Midpoint geodesic polygon / Birkhoff curve shortening

I would advise you to have a look on section 3.7 of the paper of Bowditch, Notes on locally CAT(1) spaces, in Geometric Group Theory, R.Charney, M.Davis, and M.Shapiro eds., de Gruyter (1995). In this section Bowditch says: "The convergence of the Birkhoff process seems to be an open question for Riemanninan 2-manifolds". Also he constructs in this section a 3-dimensional example where the process does not converge.

Here is the abstract of the article http://www.warwick.ac.uk/~masgak/abstracts/lco.html

You may also have a look at the paper

www.md-net.org.uk/preprints/jacobi7.pdf