关于Ahlfors-David测度在几何平均误差下的最优Voronoi划分
On the optimal Voronoi partitions for Ahlfors-David measures with respect to the geometric mean error
AI总结：
本文研究Ahlfors-David概率测度在几何平均误差下n-最优集对应Voronoi划分的性质，证明了各划分单元的测度均以常数倍n^{-1}为上下界，并给出单元包含闭球半径的几何尺寸估计。
摘要：
设$\mu$是$\mathbb{R}^q$上支集为$K$的Ahlfors-David概率测度。对每个$n\geq 1$，令$C_n(\mu)$表示$\mu$关于几何平均误差的所有$n$-最优集的集合。我们证明，存在常数$d_1,d_2>0$，使得对每个$n\geq 1$、每个$\alpha_n\in C_n(\mu)$以及关于$\alpha_n$的任意Voronoi划分$\{P_a(\alpha_n)\}_{a\in\alpha_n}$，都有\\[ d_1n^{-1}\leq\min_{a\in\alpha_n}\mu(P_a(\alpha_n))\leq\max_{a\in\alpha_n}\mu(P_a(\alpha_n))\leq d_2n^{-1}. \\]此外，我们证明每个$P_a(\alpha_n)$包含一个半径为$d_3|P_a(\alpha_n)\cap K|$的闭球，其中$d_3$是常数，$|B|$表示集合$B\subset\mathbb{R}^q$的直径。在更一般的情形下，我们建立了关于$n$-最优集的Voronoi划分元素的测度和几何尺寸的一些估计。
英文摘要：
Let $μ$ be an Ahlfors-David probability measure on $\mathbb{R}^q$ with support $K$. For every $n\geq 1$, let $C_n(μ)$ denote the collection of all the $n$-optimal sets for $μ$ with respect to the geometric mean error. We prove that, there exist constant $d_1,d_2>0$, such that for each $n\geq 1$, every $α_n\in C_n(μ)$ and an arbitrary Voronoi partition $\{P_a(α_n)\}_{a\inα_n}$ with respect to $α_n$, we have \[ d_1n^{-1}\leq\min_{a\inα_n}μ(P_a(α_n))\leq\max_{a\inα_n}μ(P_a(α_n))\leq d_2n^{-1}. \] Moreover, we prove that each $P_a(α_n)$ contains a closed ball of radius $d_3|P_a(α_n)\cap K|$, where $d_3$ is a constant and $|B|$ denotes the diameter of a set $B\subset\mathbb{R}^q$. Some estimates for the measure and the geometrical size of the elements of a Voronoi partition with respect to an $n$-optimal set are established in a more general context.
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