Sylvester问题的细化：组合类型的概率
A refinement of the Sylvester problem: Probabilities of combinatorial types
AI总结：
本文研究了Sylvester问题的细化版本，计算了随机点的凸包具有特定组合类型的概率，考虑了正态分布、Beta分布和随机游走等不同情况。
摘要：
设X₁,…,X_{d+2}是R^d中的随机点。经典的Sylvester问题要求确定这些点的凸包P：=[X₁,…,X_{d+2}]为单纯形的概率。在本文中，我们研究了一个细化的版本，要求确定P具有给定的组合类型的可能性。已知有⌊d/2⌋+1种可能的单纯形d维多面体的组合类型，最多有d+2个顶点。这些类型分别表示为T₀^d, T₁^d,…, T_{⌊d/2⌋}^d，其中T₀^d是一个有d+1个顶点的单纯形，而其余类型恰好有d+2个顶点。我们的目标是计算概率p_{d,m}：=P[P为类型T_m^d]，其中m∈{0,1,…,⌊d/2⌋}。经典的Sylvester问题对应于m=0的情况。我们将在以下情况下计算p_{d,m}：(a)X₁,…,X_{d+2}是独立同分布的正态分布；(b)X₁,…,X_{d+2}服从d维的Beta或Beta分布，其中包括在球或球面上的均匀分布作为特殊情况；(c)X₁,…,X_{d+2}形成具有可交换增量的随机游走。作为案例(a)的副产品，我们恢复了一个最近的Youden's demon问题的解决方案，该问题要求确定在一维独立同分布的正态样本ξ₁,…,ξ_n中，经验均值(ξ₁+…+ξ_n)/n位于第k个和第k+1个顺序统计量之间的概率。我们还考虑了精确定义(Sylvester问题)的锥形(或球形)版本，并在几个特殊情况下解决了它。
英文摘要：
Let $X_1,\ldots, X_{d+2}$ be random points in $\mathbb R^d$. The classical Sylvester problem asks to determine the probability that the convex hull of these points, denoted by $P:= [X_1,\ldots, X_{d+2}]$, is a simplex. In the present paper, we study a refined version of this problem which asks to determine the probability that $P$ has a given combinatorial type. It is known that there are $\lfloor d/2\rfloor+1$ possible combinatorial types of simplicial $d$-dimensional polytopes with at most $d+2$ vertices. These types are denoted by $T_0^d, T_1^d, \ldots, T_{\lfloor d/2 \rfloor}^d$, where $T_0^d$ is a simplex with $d+1$ vertices, while the remaining types have exactly $d+2$ vertices. Our aim is thus to compute the probability $$ p_{d,m} := \mathbb P[P \text{ is of type } T_{m}^d], \qquad m\in \{0,1,\ldots, \lfloor d/2 \rfloor\}. $$ The classical Sylvester problem corresponds to the case $m=0$. We shall compute $p_{d,m}$ for all $m$ in the following cases: (a) $X_1,\ldots, X_{d+2}$ are i.i.d. normal; (b) $X_1,\ldots, X_{d+2}$ follow a $d$-dimensional beta or beta prime distribution, which includes the uniform distribution on the ball or on the sphere as special cases; (c) $X_1,\ldots, X_{d+2}$ form a random walk with exchangeable increments. As a by-product of case (a) we recover a recent solution to Youden's demon problem which asks to determine the probability that, in a one-dimensional i.i.d. normal sample $ξ_1,\ldots, ξ_n$, the empirical mean $\frac 1n (ξ_1 + \ldots + ξ_n)$ lies between the $k$-th and the $(k+1)$-st order statistics. We also consider the conic (or spherical) version of the refined Sylvester problem and solve it in several special cases.
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