Universally optimal distribution of points on spheres

Cohn, Henry¹ ; Kumar, Abhinav^2, 1

¹ Microsoft Research, One Microsoft Way, Redmond, Washington 98052-6399
² Department of Mathematics, Harvard University, Cambridge, Massachusetts 02138

Journal of the American Mathematical Society, Tome 20 (2007) no. 1, pp. 99-148

Résumé

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 $m$ distances between distinct points in it and it is a spherical $(2m-1)$-design. We prove that every sharp configuration minimizes potential energy for all completely monotonic potential functions. Examples include the minimal vectors of the $E_8$ and Leech lattices. We also prove the same result for the vertices of the $600$-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.

Détail

DOI : 10.1090/S0894-0347-06-00546-7

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