Representation of convex geometries by circles on the plane

Abstract

Convex geometries are closure systems satisfying the anti-exchange axiom. Every finite convex geometry can be embedded into a convex geometry of finitely many points in an n-dimensional space equipped with a convex hull operator, by the result of K. Kashiwabara, M. Nakamura and Y. Okamoto (2005). Allowing circles rather than points, as was suggested by G. Czédli (2014), may presumably reduce the dimension for representation. This paper introduces a property, the Weak 2 × 3-Carousel rule, which is satisfied by all convex geometries of circles on a plane, and we show that it does not hold in all finite convex geometries. This raises a number of representation problems for convex geometries, which may allow us to better understand the properties of Euclidean space related to its dimension.

On the representation of finite convex geometries with convex sets

“…In general this closure will not determine a convex geometry (the critical point is the anti-exchange property). In the planar case we present a sufficient (but not necessary) condition which guarantees the anti-exchange property and in a sense it is more general than that considered in [2]. For a compact convex set K in the plane the line l is a supporting line if K ∩ l = ∅ but K is contained in one of the halfplanes of l. The line l is a common supporting line of the convex sets K and L if it is a supporting line of both K and L and both sets are in the same halfplane of l. Lemma 2.4.…”

“…Czédli proved in [3] that convex geometries of convex dimension 2 may be represented as a set of circles in the plane. Very recently Adaricheva and Bolat [2] found an obstruction for representing any convex geometries with circles. In this section we present an Erdős-Szekeres type obstruction for representing convex geometries with circles or with ellipses (in the case of circles it is different from the obstruction of Adaricheva and Bolat).…”

“…Adaricheva and Bolat [2] found an obstruction for representing general convex geometries with circles in the plane. In Section 4 we present an Erdős-Szekeres type obstruction, which answers a question of Czédli [3] negatively, that is general convex geometries cannot be represented with ellipses.…”

“…This will be obtained by giving an upper bound for the convex dimension of the geometry in terms of the number of common supporting lines of the pairs. Adaricheva and Czédli [2] raised the question whether every finite convex geometry can be represented with balls in R d . In Section 6 we prove that this can be done with ellipsoids.…”

Circles and crossing planar compact convex sets

“…This paper raised the question which finite convex geometries can be represented. Soon afterwards, Adaricheva and Bolat [2] proved that not all finite convex geometries; see also Czédli [13] for an alternative proof. The reason of this result is the Adaricheva-Bolat property, which is a convex combinatorial property that circles have but most convex geometries do not have.…”