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More percisely:

Question: Is there a (convex) polytope that has a unique realization up to, say, projective transformations?

I suppose I have to assume that it has more than $d+2$ vertices/facets if it is $d$-dimensional, to avoid some trivial cases such as simplices. By realization I mean just any other polytope that has the same combinatorics as the original one, that is, the same faces being incident in the same way.

For example, the hexagon is not such a polytope, as there are many irregular hexagons that are not projective transformations of the regular hexagon.

More generally, such a polytope cannot be simple of simplicial, as in these cases there are generic ways to move vertices or facets to change its shape while keeping the combinatorics.

I am not sure that projective transformations are the most general thing one wants to exclude to make this an interesting question. I am open for other ideas. Here is another similar question:

Question II: Is there a centrally symmetric polytope that has a unique realization up to linear transformations?

This time with $> d+1$ vertices/facets if it is $d$-dimensional.

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    $\begingroup$ One reference is the paper arxiv.org/pdf/1212.5812.pdf by Adiprasito and Ziegler. $\endgroup$
    – Richard Stanley
    Commented Dec 5, 2020 at 0:31
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    $\begingroup$ In 3d, the only polytope like this is the triangular prism. $\endgroup$
    – Yury
    Commented Dec 5, 2020 at 1:45
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    $\begingroup$ Even more is true in view of the Mnev Universality Theorem. $\endgroup$
    – Moishe Kohan
    Commented Dec 5, 2020 at 4:35
  • $\begingroup$ Title of @RichardStanley's reference: Adiprasito and Ziegler - Many projectively unique polytopes. If my name started with an 'A', I'd also have to find a collaborator whose last name started with 'Z'. :-) $\endgroup$
    – LSpice
    Commented Dec 5, 2020 at 16:03
  • $\begingroup$ @MoisheKohan Well, yes and no. Mnev does not really restrict the dimension of the realization space, just says that there are many whose realization space is contractible (or of any homotopy type you want) $\endgroup$
    – Karim Adiprasito
    Commented Jan 10, 2021 at 22:23

1 Answer 1

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What you are after seems to be projectively unique polytopes. And so the paper of Adiprasito and Ziegler may answer your question. Perhaps two papers in the references of that paper will give you many examples, as they contain operations that preserve this kind of uniqueness.

  1. A construction for projectively unique polytopes by Perles, MA and Shephard, GC
  2. Constructions for projectively unique polytopes by McMullen, P.

Regards, Guillermo

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