I would like to know under what conditions the process
of creating a midpoint piecewise geodesic polygon converges
on a surface $S \subset \mathbb{R}^3$.
$S$ may be assumed smooth, closed, and oriented;
or it may be assumed a Riemannian manifold.
Let $P$ be a closed geodesic polygon
each of whose finite number of edges is a shortest path
connecting its endpoint vertices $v_i$.
Moreover, the edges of $P$ are "short" in a sense specified
below.
$P$ need not be simple, i.e.,
it may intersect and cross itself.
Define the *midpoint polygon* $M(P)$ for $P$
to have vertices $u_i$, each the midpoint of the geodesic
segment $v_i v_{i+1}$ of $P$, where $u_i$ is connected
to $u_{i+1}$ by a shortest path.
I would like the segment $u_i u_{i+1}$ to be the unique
shortest path between those points.
So assume that $|u_i u_{i+1}|$ is smaller than the injectivity radius
of the manifold. (This might require adding new vertices to ensure
this condition holds for the next iteration.)

In the plane, iterating this
*midpoint polygon*
construction is well studied.
On a surface (or Riemannian manifold)
it plays a role in curve shortening,
in particular *Birkhoff curve shortening*, which he
used to prove the existence of a simple closed geodesic [B27].
Subsequently the technique has many uses.
For example,
Christopher Croke used Birkhoff shortening to find
the length of a shortest closed geodesic on a sphere [C88].

The uses I have seen of Birkhoff shortening are on the sphere, not on more general surfaces. Which brings me to the question:

. Does Birkoff shortening, repeatedly applied to a geodesic polygon on a closed surface $S$, always converge (perhaps to a point)? If so, always to a closed geodesic? If not, under what conditions might it not converge?Q

**References.**

[B27] George D. Birkhoff,
*Dynamical Systems*, AMS, 1927. p.135ff.

[C88]
Christopher B. Croke,
"Area and the length of the shortest closed geodesic."
*J. Differential Geom.*, Volume 27, Number 1 (1988), 1-21.