Abstract
The classical Stolarsky invariance principle connects the spherical cap L2 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.
| Original language | English (US) |
|---|---|
| Pages (from-to) | 31-60 |
| Number of pages | 30 |
| Journal | Constructive Approximation |
| Volume | 48 |
| Issue number | 1 |
| DOIs | |
| State | Published - Aug 1 2018 |
Bibliographical note
Funding Information:This work was partially supported by the Simons Foundation Collaboration Grant (Bilyk), NSERC Canada under Grant RGPIN 04702 (Dai), and the NSF Graduate Research Fellowship 00039202 (Matzke). The first two authors are grateful to CRM Barcelona: this collaboration originated during their participation in the research program “Constructive Approximation and Harmonic Analysis” (Bilyk’s trip was sponsored by NSF Grant DMS 1613790).
Publisher Copyright:
© 2018, Springer Science+Business Media, LLC, part of Springer Nature.
Keywords
- Discrepancy
- Energy minimization
- Stolarsky principle