Description of closure operators in convex geometries of segments on the line

Convex geometry is a closure space (G, phi) with the antiexchange property. A classical result of Edelman and Jamison (1985) claims that every finite convex geometry is a join of several linear subgeometries, and the smallest number of such sub-geometries necessary for representation is called the convex dimension. In our work we find necessary and sufficient conditions on a closure operator phi of convex geometry (G, phi) so that its convex dimension equals 2, equivalently, they are represented by segments on a line. These conditions, for a given convex geometry (G, phi), can be checked in polynomial time in two parameters: the size of the base set vertical bar G vertical bar and the size of the implicational basis of (G, phi).