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It seems, at least in the 2-D case, that all polytopes are going to be convex. Does this hold if the dimensions are increased?

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    I'm specifically referring to polytopes created from a combination of half spaces.2012-10-31

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Regular polygons are convex whereas regular polytopes may not be. Furthermore, the Platonic solids are indeed convex. Maybe you want to read about convex polytopes?

edit: seeing your comment I guess you are really interested in interesections of halfspaces?

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    That was it! Thanks!2012-10-31