Skip to main content
    • Aa
    • Aa
  • Get access
    Check if you have access via personal or institutional login
  • Cited by 12
  • Cited by
    This article has been cited by the following publications. This list is generated based on data provided by CrossRef.

    Beltrán, Carlos 2015. A Facility Location Formulation for Stable Polynomials and Elliptic Fekete Points. Foundations of Computational Mathematics, Vol. 15, Issue. 1, p. 125.

    Brauchart, Johann S. and Grabner, Peter J. 2015. Distributing many points on spheres: Minimal energy and designs. Journal of Complexity, Vol. 31, Issue. 3, p. 293.

    Marques, R. Bouville, C. Ribardière, M. Santos, L. P. and Bouatouch, K. 2013. Spherical Fibonacci Point Sets for Illumination Integrals. Computer Graphics Forum, Vol. 32, Issue. 8, p. 134.

    Aistleitner, C. Brauchart, J. S. and Dick, J. 2012. Point Sets on the Sphere $\mathbb{S}^{2}$ with Small Spherical Cap Discrepancy. Discrete & Computational Geometry,

    Brauchart, Johann S. and Dick, Josef 2012. Quasi–Monte Carlo rules for numerical integration over the unit sphere $${\mathbb{S}^2}$$. Numerische Mathematik, Vol. 121, Issue. 3, p. 473.

    Hou, Xiaorong and Shao, Junwei 2011. Spherical Distribution of 5 Points with Maximal Distance Sum. Discrete & Computational Geometry, Vol. 46, Issue. 1, p. 156.

    Narcowich, F.J. Sun, X. Ward, J.D. and Wu, Z. 2010. LeVeque type inequalities and discrepancy estimates for minimal energy configurations on spheres. Journal of Approximation Theory, Vol. 162, Issue. 6, p. 1256.

    Kolushov, A. V. Yudin, V. A. Колущов, А. В. and Удин, В. А. 1997. Extremal dispositions of points on the sphere. Analysis Mathematica, Vol. 23, Issue. 1, p. 25.

    Saff, E. B. and Kuijlaars, A. B. J. 1997. Distributing many points on a sphere. The Mathematical Intelligencer, Vol. 19, Issue. 1, p. 5.

    Blümlinger, Martin 1991. Slice discrepancy and irregularities of distribution on spheres. Mathematika, Vol. 38, Issue. 01, p. 105.

    Moser, William O.J. 1991. Problems, problems, problems. Discrete Applied Mathematics, Vol. 31, Issue. 2, p. 201.

    Beck, József 1987. Irregularities of distribution. I. Acta Mathematica, Vol. 159, Issue. 1, p. 1.


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

  • József Beck (a1)
  • DOI:
  • Published online: 01 February 2010

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.

Linked references
Hide All

This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

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.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

  • ISSN: 0025-5793
  • EISSN: 2041-7942
  • URL: /core/journals/mathematika
Please enter your name
Please enter a valid email address
Who would you like to send this to? *