Riesz polarization inequalities in higher dimensions

We derive bounds and asymptotics for the maximum Riesz polarization quantity M n(A) := max x1,x2,... ,xn∈A min x∈A n ∑ j=1 1 |x− xj |p (which is n times the Chebyshev constant ) for quite general sets A ⊂ R 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.