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

Sanguo Zhu, Youming Zhou

Abstract

Let $μ$ be an Ahlfors-David probability measure on $\mathbb{R}^q$ with support $K$. For every $n\geq 1$, 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 constant $d_1,d_2>0$, such that for each $n\geq 1$, every $α_n\in C_n(μ)$ and an arbitrary Voronoi partition $\{P_a(α_n)\}_{a\inα_n}$ with respect to $α_n$, 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 $P_a(α_n)$ contains a closed ball of radius $d_3|P_a(α_n)\cap K|$, 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.

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

Abstract

Let be an Ahlfors-David probability measure on with support . For every , let denote the collection of all the -optimal sets for with respect to the geometric mean error. We prove that, there exist constant , such that for each , every and an arbitrary Voronoi partition with respect to , we have Moreover, we prove that each contains a closed ball of radius , where is a constant and denotes the diameter of a set . Some estimates for the measure and the geometrical size of the elements of a Voronoi partition with respect to an -optimal set are established in a more general context.

Paper Structure

This paper contains 11 sections, 19 theorems, 121 equations.

Table of Contents

  1. Introduction
  2. Some definitions and basic facts
  3. A significant concern about the Voronoi partition
  4. Statement of the main result
  5. Preliminaries
  6. Weak estimates for measures and geometrical size of $P_a(\alpha_k)$
  7. Auxiliary measures and auxiliary integers
  8. Some subsets of $A_\omega^*$ and auxiliary measures
  9. Selection of some auxiliary integers
  10. A characterization for the $n$-optimal sets
  11. Proof of Theorem \ref{['mthm']}

Key Result

Theorem \oldthetheorem

Let $\mu$ be an Ahlfors-David probability measure on $\mathbb{R}^q$ with support $K_\mu$. There exist positive constants $d_1, d_2, d_3$, such that for each $n\geq 1$, every $\alpha_n\in C_n(\mu)$ and an arbitrary VP $\{P_a(\alpha_n)\}_{a\in\alpha_n}$, we have Moreover, for every $a\in\alpha_n$, $P_a(\alpha_n)$ contains a ball of radius $d_3|P_a(\alpha_n)\cap K_\mu|$ which is centered at $a$.

Theorems & Definitions (42)

  • Theorem \oldthetheorem
  • Lemma \oldthetheorem
  • proof
  • Remark \oldthetheorem
  • Remark \oldthetheorem
  • Lemma \oldthetheorem
  • proof
  • Lemma \oldthetheorem
  • proof
  • Lemma \oldthetheorem
  • ...and 32 more