Advertisement

Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

On a new method for constructing good point sets on spheres

  • 97 Accesses

  • 12 Citations

Abstract

We use quadrature formulas with equal weights in order to constructN point sets on spheres ind-space (d ≥ 3) which are almost optimal with respect to a discrepancy concept, based on distance functions (potentials) and distance functionals (energies). By combining this approach with the probabilistic method, we obtain almost best possible approximations of balls by zonotopes, generated byN segments of equal length.

References

  1. 1.

    J. Beck, Some upper bounds in the theory of irregularities of distribution.Acta Arith. 43 (1984), 115–130.

  2. 2.

    U. Betke and P. McMullen, Estimating the sizes of convex bodies by projections.J. London Math. Soc. (2)27 (1983), 525–538.

  3. 3.

    E. D. Bolker, A class of convex bodies.Trans. Amer. Math. Soc. 145 (1969), 323–345.

  4. 4.

    J. Bourgain and J. Lindenstrauss, Distribution of points on spheres and approximation by zonotopes.Israel J. Math. 64 (1988), 25–32.

  5. 5.

    J. Bourgain, J. Lindenstrauss, and V. Milman, Approximation of zonoids by zonotopes.Acta Math. 162 (1989), 73–141.

  6. 6.

    J. Linhart, Approximation of a ball by zonotopes using uniform distribution on the sphere.Arch. Math. 53 (1989), 82–87.

  7. 7.

    A. Lubotzky, R. Phillips, and P. Sarnak, Hecke operators and distributing points on the sphere, I.Comm. Pure Appl. Math. 39 (1986), 149–186.

  8. 8.

    P. Sjögren, Estimates of mass distributions from their potentials and energies.Ark. Mat. 10 (1972), 59–77.

  9. 9.

    K. B. Stolarsky, Sums of distances between points on a sphere II.Proc. Amer. Math. Soc. 41 (1973), 575–582.

  10. 10.

    G. Wagner, On the product of distances to a point set on the sphere.J. Austral. Math. Soc. Ser. A 47 (1989), 466–482.

  11. 11.

    G. Wagner, On means of distances on the surface of a sphere. I. Lower bounds.Pacific J. Math. 144 (1990), 389–398. II. Upper bounds.Pacific J. Math. 154 (1992), 381–396.

  12. 12.

    G. Wagner, On averaging sets.Monatsh. Math. 111 (1991), 69–78.

Download references

Author information

Additional information

Editors' note: We learned with sadness of Gerold Wagner's untimely death as a result of an avalanche in the Alps shortly after the submission of this paper. When one of the referees, Joram Lindenstrauss, suggested that Wagner's results might be extended to dimensions >6, we invited Professor Lindenstrauss to submit a paper containing that extension which we would publish alongside the Wagner paper. The result is the paper by Bourgain and Lindenstrauss that follows the present one.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Wagner, G. On a new method for constructing good point sets on spheres. Discrete Comput Geom 9, 111–129 (1993). https://doi.org/10.1007/BF02189312

Download citation

Keywords

  • Convex Body
  • Random Point
  • Discrete Comput Geom
  • Quadrature Formula
  • Main Lemma