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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.

Let $E$ be a graph-directed set associated with a di-graph $G$. Let $\mu$ be a Markov-type measure on $E$. Assuming a separation condition for $E$, we determine the exact convergence order of the $L_r$-quantization error for $\mu$. This result provides us with accurate information on the asymptotics of the quantization error, especially when the quantization coefficient is infinite.

Let $(f_i)_{i=1}^N$ be a family of contractive similitudes on $\mathbb{R}^q$ satisfying the open set condition. Let $(p_i)_{i=0}^N$ be a probability vector with $p_i>0$ for all $i=0,1,\ldots,N$. We study the asymptotic geometric mean errors $e_{n,0}(\mu),n\geq 1$, in the quantization for the in-homogeneous self-similar measure $\mu$ associated with the condensation system $((f_i)_{i=1}^N,(p_i)_{i=0}^N,\nu)$. We focus on the following two independent cases: (I) $\nu$ is a self-similar measure on $\mathbb{R}^q$ associated with $(f_i)_{i=1}^N$; (II) $\nu$ is a self-similar measure associated with another family of contractive similitudes $(g_i)_{i=1}^M$ on $\mathbb{R}^q$ satisfying the open set condition and $((f_i)_{i=1}^N,(p_i)_{i=0}^N,\nu)$ satisfies a version of in-homogeneous open set condition. We show that, in both cases, the quantization dimension $D_0(\mu)$ of $\mu$ of order zero exists and agrees with that of $\nu$, which is independent of the probability vector $(p_i)_{i=0}^N$. We determine the convergence order of $(e_{n,0}(\mu))_{n=1}^\infty$; namely, for $D_0(\mu)=:d_0$, there exists a constant $D>0$, such that \[ D^{-1}n^{-\frac{1}{d_0}}\leq e_{n,0}(\mu)\leq D n^{-\frac{1}{d_0}}, n\geq 1. \]