Uniqueness of centers of nearly spherical bodies

Abstract

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