On the Shape of a Set of Points in the Plane

A generalization of the convex hull of a finite set of points in the plane is introduced and analyzed. This generalization leads to a family of straight-line graphs, called "shapes", which seem to capture the intuitive notion of "fine shape" and "crude shape" of point sets.

Additionally, close relationships with Delaunay triangulations are revealed and, relying on these results, an optimal algorithm that constructs "shapes" is developed.