School of Mathematical & Statistical Sciences Faculty Publications
Document Type
Article
Publication Date
1-12-2026
Abstract
This paper develops a systematic and geometric theory of optimal quantization on the unit sphere 𝕊2, 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 𝕊2.
Recommended Citation
Roychowdhury, Mrinal Kanti. "Optimal Quantization of Finite Uniform Data on the Sphere." Mathematics 14, no. 2 (2026): 288. https://doi.org/10.3390/math14020288
Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License.
Publication Title
Mathematics
DOI
10.3390/math14020288
Comments
© 2026 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license.  Â