Optimal Quantization of Finite Uniform Data on the Sphere

This paper develops a systematic and geometric theory of optimal quantization on the unit sphere $\mathbb S^2$, focusing on finite uniform probability distributions supported on the spherical surface - rather than on lower-dimensional geodesic subsets such as circles or arcs. We first establish the existence of optimal sets of $n$-means and characterize them through centroidal spherical Voronoi tessellations. Three fundamental structural results are obtained. First, a cluster - purity theorem shows that when the support consists of well-separated components, each optimal Voronoi region remains confined to a single component. Second, a ring - allocation (discrete water - filling) theorem provides an explicit rule describing how optimal representatives are distributed across multiple latitudinal rings, together with closed-form distortion formulas. Third, a Lipschitz - type stability theorem quantifies the robustness of optimal configurations under small geodesic perturbations of the support. In addition, a spherical analogue of Lloyd's algorithm is presented, in which intrinsic (Karcher) means replace Euclidean centroids for iterative refinement. These results collectively provide a unified and transparent framework for understanding the geometric and algorithmic structure of optimal quantization on $\mathbb S^2$.