Balls minimize moments of logarithmic and Newtonian equilibrium measures

The $q$-th moment ($q>0$) of electrostatic equilibrium measure is shown to be minimal for a centered ball among $3$-dimensional sets of given capacity, while among $2$-dimensional sets a centered disk is the minimizer for $0<q\le 2$. Analogous results are developed for Newtonian capacity in higher dimensions and logarithmic capacity in $2$ dimensions. Open problems are raised for Riesz equilibrium moments.