  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.
