moduli space of polytopes - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T16:23:02Z http://mathoverflow.net/feeds/question/116034 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/116034/moduli-space-of-polytopes moduli space of polytopes Li Yutong 2012-12-11T00:44:45Z 2012-12-11T01:40:16Z <p>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?</p> <p>Besides, I happen to see the following paper by Kapovich:</p> <p><a href="http://www.math.utah.edu/~kapovich/EPR/plane.pdf" rel="nofollow">http://www.math.utah.edu/~kapovich/EPR/plane.pdf</a></p> <p>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?</p> http://mathoverflow.net/questions/116034/moduli-space-of-polytopes/116036#116036 Answer by Allen Knutson for moduli space of polytopes Allen Knutson 2012-12-11T01:40:16Z 2012-12-11T01:40:16Z <p>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.</p> <p>This is no good if you want to be able to turn the faces, just to breathe them in and out.</p> <p>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. </p>