On the representation of finite convex geometries with convex sets

TL;DR

This paper generalizes the representation of convex geometries using convex sets, showing limitations with ellipses in the plane and providing higher-dimensional representations with ellipsoids.

Contribution

It extends previous work by broadening the types of convex shapes used for representing convex geometries and establishes new limitations and possibilities in different dimensions.

Findings

Convex geometries can be represented by various convex shapes.

Ellipses cannot represent all convex geometries in the plane.

All convex geometries can be represented with ellipsoids in higher dimensions.

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.