Representation of convex geometries of convex dimension 3 by spheres

A convex geometry is a closure system satisfying the anti-exchange property. This paper, following the work of Adaricheva and Bolat [1] 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 convex dimension 3 is representable by circles on the plane. We show there are geometries of convex dimension 3 that cannot be represented by spheres in any \(\mathbb{R}^k\) ${\mathbb{R}}^k$ , and this connects to posets not representable by spheres from the paper of Felsner, Fishburn and Trotter [44]. On the positive side, we use the result of Kincses [55] to show that every finite poset is an ellipsoid order.