# Representation of convex geometries by convex shapes

## Kira Adaricheva, Hofstra

## April 25, 2017

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.