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

TL;DR

This paper investigates the structure of optimal Voronoi partitions for Ahlfors-David measures, establishing bounds on measure distribution and geometric properties of the partitions, with implications for geometric measure theory.

Contribution

It provides new bounds on measure and size of Voronoi cells for optimal sets under Ahlfors-David measures, extending understanding of geometric partitions.

Findings

Bounds on the measure of Voronoi cells are proportional to 1/n.

Each Voronoi cell contains a ball with radius proportional to its diameter.

Estimates for measure and size of Voronoi elements are established.

Abstract

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.