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 Erdős--Szekeres type obstruction, which answers a question of Czédli 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.

DOI: 10.14232/actasm-017-502-z

AMS Subject Classification (1991): 52A01, 52C45

Keyword(s): finite convex geometries, convex set

Received January 10, 2017, and in revised form April 10, 2017. (Registered under 2/2017.)