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

Abstract

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 \[ d_1n^{-1}\leq\min_{a\inα_n}μ(P_a(α_n))\leq\max_{a\inα_n}μ(P_a(α_n))\leq d_2n^{-1}. \] Moreover, we prove that each $P_a({\alpha }_n)$ 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.

References in corpus (2)

• Some recent developments in quantization of fractal measures
• On the quantization for self-affine measures on Bedford-McMullen carpets

Cited by in corpus (1)

• Quantization dimensions of compactly supported probability measures via Rényi dimensions