Let $P$ be a polygon ($P$ doesn't have to be regular, convex... it's just $n$ distinct points of $\mathbb{R}^2$). We construct the sequence $(P^{(n)})_n$ with $P^{(0)}=P$ and $P^{(n+1)}$ is the polygon obtained the following way : we consider the points cutting all the sides of $P^{(n)}$ in half, $P^{(n+1)}$ is the polygon with these points as vertices. Formally : $$P^{(0)}=(x_1,\dots , x_n)\in \mathbb{C}^n$$ $$ P^{(n+1)}=\left( \dfrac{x_1^{(k)}+x_2^{(k)}}{2},\dots , \dfrac{x_{n-1}^{(k)}+x_n^{(k)}}{2}, \dfrac{x_n^{(k)}+x_1^{(k)}}{2}\right) .$$
I proved that $(P^{(n)})_n$ converges to the barycenter of $P^{(0)}$ (using algebraic arguments).
I really want to learn more around this result but I can't find anything either in books that I know nor in the internet. For example a question that I would like to have an answer to is the speed of the convergence. Does this result have a name?
So if you know a book or a pdf file that studies this problem, it would be very nice of you to let me know. Thank you in advance.
I first saw that construction in the book The Mathematical Experience and it was a wonderful surprise. See also the classic book Circulant Matrices by Philip J. Davis, one of the authors.
See a detailed analysis in this paper: