Distributing many points on spheres: minimal energy and designs

Abstract

This survey discusses recent developments in the context of spherical designs and minimal energy point configurations on spheres. The recent solution of the long standing problem of the existence of spherical $t$-designs on ${\mathbb{S}}^d$ with $\mathcal{O}(t^d)$ number of points by A. Bondarenko, D. Radchenko, and M. Viazovska attracted new interest to this subject. Secondly, D. P. Hardin and E. B. Saff proved that point sets minimising the discrete Riesz energy on ${\mathbb{S}}^d$ in the hypersingular case are asymptotically uniformly distributed. Both results are of great relevance to the problem of describing the quality of point distributions on ${\mathbb{S}}^d$, as well as finding point sets, which exhibit good distribution behaviour with respect to various quality measures.

50 pages, 1 table; revision; references added and updated

References in corpus (9)

• Two-Dimensional Matter: Order, Curvature and Defects
• New upper bounds for kissing numbers from semidefinite programming
• Covering of spheres by spherical caps and worst-case error for equal weight cubature in Sobolev spaces
• Periodic Discrete Energy for Long-Range Potentials
• Correspondences between the Classical Electrostatic Thomson Problem and Atomic Electronic Structure
• Spatial low-discrepancy sequences, spherical cone discrepancy, and applications in financial modeling
• Minimal Riesz Energy Point Configurations for Rectifiable d-Dimensional Manifolds
• Discrete Transformations in the Thomson Problem
• The support of the limit distribution of optimal Riesz energy points on sets of revolution in ${\mathbb{R}}^3$

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