中心线深度Rado界的改进
An improvement on the Rado bound for the centerline depth
摘要：
令$μ$是$\mathbb R^d$中的Borel概率测度。对于$k$平坦的$α$，考虑值$\inf μ(H)$，其中$H$贯穿包含$α$的所有半空间。这个下确界被称为$α$的半空间深度。Bukh、Matoušek和Nivasch证明，对于每个$μ$和每个$0 \leq k < d$，都存在一个深度至少为$\tfrac{k + 1}{k + d + 1}$的$k$ -平面。Rado中心点定理意味着$\tfrac{1}{d + 1 - k}$的下界（Rado界），一般来说，这个下界要弱得多。当Rado界与Bukh、Matoušek和Nivasch所确定的界重合时，即，对于$k = 0$和$k = d - 1$，已知它是最佳的。在本文中，我们表明，对于所有其他对$(d, k)$可以提高Rado界。如果$k = 1$和$d \geq 3$，我们证明存在深度至少为$\tfrac{1}{d} + \tfrac{1}{3d^3}$的一维线。作为推论，对于所有满足$0 < k < d - 1$的$(d, k)$，存在深度至少为$\tfrac{1}{d + 1 - k} + \tfrac{1}{3(d + 1 - k)^3}$的$k$ -平坦。
英文摘要：
Let $μ$ be a Borel probability measure in $\mathbb R^d$. For a $k$-flat $α$ consider the value $\inf μ(H)$, where $H$ runs through all half-spaces containing $α$. This infimum is called the half-space depth of $α$. Bukh, Matoušek and Nivasch conjectured that for every $μ$ and every $0 \leq k < d$ there exists a $k$-flat with the depth at least $\tfrac{k + 1}{k + d + 1}$. The Rado Centerpoint Theorem implies a lower bound of $\tfrac{1}{d + 1 - k}$ (the Rado bound), which is, in general, much weaker. Whenever the Rado bound coincides with the bound conjectured by Bukh, Matoušek and Nivasch, i.e., for $k = 0$ and $k = d - 1$, it is known to be optimal. In this paper we show that for all other pairs $(d, k)$ one can improve on the Rado bound. If $k = 1$ and $d \geq 3$ we show that there is a 1-dimensional line with the depth at least $\tfrac{1}{d} + \tfrac{1}{3d^3}$. As a corollary, for all $(d, k)$ satisfying $0 < k < d - 1$ there exists a $k$-flat with depth at least $\tfrac{1}{d + 1 - k} + \tfrac{1}{3(d + 1 - k)^3}$.
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