On extremal point distributions in the Euclidean plane

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In this Subject: <a title="Mathematics and Statistics" href="/content/subcat;jsessionid=s53pakemkb6a.alexandra?j_subject=195&j_availability=">Mathematics and Statistics
By this author: <a href="/search;jsessionid=s53pakemkb6a.alexandra?option2=author&value2=Pillichshammer+F">Pillichshammer F.

Author: Pillichshammer F.

Source: Acta Mathematica Hungarica, Volume 98, Number 4, 2003 , pp. 311-321(11)

Publisher: Springer

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Abstract:

We ask for the maximum sigma _n of Sigma ⁿ_i,j₌₁||x_it-x_j||, where x₁,…,x_n are points in the Euclidean plane R² with ||x_i-x_j|| 1 for all 1 i,j n and where ||.|| denotes the gamma -th power of the Euclidean norm, gamma 1. (For gamma =1 this question was stated by L. Fejes Tóth in [1].) We calculate the exact value of sigma _n for all gamma gamma 1,0758… and give the distributions which attain the maximum sigma _n. Moreover we prove upper bounds for sigma _n for all gamma 1 and calculate the exact value of sigma ₄ for all gamma 1.