点集分裂的Helly型定理
Helly-Type Theorems for Splitting Point Sets
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中文总结 AI 辅助
该研究针对ℝ^d中有限点集的分裂族建立Helly型定理，突破经典准则仅适用于点和超平面的局限，可扩展至任意维数0≤k≤d-1的k维平面族分裂点集的情形。
中文摘要 AI 辅助
设0 < α ≤ 1/2，我们称ℝ^d中的有限点集P被超平面h α-分裂，当且仅当h确定的每个闭半空间都包含至少α|P|个P中的点；进一步称P被k维平面τ α-分裂，当且仅当P被过τ的任意超平面α-分裂。按标准记号（k=0时与Tukey深度一致），k维平面τ相对于P的深度为α。我们针对ℝ^d中有限点集的分裂族建立了有趣的Helly型定理。与仅存在点和超平面的紧凸集族横截的经典充分Helly型准则不同，我们的结果可扩展至用任意维数0 ≤ k ≤ d-1的k维平面族分裂点集的情形。
英文摘要
Let $0 < α\leq 1/2$. We say that a finite point set $P$ in $\mathbb{R}^d$ is $α$-split by a hyperplane $h$ if each of the closed half-spaces determined by $h$, contains at least $α|P|$ of the points of $P$. We further say $P$ is $α$-split by a $k$-dimensional flat $τ$ if $P$ is $α$-split by any hyperplane through $τ$. In the standard notation (which coincides with Tukey depth for $k= 0$), the $k$-flat $τ$ has depth $α$ with respect to $P$. We establish interesting Helly-type theorems for splitting families of finite point sets in $\mathbb{R}^d$. Unlike the classical sufficient Helly-type criteria for transversals to families of compact convex sets, which exist only for point and hyperplanes, our results extend to splitting families of point sets by collections of $k$-flats of arbitrary dimensionality $ 0 \leq k \leq d-1$.
发表机构
- Ben-Gurion University of the Negev（内盖夫本-古里安大学）
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