On the sum of squared distances in the Euclidean plane

Abstract.

Let x ₁,..., x _n be points in the d-dimensional Euclidean space E ^d with $\| x_{i}-x_{j}\| \le 1$ for all $1 \leqq i,j \leqq n$, where $\| .\|$ denotes the Euclidean norm. We ask for the maximum M(d,n) of $\textstyle\mathop\sum\limits _{i,\,j=1}^{n}\| x_{i}-x_{j}\| ^{2}$ (see [4]). This paper deals with the case d = 2. We calculate M(2, n) and show that the value M(2, n) is attained if and only if the points are distributed as evenly as possible among the vertices of a regular triangle of edge-length 1. Moreover we give an upper bound for the value $\textstyle\mathop\sum\limits _{i,\,j=1}^{n}\| x_{i}-x_{j}\|$, where the points x ₁,...,x _n are chosen under the same constraints as above.