Numerical Experiments

Definition

Let k(n) denote the minimum number of iterations deeded to make sure that every polygon with n side becomes convex after at most n midpoint iterations.

Triangle

It is clear that:

Quadrilateral

Further, it is not hard to see that

which we may check as follows:

To get an idea as to what is the probablity that a random 4-gon is convex:

So this seems to suggest that a random 4-gon is convex about 30% of the time.

Pentagons

Hexagons