Universally optimal distribution of points on spheres

@article{Cohn2006UniversallyOD,
  title={Universally optimal distribution of points on spheres},
  author={Henry Cohn and Abhinav Kumar},
  journal={arXiv: Metric Geometry},
  year={2006}
}
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… 

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References

SHOWING 1-10 OF 86 REFERENCES
Positive Definite Functions on Spheres
In this paper we study strictly positive definite functions on the unit sphere of the m-dimensional Euclidean space. Such functions can be used for solving a scattered data interpolation problem on
TWO-POINT HOMOGENEOUS SPACES
Most of the classical metric spaces with metric "d" have the property that given any two pairs of points a, , a2 , bi, b2 of the space with d(al , a2) = d(b1, b2), there is an isometry of the space
On Korkin-Zolotarev’s construction
It is proved that the minimum of the potential energy of 24 unit charges placed on the sphere in R is equal to 637975/72. This minimum is attained on the minimal vectors of the lattice ES. Let χ =
An Extremal Property Of The Icosahedron
The purpose of this article is to prove that the product of all possible pairwise distances between twelve points located on the unit sphere in a 3-dimensional Euclidean space is not greater than 2
Packing Lines, Planes, etc.: Packings in Grassmannian Spaces
TLDR
A reformulation of the problem gives a way to describe n-dimensional subspaces of m-space as points on a sphere in dimension ½(m–l)(m+2), which provides a (usually) lowerdimensional representation than the Plucker embedding and leads to a proof that many of the new packings are optimal.
Orthogonal Polynomials and Special Functions
Asymptotics of Jacobi matrices for a family of fractal measures GÖKALP APLAN BILKENT UNIVERSITY, TYRKEY There are many results concerning asymptotics of orthogonal polynomials and Jacobi matrices
The nonexistence of certain tight spherical designs
In this paper, the nonexistence of tight spherical designs is shown in some cases left open to date. Tight spherical 5-designs may exist in dimension n = (2m + 1)2 − 2, and the existence is known
The D 4 Root System Is Not Universally Optimal
TLDR
It is proved that the D 4 root system (equivalently, the set of vertices of the regular 24-cell) is not a universally optimal spherical code, and it is conjectured that every 5-design consisting of 24 points in S 3 is in a 3-parameter family.
...
1
2
3
4
5
...