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Acta Mathematica Hungarica

, Volume 98, Issue 4, pp 311–321 | Cite as

On extremal point distributions in the Euclidean plane

  • F. Pillichshammer
Article

Abstract

We ask for the maximum σ n γ of Σ i,j=1 n x i-x jγ, where x 1,χ,x n are points in the Euclidean plane R 2 with ‖xi-xj‖ ≦1 for all 1≦ i,jn and where ‖.‖γ denotes the γ-th power of the Euclidean norm, γ ≧ 1. (For γ =1 this question was stated by L. Fejes Tóth in [1].) We calculate the exact value of σ n γ for all γ γ 1,0758χ and give the distributions which attain the maximum σ n γ . Moreover we prove upper bounds for σ n γ for all γ ≧ 1 and calculate the exact value of σ 4 γ for all γ ≧ 1.

Euclidean norm sum of distances 

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References

  1. [1]
    L. Fejes Tóth, Über eine Punktverteilung auf der Kugel, Acta Math. Acad. Sci. Hungar., 10 (1959), 13-19.zbMATHMathSciNetCrossRefGoogle Scholar
  2. [2]
    F. Pillichshammer, On the sum of squared distances in the Euclidean plane, Arch. Math., 74 (2000), 472-480.zbMATHMathSciNetCrossRefGoogle Scholar
  3. [3]
    F. Pillichshammer, A note on the sum of distances in the Euclidean plane, Arch. Math., 77 (2001), 195-199.zbMATHMathSciNetCrossRefGoogle Scholar
  4. [4]
    K. Reinhardt, Extremale Polygone gegebenen Durchmessers, Jahresbericht der deutschen Mathematiker Vereinigung, 31 (1922), 251-270.Google Scholar
  5. [5]
    H. S. Witsenhausen, On the maximum of the sum of squared distances under a diameter constraint, Amer. Math. Monthly, 81 (1974), 1100-1101.zbMATHMathSciNetCrossRefGoogle Scholar
  6. [6]
    R. Wolf, Averaging distances in real quasihypermetric Banach spaces of finite dimension, Israel J. Math., 110 (1999), 125-152.zbMATHMathSciNetGoogle Scholar

Copyright information

© Kluwer Academic Publishers/Akadémiai Kiadó 2003

Authors and Affiliations

  • F. Pillichshammer
    • 1
  1. 1.Institut für AnalysisUniversität, Linz Altenbergerstrasse 69LinzAustria

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