Title: On the representation of finite convex geometries with convex sets,
Abstract: Very recently Richter and Rogers proved that any convex geometry can be
represented by a family of convex polygons in the plane. We shall generalize
their construction and obtain a wide variety of convex shapes for representing
convex geometries. We present an Erdos-Szekeres type obstruction, which answers
a question of Czedli negatively, that is general convex geometries cannot be
represented with ellipses in the plane. Moreover, we shall prove that one
cannot even bound the number of common supporting lines of the pairs of the
representing convex sets. In higher dimensions we prove that all convex
geometries can be represented with ellipsoids.