Representation of convex geometries of convex dimension 3 by spheres

TL;DR

This paper investigates the representability of convex geometries with convex dimension 3 using spheres and circles, showing limitations and connections to poset representations.

Contribution

It proves that not all convex geometries of cdim=3 can be represented by spheres in any dimension, answering an open question.

Findings

Some convex geometries of cdim=3 cannot be represented by spheres.

Not all convex geometries of cdim=3 are representable by circles on the plane.

Every finite poset can be represented as an ellipsoid order.

Abstract

A convex geometry is a closure system satisfying the anti-exchange property. This paper, following the work of Adaricheva and Bolat (2019) and the Polymath REU (2020), continues to investigate representations of convex geometries with small convex dimension by convex shapes on the plane and in spaces of higher dimension. In particular, we answer in the negative the question raised by Polymath REU (2020): whether every convex geometry of $cdim=3$ is representable by the circles on the plane. We show there are geometries of $cdim=3$ that cannot be represented by spheres in any $\mathbb{R}^k$, and this connects to posets not representable by spheres from the paper of Felsner, Fishburn and Trotter (1999). On the positive side, we use the result of Kincses (2015) to show that every finite poset is an ellipsoid order.