论文标题
关于AHLFORS-DAVID措施的最佳Voronoi分区，相对于几何均值误差
On the optimal Voronoi partitions for Ahlfors-David measures with respect to the geometric mean error
论文作者
论文摘要
令$μ$为$ \ mathbb {r}^q $的ahlfors-david概率度量，并带有支持$ k $。对于每$ n \ geq 1 $，让$ c_n（μ）$表示相对于几何平均值，以$μ$的价格为$ n $最佳的集合的集合。我们证明，存在常数$ d_1，d_2> 0 $，因此，对于每种$ n \ geq 1 $，c_n（μ）$中的每个$α_n\和任意的voronoi分区$ \ {p_a（p_a（α_n）\} d_1n^{ - 1} \ leq \ min_ {a \inα_n}μ（p_a（α_n））\ leq \ max_ {a \inα_n}μ（p_a（p_a（α_n））此外，我们证明每个$ p_a（α_n）$都包含一个封闭的半径$ d_3 | p_a（α_n）\ cap k | $，其中$ d_3 $是常数，$ | b | $表示$ b \ subset \ subset \ subset \ subset \ subbb \ nathbb {r} r}^q $。在更一般的环境中，建立了有关Voronoi分区元素元素元素元素元素元素的一些估计。
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.
