Sums of Distances Between Points on a Sphere

Stolarsky, Kenneth B.

doi:10.2307/2037644

Cited by 49 publications

(56 citation statements)

References 10 publications

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“…R. Alexander [1], J. Beck [3], and K. B. Stolarsky [23] have obtained results for the maximal distance sums. A first step toward this goal is to determine precise asymptotics for the extremal energies.…”

Section: Asymptotics For Extremal Energiesmentioning

confidence: 99%

Distributing many points on a sphere

Saff

Kuijlaars

1997

The Mathematical Intelligencer

962

676

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he problem of distributing a large number of points uniformly over the surface of a tsphere has not only inspired mathematical researchers, it has the attracted attention ~f:/t/~176 ::t:u;hifi;l:: ::/i~:l :i~t;~ ogous problem is simply that of uniformly distributing points on the circumference of a disk, and equally spaced points provide an obvious answer. So we are faced with this question: What sets of points on the sphere imitate the role of the roots of unity on the unit circle? One way such points can be generated is via optimization with respect to a suitable criterion such as "generalized energy." Although there is a large and growing literature concerning such optimal spherical configurations of N points when N is "small," here we shall focus on this question from an asymptotic perspective (N--~ ~).The discovery of stable carbon-60 molecules (Kroto, et al., 1985)* with atoms arranged in a spherical (soccer ball) pattern has had a considerable influence on current scientific pursuits. The study of this C60 buckminsterfifllerene also has an elegant mathematical component, revealed by F.R.K. Chung, B. Kostant, and S. Sternberg [5]. Now the search is on for much larger stable carbon molecules! Although such molecules are not expected to have a strictly spherical structure (due to bonding constraints), the construction of large stable configurations of spherical points is of interest here, as an initial step in hypothesizing more complicated molecular net structures.In electrostatics, locating identical point charges on the sphere so that they are in equilibrium with respect to a Coulomb potential law is a challenging problem, sometimes referred to as the dual problem for stable molecules.Certainly, uniformly distributing many points on the sphere has important applications to the field of computation. Indeed, quadrature formulas rely on appropriately chosen sampled data-points in order to approximate area integrals by taking averages in these points. Another example arises in the study of computational complexity, where M. Shub and S. Smale [22] encountered the problem of determining spherical points that maximize the product of their mutual distances.These various points of view clearly lead to different extremal conditions imposed on the distribution of N points. Except for some special values of N (e.g., N = 2, 3, 6, 12, 24) these various conditions yield different optimal configurations. However, and this is the main theme of this article, the general pattern for optimal configurations is the same: points (for N large) appear to arrange themselves according to a hexagonal pattern that is slightly perturbed in order to fit on the sphere.To make this more precise, we introduce some notation. We denote by S 2 the unit sphere in the Euclidean space R3: S 2 = {x E R 3 : Jxt = 1}.Lebesgue (area) measure on S 2 is denoted by ~, so that *

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Section: Asymptotics For Extremal Energiesmentioning

confidence: 99%

Distributing many points on a sphere

Saff

Kuijlaars

1997

The Mathematical Intelligencer

962

676

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“…Γppn`1q{2q is a universal constant that depends only on the dimension of the sphere. This relation was proved by Stolarsky [35] and is currently known as Stolarsky's invariance principle. The average distance on the sphere is given by ş π 0 2 sinpθ{2q sin n´1 θdθ{ ş π 0 sin n´1 θdθ, which evaluates to W pS n q " 2 n Γppn `1q{2q Since D L2 pZ N q ě 0, the following bound is immediate: For any code…”

Section: Introduction: Sum Of Distances and Related Problemsmentioning

confidence: 89%

“…Formula (35) is instrumental in the representation (5) of the ULB and the proof of its optimality in [11]. The quantity f 0 N ´f p1q which provides lower linear programming bounds for the energy E h pZ N q (assuming that certain conditions on the polynomial f ptq are met) is computed by (35) to give the right-hand side of (5). The reason for this is that f pα i q " hpα i q from Hermite interpolation.…”

Section: Derivation Of the Necessary Parametersmentioning

confidence: 99%

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Bounds for the sum of distances of spherical sets of small size

Barg¹,

Boyvalenkov²,

Stoyanova³

2021

Preprint

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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].…”

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confidence: 99%

Extremal Problems of Distance Geometry Related to Energy Integrals

Alexander¹,

Stolarsky²

1974

Transactions of the American Mathematical Society

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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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Sums of Distances Between Points on a Sphere

Abstract: Abstract.An upper bound for the sum of the Ath powers of all distances determined by N points on a unit sphere is given for

Cited by 49 publications

References 10 publications

Distributing many points on a sphere

Distributing many points on a sphere

Bounds for the sum of distances of spherical sets of small size

Extremal Problems of Distance Geometry Related to Energy Integrals

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