Uniqueness of centers of nearly spherical bodies

An $r^{\an}$-center of a compact body $\Om$ in an $n$ dimensional Euclidean space is a point that gives an extremal value of the regularized Riesz potential, which is the (Hadamard regularization of) integration on $\Om$ of the distance from the point to the power $\an$. We show that for any real number $\la$ if a compact body is sufficiently close to a ball in the sense of asphericity then the $r^{\an}$-center is unique. We also study the regularized potentials of a unit ball.