Riesz polarization inequalities in higher dimensions (Q390504)

This paper is concerned with bounds and asymptotics for the maximum Riesz polarization quantity NEWLINE\[NEWLINE M_n^p(A):=\max_{x_1,x_2,\dots,x_n\in A}\min_{x\in A}\sum_{j=1}^n\frac{1}{|x-x_j|^p}, NEWLINE\]NEWLINE where \(A\subset {\mathbb R}^m\). A particular attention is paid to the case where \(A\) is the unit sphere and the unit ball case in which the authors obtain various upper and lower bounds. The approach combines elementary techniques with properties of Riesz potentials.