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Consider an arbitrary polygon $p$, and then take its midpoint polygon $p'$. Repeat this process to create $p''$, $p'''$, etc. Is there always a convex polygon in this series?

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See the MO question, "'Derived' polyhedra and polytopes." See especially the answer by Gjergji Zaimi, who says:

"the limiting shape ... is in fact the affine image of a regular $n$-gon"


          DerivedPolygons3x5


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