Realization of abstract convex geometries by point configurations

The following classical example of convex geometries shows how they earned their name. Given a set of points X in Euclidean space R, one defines a closure operator on X as follows: for any Y ⊆ X, Y = convex hull(Y ) ∩ X. One easily verifies that such an operator satisfies the anti-exchange axiom. Thus, (X,−) is a convex geometry. Denote by Co(R, X) the closure lattice of this closure space, namely, the lattice of convex sets relative to X. The current work was motivated by the following problem raised in [1]: which lattices can be embedded into Co(R, X) for some n ∈ ω and some finite X ⊆ R? Is this the class of all finite join-semidistributive lattices? On the way to answer the above questions, one can address the associated problem raised in [2], and known as the Edelman− Jamison Problem : Characterize those finite convex geometries that are realizable as Co(R, X). In the current paper we restrict ourselves to the case of n = 2 and point configurations in general position, i.e. where no 3 different points belong to one line. We formulate the hypothesis that a finite convex geometry is realizable by a point configuration on a plane, if two properties of very lucid geometrical nature hold: the so-called splitting rule and the carousel rule. In one of major results of the paper we prove the hypothesis for all point configurations that have at most 2 points inside the n-gon. This extends the description of Co(R, X) for the point configurations X that have one point inside a n-gon, given in [3]. We also confirm the hypothesis for all 6-point configurations on the plane. In another part of our paper we discuss the connection between the EdelmanJamison Problem and the Order Type Problem. Following [4], call t : J [3] → {1,−1} an order type on J , if there is a function f : J → R such that for all (a, b, c) in J [3] one has