Experimental investigation on the uniqueness of a center of a body

TL;DR

This paper investigates the conditions under which a unique center, defined as the point maximizing a potential with a radially symmetric kernel, exists within a body, focusing on new sufficient conditions for uniqueness.

Contribution

It introduces new sufficient conditions that guarantee the uniqueness of a center for bodies with radially symmetric kernels, extending previous theoretical results.

Findings

Established new sufficient conditions for uniqueness

Extended understanding of centers with Riesz and Poisson kernels

Provided theoretical criteria for the existence and uniqueness

Abstract

The object of our investigation is a point that gives the maximum value of a potential with a strictly decreasing radially symmetric kernel. It defines a center of a body in Rm. When we choose the Riesz kernel or the Poisson kernel as the kernel, such centers are called a radial center or an illuminating center, respectively. The existence of a center is easily shown but the uniqueness does not always hold. Sufficient conditions of the uniqueness of a center have been studied by some researchers. The main results in this paper are some new sufficient conditions for the uniqueness of a center of a body.