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.

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!