Uniqueness of centers of nearly spherical bodies

TL;DR

This paper proves that for bodies nearly spherical in shape, the extremal point of a specific potential function is unique, and it explores properties of this potential for a unit ball.

Contribution

It establishes the uniqueness of the $r^{ ext{a}}$-center for nearly spherical bodies and analyzes the regularized potentials of a unit ball.

Findings

Uniqueness of the $r^{ ext{a}}$-center for bodies close to a sphere.

Characterization of regularized potentials for a unit ball.

Results hold for any real number $ ext{a}$ with bodies sufficiently close to a sphere.

Abstract

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.