On polarization of spherical codes and designs

In this article we investigate the N -point min-max and max-min polarization problems on the sphere for a large class of potentials in Rn. We derive universal lower and upper bounds on the polarization of spherical designs of fixed dimension, strength, and cardinality. The bounds are universal in the sense that they are a convex combination of potential function evaluations with nodes and weights independent of the class of potentials. As a consequence of our lower bounds, we obtain the Fazekas-Levenshtein bounds on the covering radius of spherical designs. Utilizing the existence of spherical designs, our polarization bounds are extended to general configurations. As examples we completely solve the min-max polarization problem for 120 points on S3 and show that the 600-cell is universally optimal for that problem. We also provide alternative methods for solving the max-min polarization problem when the number of points N does not exceed the dimension n and when N = n + 1. We further show that the cross-polytope has the best max-min polarization constant among all spherical 2-designs of N = 2n points for n = 2, 3, 4; for n >= 5, this statement is conditional on a well-known conjecture that the cross-polytope has the best covering radius. This max-min optimality is also established for all so-called centered codes.(c) 2023 Elsevier Inc. All rights reserved.