论文标题
凸组的方向
Orientation of convex sets
论文作者
论文摘要
我们在成对相交的平面凸组集合并研究其特性的三元组中引入了一个新的定义定义。特别是，我们将其与满足所谓内部条件的三倍的其他方向系统进行了比较：$ \ circlearRowleft（abd）=〜\ circlearRowleflf（bcd）=〜\ circlearrowlefft（cad）= 1 $ $ $ \ circlearrowlefft（abc）我们将这种取向称为P3O（部分3阶），这是Poset的自然概括，具有几个有趣的特殊情况。例如，平面点集的顺序类型（可以具有共线三元）为p3O；我们将p3O表示为p-p3O。 如果我们不允许$ \ circlearRowleft（ABC）= 0 $，我们将获得T3O（总计3阶）。与线性顺序相反，T3O可以具有丰富的结构。 T3O可按点（P-T3O）实现的t3O是一般位置设置的点的顺序类型。尽管与订单类型相似，但p3O和p-t3O可能是由成对相交的凸组的方向引起的，该集合用C-P3O和C-T3O表示，但事实证明与订单类型完全不同：所有C-P3O的家族都没有c-p3o的家族以及C-P3O的所有C-P3O家族中C-T3O的家族中的家族和P3O的家族中的家族和P3O的家族。 最后，如果我们还要求基础凸组的家族满足（4,3）属性，我们研究这些方向的属性。
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.
