On extremal point disributions in the Euclidean plane (Q1410418)

For \(n\geq 1\) points \(x_1,\dots,x_n\in R^2\) with \(\|x_i-x_j\|\leq 1\), \(1\leq i\), \(j\leq n\), and a real number \(\gamma\leq 1\) the author investigates the maximum value \(\delta^\gamma_n\) of \[ \sum^n_{i=1}\|x_i-x_j\|^\gamma. \] There exist several results on \(\gamma=1\) and \(\gamma\leq 2\), but nothing seems to be known about \(1< \gamma<2\). In the present paper the exact values of \(\delta^\gamma_n\) for any \(\gamma\geq 1,0758\dots\) is given, and for all \(1\leq \gamma \leq 1,0758\dots\) related upper bounds are derived. Some further (exact) results are given for special values of \(n\).