Two geometric representation theorems for separoids

Summary {\it 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 <InlineEquation ID=IE"1"><EquationSource Format="TEX"><!CDATA<InlineEquation ID=IE"2"><EquationSource Format="TEX"><!CDATA<InlineEquation ID=IE"3"><EquationSource Format="TEX"><!CDATA$]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>S$ can be represented by a family of convex sets in the $(|S|-1)$-dimensional Euclidian space. The {\it 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 {\it combinatorial dimension\/} and {\it general position\/} which are purely combinatorial in nature. In this note we also prove that: {\it a separoid in general position can be represented by a family of points if and only if its geometric and combinatorial dimensions coincide\/}.