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Sums of distances between points on a sphere — an application of the theory of irregularities of distribution to discrete Geometry

  • József Beck (a1)
Abstract

This paper is concerned with the solution of the following interesting geometrical problem. For what set of n points on the sphere is the sum of all Euclidean distances between points maximal, and what is the maximum?

Our starting point is the following surprising “invariance principle” due to K. B. Stolarsky: The sum of the distances between points plus the quadratic average of a discrepancy type quantity is constant. Thus the sum of distances is maximized by a well distributed set of points. We now introduce some notation to make the statement more precise.

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1.Beck J.. Some upper bounds in the theory of irregularities of distribution. Acta Arith., 43 (1983), 115130.
2.Beck J.. On a problem of K. F. Roth concerning irregularities of point distribution. Inventiones Math., 74 (1983). 477487.
3.Fejes-Tóth L.. On the sum of distances determined by a pointset. Acta Math. Acad. Sci. Hungar., 7 (1956), 397401.
4.Harman G.. Sums of distances between points on a sphere. Internal. J. Math., 5 (1982), 707714.
5.Olver F. W. J.. Asymptotics and Special Functions (Academic Press, New York and London, 1974).
6.Roth K. F.. Remark concerning integer sequences. Acta Arithmetica, 9 (1964), 257260.
7.Schmidt W. M.. Irregularities of distribution IV. Inventiones Math., 7 (1969), 5582.
8.Stolarsky K. B.. Sums of distances between points on a sphere II. Proc. Amer. Math. Soc., 41 (1973), 575582.
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Mathematika
  • ISSN: 0025-5793
  • EISSN: 2041-7942
  • URL: /core/journals/mathematika
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