Finite convex geometries of circles

TL;DR

This paper explores how finite sets of circles arranged in specific patterns form convex geometries, linking geometric configurations with combinatorial structures and establishing representation results for low convex dimension cases.

Contribution

It demonstrates that collinear, concavely arranged circles form convex geometries of dimension at most 2, and all such geometries can be represented by this configuration.

Findings

Convex closure of certain circle arrangements forms convex geometries.

Representation of all convex geometries of dimension ≤ 2 by circle arrangements.

Use of lattice theory results in geometric context.

Abstract

Let F be a finite set of circles in the plane. We point out that the usual convex closure restricted to F yields a convex geometry, that is, a combinatorial structure introduced by P. H Edelman in 1980 under the name "anti-exchange closure system". We prove that if the circles are collinear and they are arranged in a "concave way", then they determine a convex geometry of convex dimension at most 2, and each finite convex geometry of convex dimension at most 2 can be represented this way. The proof uses some recent results from Lattice Theory, and some of the auxiliary statements on lattices or convex geometries could be of separate interest. The paper is concluded with some open problems.