Convex hull lattices point generated

TL;DR

This paper investigates the structure of convex hull lattices generated from initial point sets, identifying conditions for finiteness and classifying configurations in the plane and higher dimensions.

Contribution

It introduces a notion of point configuration to classify which initial sets produce finite convex hull lattices, including four regular families and one sporadic case.

Findings

Finite convex hull lattices are characterized by specific point configurations.

Four regular families and one sporadic configuration generate all finite lattices.

The classification extends to higher dimensions and various configurations.

Abstract

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.