Potential Analysis

, Volume 47, Issue 2, pp 235–244 | Cite as

A Minimum Principle for Potentials with Application to Chebyshev Constants

  • A. ReznikovEmail author
  • E. B. Saff
  • O. V. Vlasiuk


For “Riesz-like” kernels K(x,y) = f(|xy|) on A×A, where A is a compact d-regular set \(A\subset \mathbb {R}^{p}\), we prove a minimum principle for potentials \(U_{K}^{\mu }=\int K(x,y)\textup {d}\mu (x)\), where μ is a Borel measure supported on A. Setting \(P_{K}(\mu )=\inf _{y\in A}U^{\mu }(y)\), the K-polarization of μ, the principle is used to show that if {ν N } is a sequence of measures on A that converges in the weak-star sense to the measure ν, then P K (ν N )→P K (ν) as \(N\to \infty \). The continuous Chebyshev (polarization) problem concerns maximizing P K (μ) over all probability measures μ supported on A, while the N-point discrete Chebyshev problem maximizes P K (μ) only over normalized counting measures for N-point multisets on A. We prove for such kernels and sets A, that if {ν N } is a sequence of N-point measures solving the discrete problem, then every weak-star limit measure of ν N as \(N \to \infty \) is a solution to the continuous problem.


Maximal Riesz polarization Chebyshev constant Hausdorff measure Riesz potential Minimum principle 

Mathematics Subject Classification (2010)

Primary 31C15, 31C20 Secondary 30C80 


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Copyright information

© Springer Science+Business Media Dordrecht 2017

Authors and Affiliations

  1. 1.Center for Constructive Approximation, Department of MathematicsVanderbilt UniversityNashvilleUSA

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