On the optimal Voronoi partitions for Ahlfors-David measures with respect to the geometric mean error

Let $\mu$ be an Ahlfors-David probability measure on ${\mathbb{R}}^q$ with support $K$. For every $n\ge 1$, let $C_n(\mu )$ denote the collection of all the $n$-optimal sets for $\mu$ with respect to the geometric mean error. We prove that, there exist constant $d_1,d_2>0$, such that for each $n\ge 1$, every ${\alpha }_n\in C_n(\mu )$ and an arbitrary Voronoi partition $\{P_a({\alpha }_n){\}}_{a\in {\alpha }_n}$ with respect to ${\alpha }_n$, we have $$d_1{n}^{-1}\le \underset{a\in {\alpha }_n}{\min }\mu (P_a({\alpha }_n))\le \underset{a\in {\alpha }_n}{\max }\mu (P_a({\alpha }_n))\le d_2{n}^{-1}\mathrm{.}$$ Moreover, we prove that each $P_a({\alpha }_n)$ contains a closed ball of radius $d_3\mathrm{\mid }{P}_a({\alpha }_n)\cap K\mathrm{\mid }$, where $d_3$ is a constant and $\mathrm{\mid }B\mathrm{\mid }$ 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.