Caratheodory深度定理
Caratheodory's Theorem in Depth
摘要：
设$X$是$\mathbb{R}^d$中的有限点集。点$q$相对于$X$的Tukey深度是在包含$q$的半空间中$X$的点的最小数目$τ_X(q)$。在本文中，我们证明了Carathéodory定理的一个深度版本。特别是，我们证明了存在一个常数$c$（这只取决于$d$和$τ_X(q)$）和成对不相交的集合$X_1,\dots, X_{d+1} \subset X$，使以下成立。每个$X_i$至少具有$c|X|$点，并且对于$X_i$中的点$x_i$的每个选择，$q$是$x_1,\dots, x_{d+1}$的凸组合。我们还证明了深度版本的Helly的和Kirchberger的定理。
英文摘要：
Let $X$ be a finite set of points in $\mathbb{R}^d$. The Tukey depth of a point $q$ with respect to $X$ is the minimum number $τ_X(q)$ of points of $X$ in a halfspace containing $q$. In this paper we prove a depth version of Carathéodory's theorem. In particular, we prove that there exists a constant $c$ (that depends only on $d$ and $τ_X(q)$) and pairwise disjoint sets $X_1,\dots, X_{d+1} \subset X$ such that the following holds. Each $X_i$ has at least $c|X|$ points, and for every choice of points $x_i$ in $X_i$, $q$ is a convex combination of $x_1,\dots, x_{d+1}$. We also prove depth versions of Helly's and Kirchberger's theorems.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
