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 natural interiority condition, introduced in this form by Knuth. Such systems, P3O (partial 3-order), are a natural generalization of posets, and include the order types of planar point sets. Our main result is that P3O that emerge from points sets, and P3O that emerge from convex sets, do not contain each other. We also extend our orientation to other good covers from convex sets and study the resulting P3O's.

Based on joint work with Ágoston, Damásdi, and Keszegh:

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