Let be a collection of vertices in the plane. Define themaximum convex hullof 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.