Renormalization of potentials and generalized centers

TL;DR

This paper introduces a renormalized potential for compact domains in Euclidean space, generalizing classical concepts like dual mixed volumes and centers, and analyzes the extremal points, showing balls uniquely attain extreme values for fixed volume.

Contribution

It extends the Riesz potential to a renormalized form for non-positive parameters and explores the properties of generalized centers, including their uniqueness for balls.

Findings

Renormalized potentials generalize classical Riesz potentials.

Generalized centers extend the concept of the center of mass.

Balls uniquely attain extremal potential values among bodies of equal volume.

Abstract

We generalize the Riesz potential of a compact domain in $\mathbb{R}^{m}$ by introducing a renormalization of the $r^{\alpha-m}$-potential for $\alpha\le0$. This can be considered as generalization of the dual mixed volumes of convex bodies as introduced by Lutwak. We then study the points where the extreme values of the (renormalized) potentials are attained. These points can be considered as a generalization of the center of mass. We also show that only balls give extreme values among bodied with the same volume.