Universally Optimal Distribution of Points on Spheres

We study configurations of points on the unit sphere that minimize potential energy for a broad class of potential functions (viewed as functions of the squared Euclidean distance between points). Call a configuration sharp if there are m m distances between distinct points in it and it is a spheric...

Full description

Saved in:
Bibliographic Details
Published in:Journal of the American Mathematical Society 2007-01, Vol.20 (1), p.99-148
Main Authors: Cohn, Henry, Kumar, Abhinav
Format: Article
Language:English
Subjects:
Degrees of polynomials
Harmonic functions
Inner products
Kissing
Mathematical lattices
Mathematical monotonicity
Polynomials
Potential energy
Uniqueness
Vertices
Citations: Items that this one cites
Items that cite this one
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary:We study configurations of points on the unit sphere that minimize potential energy for a broad class of potential functions (viewed as functions of the squared Euclidean distance between points). Call a configuration sharp if there are m m distances between distinct points in it and it is a spherical ( 2 m − 1 ) (2m-1) -design. We prove that every sharp configuration minimizes potential energy for all completely monotonic potential functions. Examples include the minimal vectors of the E 8 E_8 and Leech lattices. We also prove the same result for the vertices of the 600 600 -cell, which do not form a sharp configuration. For most known cases, we prove that they are the unique global minima for energy, as long as the potential function is strictly completely monotonic. For certain potential functions, some of these configurations were previously analyzed by Yudin, Kolushov, and Andreev; we build on their techniques. We also generalize our results to other compact two-point homogeneous spaces, and we conclude with an extension to Euclidean space.
ISSN:0894-0347
1088-6834
DOI:10.1090/s0894-0347-06-00546-7