Let be a collection of vertices in the plane. Define the maximum convex hull of to be the convex polygon with vertices in such that every point in is either an interior point or a vertex of .
Define to be the maximum convex hull of relabling if necessary. With the remaining points recursively define maximum convex hulls .
The picture should be a series of shrinking polygons each completely contained in the previous. Some of this requires proof but I don't think it is too difficult to show that given any finite collection of points in the plane there exists a unique maximum convex hull.
What I would like to do is connect the convex hulls in some minimum way (minimum number of edges) to ensure the resulting graph contains a complete cycle.
Idea:
connect such that
In other words connect each vertex on each hull to the closest vertex on the next hull.