Communications of the
Korean Mathematical Society

ISSN(Print) 1225-1763 ISSN(Online) 2234-3024



Commun. Korean Math. Soc. 2023; 38(2): 431-450

Online first article April 24, 2023      Printed April 30, 2023

Copyright © The Korean Mathematical Society.

Uniform distributions on curves and quantization

Joseph Rosenblatt, Mrinal Kanti Roychowdhury

1409 W. Green Street; 1201 West University Drive


The basic goal of quantization for probability distribution is to reduce the number of values, which is typically uncountable, describing a probability distribution to some finite set and thus to make an approximation of a continuous probability distribution by a discrete distribution. It has broad application in signal processing and data compression. In this paper, first we define the uniform distributions on different curves such as a line segment, a circle, and the boundary of an equilateral triangle. Then, we give the exact formulas to determine the optimal sets of $n$-means and the $n$th quantization errors for different values of $n$ with respect to the uniform distributions defined on the curves. In each case, we further calculate the quantization dimension and show that it is equal to the dimension of the object; and the quantization coefficient exists as a finite positive number. This supports the well-known result of Bucklew and Wise \cite{BW}, which says that for a Borel probability measure $P$ with non-vanishing absolutely continuous part the quantization coefficient exists as a finite positive number.

Keywords: Uniform distribution, optimal quantizers, quantization error, quantization dimension, quantization coefficient

MSC numbers: Primary 60Exx, 94A34