Dömötör Pálvölgyi, C-P3O: Orientation of convex sets and other good covers

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. 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, p-P3O, and P3O that emerge from convex sets, C-P3O, 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 Agoston, Damasdi, and Keszegh:
https://arxiv.org/abs/2206.01721
https://arxiv.org/abs/2206.01723