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 $n\geq 1$, 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 $n\geq 1$, 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 $$d_1n^{-1}\leq\min_{a\in\alpha_n}\mu(P_a(\alpha_n))\leq\max_{a\in\alpha_n}\mu(P_a(\alpha_n))\leq d_2n^{-1}.$$ Moreover, we prove that each $P_a(\alpha_n)$ contains a closed ball of radius $d_3|P_a(\alpha_n)\cap K|$, 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.