Skip to main content
SpringerLink
Log in

On extremums of sums of powered distances to a finite set of points

  • Original Paper
  • Published:
Geometriae Dedicata Aims and scope Submit manuscript

Abstract

In this paper we investigate the extremal properties of the sum

$$\begin{array}{ll} \sum\limits_{i=1}^n|MA_i|^{\lambda}, \end{array}$$

where A i are vertices of a regular simplex, a cross-polytope (orthoplex) or a cube and M varies on a sphere concentric to the sphere circumscribed around one of the given polytopes. We give full characterization for which points on Γ the extremal values of the sum are obtained in terms of λ. In the case of the regular dodecahedron and icosahedron in \({\mathbb{R}^3}\) we obtain results for which values of λ the corresponding sum is independent of the position of M on Γ. We use elementary analytic and purely geometric methods.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Ambros, G., Ball, K., Erdélyi, T.: Chebyshev constants for the unit circle. J. Lond. Math. Soc. (to appear).

  2. Armentano D., Beltran C., Schub M.: Minimazing the discrete logarithmic energy on the sphere: the role of random polynomials. Trans. Am. Math. Soc. 363, 2955–2965 (2011)

    Article  MATH  Google Scholar 

  3. Ballinger B., Blekherman G., Cohn H., Giansiracusa N., Kelly E., Schuermann A.: Experimental study of energy-minimizing point configurations on spheres. Exp. Math. 18, 257–283 (2009)

    Article  MATH  Google Scholar 

  4. Cohn H., Kumar A.: Universally optimal distribution of points on a sphere. J. Am. Math. Soc. 20, 99–148 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  5. Erdélyi, T., Saff, E.B.: Polarization inequalities in higher dimensions. arXiv:1206.4729

  6. Hardin, D., Kendall, A., Saff, E.B.: Polarization optimality of equally spaced points on the circle for discrete potentials. arXiv:1208.5261

  7. Nikolov N., Rafailov R.: On the sum of powered distances to certain sets of points on the circle. Pac. J. Math. 253, 157–168 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  8. Stolarsky K.: The sum of the distances to certain pointsets on the unit circle. Pac. J. Math 59, 241–251 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  9. Stolarsky K.: The sum of the distances to N poinst on a sphere. Pac. J. Math 57, 563–573 (1975)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad. G. Bonchev 8, 1113, Sofia, Bulgaria

    Nikolai Nikolov

  2. University of California, Berkeley, CA, USA

    Rafael Rafailov

Authors
  1. Nikolai Nikolov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Rafael Rafailov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Nikolai Nikolov.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Nikolov, N., Rafailov, R. On extremums of sums of powered distances to a finite set of points. Geom Dedicata 167, 69–89 (2013). https://doi.org/10.1007/s10711-012-9804-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10711-012-9804-3

Keywords

Mathematics Subject Classification (2000)

Search

Navigation