Representation of convex geometries by circles on a plane

Convex geometries are closure systems satisfying the anti-exchange axiom. Every finite convex geometry can be embedded into a convex geometry of finitely many points in an n-dimensional space equipped with a convex hull operator, by the result of K. Kashiwabara, M.Nakamura and Y.Okamoto (2005). Allowing circles rather than points, as was suggested by G.Cz\'edli (2014), may presumably reduce the dimension for representation. This paper introduces a property, the Weak $2\times 3$-Carousel rule, which is satisfied by all convex geometries of circles on a plane, and we show that it does not hold in all finite convex geometries. This raises a number of representation problems for convex geometries, which may allow us to better understand the properties of Euclidean space related to its dimension.