Tamas Erdélyi, Edward B. Saff
We derive bounds for the size of the maximum polarization quantity $$M_n^p(A) := \max_{{\bold x}_1, {\bold x}_2, ..., {\bold x}_n \in A} {\min_{{\bold x} \in A}{\sum_{j=1}^n{\frac{1}{|{\bold x} - {\bold x}_j|^{p}}}}}$$ for quite general sets $A \subset {\Bbb R}^m$ 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 establish the sharp polarization inequality on the unit circle for the case $p=4$ by exploiting some interesting polynomial inequalities. Furthermore, we raise some challenging conjectures.
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http://arxiv.org/abs/1206.4729
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