Quantization coefficients for uniform distributions on the boundaries of regular polygons

TL;DR

This paper derives a general formula for quantization coefficients of uniform distributions on regular polygon boundaries, showing they increase with the number of sides and approach the circle's coefficient as the polygon becomes more circular.

Contribution

It provides a novel formula linking polygon boundary distributions to quantization coefficients, bridging polygonal and circular cases.

Findings

Quantization coefficient increases with number of polygon sides.

Coefficient approaches that of the circle as sides tend to infinity.

Provides a unified formula for various regular polygons.

Abstract

In this paper, we give a general formula to determine the quantization coefficients for uniform distributions defined on the boundaries of different regular $m$-sided polygons inscribed in a circle. The result shows that the quantization coefficient for the uniform distribution on the boundary of a regular $m$-sided polygon inscribed in a circle is an increasing function of $m$, and approaches to the quantization coefficient for the uniform distribution on the circle as $m$ tends to infinity.