Uniqueness of centers of nearly spherical bodies

2021/03/13 by Jun O’Hara, O'Hara, Jun
Mathematics · #31B99 #51F99 #51M16 #52A40 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Mathematical Approximation and Integration #Metric Geometry (math.MG) #Point processes and geometric inequalities

paper · pdf · doi:10.48550/arxiv.2103.07685

openalex publication_date 2021/03/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

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