Riesz polarization inequalities in higher dimensions
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abstract
We derive bounds and asymptotics for the maximum Riesz polarization quantity (which is n times the Chebyshev constant) for quite general sets ARm 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. 2013 Elsevier Inc.