Quantization for uniform distributions on hexagonal, semicircular, and elliptical curves

Abstract

In this paper, first we have defined the uniform distribution on the boundary of a regular hexagon, and then investigate the optimal sets of n-means and the nth quantization errors for all positive integers n. We give an exact formula to determine them if n is of the form n = 6k + 6 for some positive integer k. We further calculate the quantization dimension, and the quantization coefficient, and show that the quantization dimension is equal to the dimension of the object, and the quantization coefficient exists as a finite positive number, which supports the well-known result of Bucklew and Wise (1982), which says that for a Borel probability measure P with non-vanishing absolutely continuous part the quantization coefficient exists as a finite positive number. Then, we define the uniform distribution on the boundary of a semicircular disc, and obtain a sequence and an algorithm, which help us to determine the optimal sets of n-means and the nth quantization errors for all positive integers n with respect to the uniform distribution. Finally, for a uniform distribution defined on an elliptical curve, we investigate the optimal sets of n-means and the nth quantization errors for all positive integers n.