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Let $P=P_0$ be a convex polygon of $n$ vertices $v_k$. Let $P_{i+1}$ be the convex polygon obtained by intersecting the halfplanes determined by the lines through every other vertex. Below, $P_0$ is an octagon. $P_1$ is the green-shaded octagon bound by (red) lines through $v_1 v_3$, $v_2 v_4$, $v_3 v_5$, etc. $P_2$ is the tan-shaded octagon bound by green lines.


          OctagonInscribed


For generic $P_0$, does $P_k$ for $k \to \infty$ approach some identifiable limit shape, when rescaled?

I feel certain this process has been studied but I am not remembering where I may have seen it, or what search terms would lead me to it. Thanks for any help!

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up vote 8 down vote accepted

This is the pentagram map, about which much is known. Not by me, or I would give a nicer summary. But the Wikipedia article has plenty of references. In a quick scan of it I don't see a direct answer to your specific question about the limit after rescaling. It seems that usually the polygons are considered up to projective equivalence, and then the pentagram map is integrable.

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Thank you! I have much now to investigate. –  Joseph O'Rourke Mar 29 at 13:06
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