On the sum of squared distances in the Euclidean plane (Q1581203)

Let \(x_1, \dots , x_n\) be points in the Euclidean plane with \(\|x_i-x_j\|\leq 1\) for all \(1\leq i,j \leq n\) where \(\|.\|\) denotes the euclidean norm. The author proves that the maximum of \(\sum_{i,j=1}^n \|x_i-x_j\|^2\) 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. This answers a conjecture stated by Witsenhausen in 1974; the conjecture still remains open in \(d\)-dimensional Euclidean space for \(d>2\). The author gives also an upper bound for the value \(\sum_{i,j=1}^n \|x_i-x_j\|\) where the points \(x_1, \dots ,x_n\) are chosen under the same constraints as above.