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?