The classical example of convex geometry, a closure system with the anti-exchange axiom, is realized by a configuration of points in a $n$-dimensional space and convex hull operator, such example being called an affine convex geometry.
It was proved in 2005 by K. Kashiwabara et al. that every finite convex geometry is a subgeometry in some affine geometry, for some space dimension $n$.
In 2013 G. Czedli introduced an idea to generalize from points to the spheres in affine representations in order to make representation exact and, possibly, to reduce the dimension of representation. In particular, it was speculated that representation could be achieved in dimension 2, i.e., by circles in the plane. In our work we disprove this hypothesis, exhibiting a property that all convex geometries of circles on the plane possess, but which fails in some convex geometries.
The result generated a number of new questions about convex geometries representation, and a remarkably swift response in several follow-up papers written by others, which we will survey in the talk.
Joint work with Madina Bolat, Nazarbayev University.