The Stolarsky Principle and Energy Optimization on the Sphere

The classical Stolarsky invariance principle connects the spherical cap discrepancy of a finite point set on the sphere to the pairwise sum of Euclidean distances between the points. In this paper, we further explore and extend this phenomenon. In addition to a new elementary proof of this fact, we establish several new analogs, which relate various notions of discrepancy to different discrete energies. In particular, we find that the hemisphere discrepancy is related to the sum of geodesic distances. We also extend these results to arbitrary measures on the sphere and arbitrary notions of discrepancy and apply them to problems of energy optimization and combinatorial geometry and find that, surprisingly, the geodesic distance energy behaves differently than its Euclidean counterpart.