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Let ( f i ) i = 1 N be a family of contractive similitudes on 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 , … , N . We study the asymptotic geometric mean errors e n , 0 ( μ ) , n ≥ 1 , in the quantization for the in-homogeneous self-similar measure μ associated with the condensation system ( ( f i ) i = 1 N , ( p i ) i = 0 N , ν ) . We focus on the following two independent cases: (I) ν is a self-similar measure on R q associated with ( f i ) i = 1 N ; (II) ν is a self-similar measure associated with another family of contractive similitudes ( g i ) i = 1 M on R q satisfying the open set condition and ( ( f i ) i = 1 N , ( p i ) i = 0 N , ν ) satisfies a version of in-homogeneous open set condition. We show that, in both cases, the quantization dimension D 0 ( μ ) of μ of order zero exists and agrees with that of ν , which is independent of the probability vector ( p i ) i = 0 N . We determine the exact convergence order of ( e n , 0 ( μ ) ) n = 1 ∞ ; namely, for D 0 ( μ ) = : d 0 , there exists a constant D > 0 , such that D − 1 n − 1 d 0 ≤ e n , 0 ( μ ) ≤ D n − 1 d 0 , n ≥ 1 . [ABSTRACT FROM AUTHOR]

Let μ be an Ahlfors-David probability measure on R q with support K μ. For every n ≥ 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 constants d 1 , d 2 > 0 , such that for each n ≥ 1 , every α n ∈ C n (μ) and an arbitrary Voronoi partition { P a (α n) } a ∈ α n with respect to α n , we have d 1 n − 1 ≤ min a ∈ α n ⁡ μ (P a (α n)) ≤ max a ∈ α n ⁡ μ (P a (α n)) ≤ d 2 n − 1. Moreover, we prove that for each a ∈ α n , the set P a (α n) contains a closed ball of radius d 3 | P a (α n) ∩ K μ | which is centered at a , where d 3 is a constant and | B | denotes the diameter of a set B ⊂ 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. These estimates also allow us to drop an additional condition in our previous work on self-similar measures. [ABSTRACT FROM AUTHOR]