Conditional Constrained and Unconstrained Quantization for Uniform Distributions on Regular Polygons

We consider a uniform distribution on a regular polygon with \(k\) sides for some \(k\geqslant 3\) and the set of all its \(k\) vertices as a conditional set. For the uniform distribution under a given conditional set, we first obtain the conditional optimal sets of \(n\) points and the corresponding \(n\)th conditional quantization errors for all positive integers \(n\geqslant k\). Then, we calculate the conditional quantization dimension and the conditional quantization coefficient in the unconstrained scenario. Next, for the uniform distribution on a polygon with the same conditional set, we investigate the conditionally constrained optimal sets of \(n\) points and the conditional constrained quantization errors for all \(n \geqslant 6\), under constraints such as the circumcircle, the incircle, and various diagonals of the polygon.