Convex dimension of locally planar convex geometries (Q1592515)

Let \(X\) be a finite set and \({\mathcal L}= \{L_e\}_{e\in E}\) a finite collection of linear orders of \(X\). The set \({\mathcal L}\) generates a convex geometry \({\mathcal C}\) on \(X\), that is a collection \({\mathcal C}\) of subsets of \(X\) such that \(A\in {\mathcal C}\) if and only if there is no \(x\in X\setminus A\) so that \(\bigcap_{y\in A}\{e\in E:y<x\} =\emptyset\). The convex dimension of the convex geometry \((X,{\mathcal C})\) is defined to be the smallest number of linear orders needed to generate \({\mathcal C}\). The author studies the following problem: Determine the largest number of points that a set \(X\) in the plane may have if no three points of \(X\) are on a line and the convex dimension of \(X\) is \(n\). It is proved that such a number is not greater than \(2^{n-1}\) for a particular class of convex geometries, namely locally planar convex geometries. Examples of locally planar convex geometries which are not realizable in the plane, but are realizable by uniform rank 3 oriented matroids are also given.