7.4 of Ziegler's "Lectures on Polytopes." $\endgroup$
    – Sam Hopkins
    Commented Nov 25, 2020 at 18:00
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In retrospect, there are easy counterexamples.

For example, the truncatd octahedron (or permutahedron) is a zonotope. However, below (on the right) is a centrally symmetric realization that is not a zonotope (it has non-centrally symmetric faces):

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  • $\begingroup$ Nice! This is one kind of belt polytope that's not a zonotope (an equivalent condition for belt polytopes is that every 2-face has an even number of edges and opposite edges are parallel, see the Ziegler reference I mentioned before). $\endgroup$
    – Sam Hopkins
    Commented Nov 27, 2020 at 15:21

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