Center of mass and the optimal quantizers for some continuous and discrete uniform distributions

TL;DR

This paper investigates the properties of the center of mass and optimal quantizers for uniform distributions over various regions, deriving explicit solutions for optimal means and quantization errors in both continuous and discrete cases.

Contribution

It establishes the relationship between the center of mass and the centroid for uniform regions and derives explicit optimal quantizers for specific geometric shapes and discrete distributions.

Findings

Voronoi regions of two-means on a disc are semicircular.

Centers of mass in certain triangles form centroidal Voronoi tessellations.

Explicit optimal sets and quantization errors are determined for rhombus and discrete uniform distributions.

Abstract

In this paper, we first consider a flat plate (called a lamina) with uniform density $\rho$ that occupies a region $\mathfrak R$ of the plane. We show that the location of the center of mass, also known as the centroid, of the region equals the expected vector of a bivariate continuous random variable with a uniform probability distribution taking values on the region $\mathfrak R$. Using this property, we prove that the Voronoi regions of an optimal set of two-means with respect to the uniform distribution defined on a disc partition the disc into two regions bounded by the semicircles. Besides, we show that if an isosceles triangle is partitioned into an isosceles triangle and an isosceles trapezoid in the Golden ratio, then their centers of mass form a centroidal Voronoi tessellation of the triangle. In addition, using the properties of center of mass we determine the optimal sets of two-means and the corresponding quantization error for a uniform distribution defined on a region with uniform density bounded by a rhombus. Further, we determine the optimal sets of $n$-means, and the $n$th quantization errors for two different discrete uniform distributions for some positive integers $n\leq \text{card(supp}(P))$.