Riesz Polarization Inequalities in Higher Dimensions

2012/06/20 by Tamás Erdélyi, Edward B. Saff, Erdélyi, Tamas +1
Mathematics · #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #FOS: Physical sciences #Mathematical Approximation and Integration #Mathematical Physics (math-ph) #Mathematical functions and polynomials #Point processes and geometric inequalities

paper · pdf · doi:10.48550/arxiv.1206.4729

openalex publication_date 2012/06/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We derive bounds and asymptotics for the maximum Riesz polarization quantity M_n^p(A) := max_\bold x₁, \bold x₂, …, \bold x_n ∈ A min_\bold x ∈ A∑_j=1ⁿ\frac1|\bold x - \bold x_j|^p (which is n times the Chebyshev constant) for quite general sets A ⊂ \Bbb R^m with special focus on the unit sphere and unit ball. We combine elementary averaging arguments with potential theoretic tools to formulate and prove our results. We also give a discrete version of the recent result of Hardin, Kendall, and Saff which solves the Riesz polarization problem for the case when A is the unit circle and p>0, as well as provide an independent proof of their result for p=4 that exploits classical polynomial inequalities and yields new estimates. Furthermore, we raise some challenging conjectures.

Related

• Unconstrained polarization (Chebyshev) problems: basic properties and Riesz kernel asymptotics
• Optimal discrete measures for Riesz potentials
• Asymptotics of discrete Riesz d-polarization on subsets of d-dimensional manifolds
• Reverse Triangle Inequalities for Riesz Potentials and Connections with Polarization
• Min-Max Polarization for Certain Classes of Sharp Configurations on the Sphere