Quantization for uniform distributions on equilateral triangles

TL;DR

This paper studies how to best approximate a uniform distribution on an equilateral triangle using a finite set of points, finding optimal point configurations and quantization errors for small n and providing bounds for large n.

Contribution

It determines optimal point sets and quantization errors for the uniform measure on an equilateral triangle for small n and offers bounds for large n, extending to general piecewise smooth sets.

Findings

Optimal n-means for n ≤ 4 identified

Numerical results for n ≤ 21 provided

Quantization error bounds established for large n

Abstract

We approximate the uniform measure on an equilateral triangle by a measure supported on $n$ points. We find the optimal sets of points ($n$-means) and corresponding approximation (quantization) error for $n\leq4$, give numerical optimization results for $n\leq 21$, and a bound on the quantization error for $n\to\infty$. The equilateral triangle has particularly efficient quantizations due to its connection with the triangular lattice. Our methods can be applied to the uniform distributions on general sets with piecewise smooth boundaries.