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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)

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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2. J. Beck . On a problem of K. F. Roth concerning irregularities of point distribution. Inventiones Math., 74 (1983). 477487.

3. L. Fejes-Tóth . On the sum of distances determined by a pointset. Acta Math. Acad. Sci. Hungar., 7 (1956), 397401.

7. W. M. Schmidt . Irregularities of distribution IV. Inventiones Math., 7 (1969), 5582.

8. K. B. Stolarsky . Sums of distances between points on a sphere II. Proc. Amer. Math. Soc., 41 (1973), 575582.

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  • ISSN: 0025-5793
  • EISSN: 2041-7942
  • URL: /core/journals/mathematika
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