Balanced configurations of points in the plane

TL;DR

This paper classifies all balanced point configurations in the Euclidean plane that meet a minimal distance criterion, confirming a conjecture that such configurations are inherently group-balanced.

Contribution

It provides a complete classification of minimal-distance balanced configurations in the plane, supporting the conjecture that all are group-balanced.

Findings

All minimal-distance balanced configurations in the plane are group-balanced.

The classification confirms the conjecture for Euclidean plane configurations.

Extends the understanding of symmetry and equilibrium in geometric point arrangements.

Abstract

A balanced configuration of points on the sphere $S^2$ is a (finite) set of points which are in equilibrium if they act on each other according any force law dependent only on the distance between two points. The configuration is additionally group-balanced if for each point in a configuration $\mathcal{C}$, there is a symmetry of $\mathcal{C}$ fixing only that point and its antipode. Leech showed that these definitions are equivalent on the sphere $S^2$ by classifying all possible balanced configurations. On the other hand, Cohn, Elkies, Kumar, and Sch\"urmann showed that for $n\geq 7,$ there are examples of balanced configurations in $S^{n-1}$ which are not group balanced. They also suggested extending the notion of balanced configurations to Euclidean space, and conjectured that at least in the case of the plane, all discrete balanced configurations in $\mathbb{R}^n$ are group-balanced. We verify a reformulation of this conjecture by providing a complete classification of the balanced configurations in $\mathbb{R}^2$ satisfying a certain minimal distance property.