Orientation of Convex Sets

Abstract

We introduce a novel definition of orientation on the triples of a family of pairwise intersecting planar convex sets and study its properties. In particular, we compare it to other systems of orientations on triples that satisfy a so-called interiority condition: $\circlearrowleft(ABD)=\circlearrowleft(BCD)=\circlearrowleft(CAD)=1$ imply $\circlearrowleft(ABC)=1$ for any $A, B, C, D$. We call such an orientation a P3O (partial 3-order), a natural generalization of a poset, that has several interesting special cases. For example, the order type of a planar point set (that can have collinear triples) is a P3O but not every P3O is the order type of some planar point set; a P3O that is realizable by points is called a p-P3O. If the family is non-degenerate with respect to the orientation, i.e., always $\circlearrowleft(ABC)\ne 0$, we obtain a T3O (total 3-order). Contrary to linear orders, a T3O can have a rich structure. A T3O realizable by points, a p-P3O, is the order type of a point set in general position. Despite these similarities to order types, P3O's and T3O's that can arise from the orientation of pairwise intersecting convex sets, denoted by C-P3O and C-T3O, turn out to be quite different from order types: there is no containment relation among the family of all C-P3O's and the family of all p-P3O's, or among the families of C-T3O's and p-T3O's.Finally, we study properties of these orientations if we also require that the family of the underlying convex sets satisfies the (4,3) property, as a first step towards obtaining better $(p,q)$-theorems.