Finite convex geometries of circles

Abstract

Let F be a finite set of circles in the plane. The usual convex closure restricted to F yields a convex geometry, which 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 concludes with some open problems. © 2014 Elsevier B.V. All rights reserved.