Two geometric representation theorems for separoids

Separoids capture the combinatorial structure which arises from the separations by hyperplanes of a family of convex sets in some Euclidian space. Furthermore, as we prove in this note, every abstract separoid S can be represented by a family of convex sets in the (|S| − 1)dimensional Euclidian space. The geometric dimension of the separoid is the minimum dimension where it can be represented and the upper bound given here is tight. Separoids have also the notions of combinatorial dimension and general position which are purely combinatorial in nature. In this note we also prove that: a separoid in general position can be represented by a family of points if and only if its geometric and combinatorial dimensions coincide.