We establish the following two main results on order types of points in general position in the plane (realizable simple planar order types, realizable uniform acyclic oriented matroids of rank $3$): (a) The number of extreme points in an $n$-point order type, chosen uniformly at random from all such order types, is on average $4+o(1)$. For labeled order types, this number has average $4- \frac{8}{n^2 - n +2}$ and variance at most $3$. (b) The (labeled) order types read off a set of $n$ points sampled independently from the uniform measure on a convex planar domain, smooth or polygonal, or from a Gaussian distribution are concentrated, i.e. such sampling typically encounters only a vanishingly small fraction of all order types of the given size. Result (a) generalizes to arbitrary dimension $d$ for labeled order types with the average number of extreme points $2d+o(1)$ and constant variance. We also discuss to what extent our methods generalize to the abstract setting of uniform acyclic oriented matroids. Moreover, our methods allow to show the following relative of the Erd\H{o}s-Szekeres theorem: for any fixed $k$, as $n \to \infty$, a proportion $1 - O(1/n)$ of the $n$-point simple order types contain a triangle enclosing a convex $k$-chain over an edge. For the unlabeled case in (a), we prove that for any antipodal, finite subset of the $2$-dimensional sphere, the group of orientation preserving bijections is cyclic, dihedral or one of $A_4$, $S_4$ or $A_5$ (and each case is possible). These are the finite subgroups of $SO(3)$ and our proof follows the lines of their characterization by Felix Klein.

中文翻译：

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随机顺序类型的凸包

我们在平面中一般位置的点的阶类型（可实现的简单平面阶类型，可实现的秩为 $3$ 的均匀无环定向拟阵）上建立以下两个主要结果： (a) $n$-从所有此类订单类型中随机统一选择的点订单类型平均为 $4+o(1)$。对于带标签的订单类型，这个数字的平均值为 $4- \frac{8}{n^2 - n +2}$，方差最多为 $3$。（b）（标记的）顺序类型读取一组 $n$ 点独立于凸平面域上的均匀度量采样，平滑或多边形，或从高斯分布集中，即这种采样通常只遇到一个消失给定大小的所有订单类型的一小部分。结果 (a) 泛化到任意维度 $d$，用于标记订单类型，平均极值点数 $2d+o(1)$ 和恒定方差。我们还讨论了我们的方法在多大程度上推广到统一非循环定向拟阵的抽象设置。此外，我们的方法允许显示 Erd\H{o}s-Szekeres 定理的以下相对：对于任何固定的 $k$，作为 $n \to \infty$，比例 $1 - O(1/n)$ $n$-point 简单订单类型中的一个包含一个三角形，该三角形在边缘上包围一个凸形 $k$-chain。对于 (a) 中未标记的情况，我们证明对于 $2$ 维球体的任何对映有限子集，方向保持双射组是循环的、二面体的或 $A_4$、$S_4$ 或 $A_5$ 之一（并且每种情况都是可能的）。