{
  "id": "2206.01721",
  "version": "v1",
  "published": "2022-06-03T17:57:43.000Z",
  "updated": "2022-06-03T17:57:43.000Z",
  "title": "Orientation of convex sets",
  "authors": [
    "Péter Ágoston",
    "Gábor Damásdi",
    "Balázs Keszegh",
    "Dömötör Pálvölgyi"
  ],
  "categories": [
    "math.CO"
  ],
  "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; we denote a P3O realizable by points as p-P3O. If we do not allow $\\circlearrowleft(ABC)=0$, 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-T3O, is the order type of a point set in general position. Despite these similarities to order types, P3O and p-T3O 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 underlying convex sets satisfies the (4,3) property.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-06-03T17:57:43.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "52C99"
    ],
    "keywords": [
      "orientation",
      "order type",
      "pairwise intersecting planar convex sets",
      "convex sets satisfies",
      "planar point set"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}