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On extremal point distributions in the Euclidean plane

Abstract

We ask for the maximum σ γn of Σ ni,j=1 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 γn for all γ γ 1,0758χ and give the distributions which attain the maximum σ γn . Moreover we prove upper bounds for σ γn for all γ ≧ 1 and calculate the exact value of σ γ4 for all γ ≧ 1.

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Pillichshammer, F. On extremal point distributions in the Euclidean plane. Acta Mathematica Hungarica 98, 311–321 (2003). https://doi.org/10.1023/A:1022838328562

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  • Euclidean norm
  • sum of distances