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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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Do you mean regular polygons? –  M.B. Oct 31 '12 at 14:53
    
I'm specifically referring to polytopes created from a combination of half spaces. –  Wesley Tansey Oct 31 '12 at 14:57

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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! –  Wesley Tansey Oct 31 '12 at 15:01

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