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When considering classification problems about polytopes, I sometimes has the feeling that one need to talk about certain parametrized families, i.e. moduli space of such polytopes. But neither do I have a concrete example on hand nor do I know how to formulate the definition of such moduli space. Does anyone know the concept along this line?

Besides, I happen to see the following paper by Kapovich:

http://www.math.utah.edu/~kapovich/EPR/plane.pdf

Which at least from its title has some relation to do with this moduli space. But I am not the experts on this field, so can anyone explain to me if this do has the relation with "moduli space" of polytopes with certain properties?

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Perhaps the website en.wikipedia.org/wiki/Mnev%27s_universality_theorem is relevant. –  Richard Stanley Dec 11 '12 at 1:39
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The monograph "Realization spaces of polytopes" by Jürgen Richter-Gebert covers much of what is known about such spaces. It is published by Springer though a PDF used to be available on his website. There are still copies archived on citeseer and other sites. –  j.c. Dec 11 '12 at 1:54
    
@jc Thank you for point it to me! –  Li Yutong Dec 11 '12 at 3:56
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up vote 5 down vote accepted

One thing that is commonly done is to fix an initial polytope $P$, and consider all the polytopes whose fans are coarsenings of $P$'s fan. You can parametrize these by the space of convex piecewise-linear functions on $P$'s fan, to see that the moduli space itself forms a polyhedral cone.

This is no good if you want to be able to turn the faces, just to breathe them in and out.

I don't think the paper you cite will be of much use to you, unless you want a moduli space of polygons. The paper considers a space of polygons in 3-d with fixed edge lengths, but that comes with an involution "flip" whose fixed points are polygons in 2-d. But they're not convex, they self-intersect, etc.

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@Allen Knutson Thank you for your answer! According to my understanding, the moduli space you are talking about is truely the moduli space of corresponding toric varities. Besides, what do you mean by "polygon", is it used as a synonym as polytope? –  Li Yutong Dec 11 '12 at 3:59
    
2-d polytope <-> polygon in R^2. –  Allen Knutson Dec 13 '12 at 1:21
    
Also, there isn't really a moduli space of toric varieties, just of symplectic structures on a fixed toric variety, which is what I gave above. –  Allen Knutson Dec 13 '12 at 1:21
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