The Stolarsky Principle and Energy Optimization on the Sphere

Abstract

The classical Stolarsky invariance principle connects the spherical cap L 2 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.

Bounds for $$L_p$$-Discrepancies of Point Distributions in Compact Metric Measure Spaces

“…At the same time, the lower bound in (1.7) fails, even for the spheres S d , if the measure ξ is singular. The corresponding example can be found in [5,12,13]. In this example the discrepancy λ p [ξ, N ] is bounded from above by a constant independent of N and p.…”

Numerical integration on graphs: Where to sample and how to weigh

“…In the ideal setting with perfectly distributed heat balls all the weights would be identical (as can be seen in examples with lots of symmetry, see [16]); this nicely mirrors an unofficial guideline in numerical integration stating that how far the weights deviate from constant weights can be used as measure of quality and stability of the method. We summarize that (1) it is desirable to construct W ⊂ V equipped with weights a w such that random walks, starting in w ∈ W and weighted by a w , intersect each other as little as possible.…”

General and refined Montgomery Lemmata

“…where C ′ is a large constant depending only on the dimension, and I F (σ) = It is known (see e.g. [7,8]) that for positive definite functions F , the uniform surface measure σ minimizes the energy with potential F over all Borel probability measures on S d . Thus Corollary 2 states, in a quantitative way, that the energy of finite atomic measures with equal weights cannot be too close to the minimum.…”