Sums of distances between points on a sphere

Stolarsky, Kenneth B.

doi:10.1090/s0002-9939-1972-0303418-3

Cited by 34 publications

(42 citation statements)

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“…Several conjectures already exist concerning the asymptotic behavior of E(α, N ) for special values of α: see Alexander [1], Stolarsky [26,27] and Beck [2] for α = 1; Glasser [17] and Erber [13] for α = −1. Here we formulate a general conjecture for E(α, N ) and discuss our numerical experiments for the cases N ≤ 200, α = 0, ±1, that support it.…”

Section: Conjectures For Asymptotics Of E(α N )mentioning

confidence: 99%

Minimal Discrete Energy on the Sphere

Rakhmanov¹,

Saff²,

Zhou³

1994

Mathematical Research Letters

297

237

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Abstract. We investigate the energy of arrangements of N points on the surface of a sphere in R 3 , interacting through a power law potential V = r α , −2 < α < 2, where r is Euclidean distance. For α = 0, we take V = log(1/r). An area-regular partitioning scheme of the sphere is devised for the purpose of obtaining bounds for the extremal (equilibrium) energy for such points. For α = 0, finer estimates are obtained for the dominant terms in the minimal energy by considering stereographical projections on the plane and analyzing certain logarithmic potentials. A general conjecture on the asymptotic form (as N → ∞) of the extremal energy, along with its supporting numerical evidence, is presented. Also we introduce explicit sets of points, called "generalized spiral points", that yield good estimates for the extremal energy. At least for N ≤ 12, 000 these points provide a reasonable solution to a problem of M. Shub and S. Smale arising in complexity theory.

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Section: Conjectures For Asymptotics Of E(α N )mentioning

confidence: 99%

Minimal Discrete Energy on the Sphere

Rakhmanov¹,

Saff²,

Zhou³

1994

Mathematical Research Letters

297

237

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“…To obtain our inequality, the method of Pólya and Szegö is extended to ultraspherical harmonics, and further refined. For X = 1 very good estimates of I(K) are available ([l], [33]); for 0 < A < 1 see also [32].…”

Section: In This Case We Show That I(k) Has Amentioning

confidence: 99%

Extremal problems of distance geometry related to energy integrals

Alexander¹,

Stolarsky²

1974

Trans. Amer. Math. Soc.

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ABSTRACT. Let K be a compact set, % a prescribed family of (possibly signed) Borel measures of total mass one supported by K, and / a continuous real-valued function on Kx K. We study the problem of determining for which /x e% (if any) the energy integral 1(K, fi) = Siefi(ßx' y)df4.x)df/.y) is maximal, and what this maximum is. The more symmetry K has, the more we can say; our results are best when K is a sphere.In particular, when % is atomic we obtain good upper bounds for the sums of powers of all (?) distances determined by n points on the surface of a sphere. We make use of results from Schoenberg's theory of metric embedding, and of techniques devised by Polya and Szegö for the calculation of transfinite diameters.

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“…When α = −1 this is, except for some small values of N , a long standing open problem in discrete geometry. We refer the reader to [6], [20], [21], [11] for more on this problem. For generalizations of the α-energy problems to higher dimension see [13], [3].…”

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confidence: 99%