2022/08/04 by Laura Pierson, Pierson, Laura, Julian Wellman +1
Computer Science · Engineering · Mathematics · #3D Shape Modeling and Analysis #Combinatorics (math.CO) #Computational Geometry and Mesh Generation #FOS: Mathematics #Mathematics and Applications #Metric Geometry (math.MG)
paper · pdf · doi:10.48550/arxiv.2208.02426
openalex publication_date 2022/08/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A balanced configuration of points on the sphere S2 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 C, there is a symmetry of C fixing only that point and its antipode. Leech showed that these definitions are equivalent on the sphere S2 by classifying all possible balanced configurations. On the other hand, Cohn, Elkies, Kumar, and Schürmann showed that for n≥ 7, there are examples of balanced configurations in Sn-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 ℝn are group-balanced. We verify a reformulation of this conjecture by providing a complete classification of the balanced configurations in ℝ2 satisfying a certain minimal distance property.