Characterization of balls by generalized Riesz energy

TL;DR

This paper demonstrates that geometric shapes like balls, circles, and spheres can be uniquely identified using a generalized Riesz energy, which involves integrating powers of distances between points, leading to a method for shape recognition.

Contribution

The authors establish that generalized Riesz energy uniquely characterizes certain geometric shapes among compact submanifolds in Euclidean space, providing a new shape identification technique.

Findings

Balls, circles, and 2-spheres are uniquely characterized by their Riesz energy.

Riesz energy can be used to identify shapes via interpoint distance distributions.

The method applies to closed or codimension-0 submanifolds in Euclidean space.

Abstract

We show that balls, circles and 2-spheres can be identified by generalized Riesz energy among compact submanifolds of the Euclidean space that are either closed or with codimension 0, where the Riesz energy is defined as the double integral of some power of the distance between pairs of points. As a consequence, we obtain the identification by the interpoint distance distribution.