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.Comment: 22 pages, 7 figure