# moduli space of polytopes

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
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

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.