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I'm going by the Wikipedia definition for a 4-polytope. Do by definition, cells of 4-polytopes have to be a convex polyhedra? If not, then are there polyhedra with non-convex faces? If yes, is it the ...

One of the most useful tools in the study of convex polytopes is to move from polytopes (through their fans) to toric varieties and see how properties of the associated toric variety reflects back on ...