Convex hull lattices point generated

The simplest way to generate a lattice of convex sets is to consider an initial set of points and draw segments, triangles, and any convex hull from it, then intersect them to obtain new points, and so forth. The result is an infinite lattice for most sets, while only a few initial sets of points perform a finite lattice. By giving an adequate notion of the configuration of points, we identify which sets in the plane define a finite convex hull lattice: four regular families and one sporadic configuration. We explore configurations in the space and higher dimensions.