The sum of squared distances under a diameter constraint, in arbitrary dimension
Let \(X=\{ x_1, \dots , x_n\}\) be a set of \(n\) points in the \(d\)-dimensional Euclidean space, with unit diameter. In 1970, Witsenhausen conjectured that the maximum \(M(d,n)\) of \(\sum _{i,j=1}^n \| x_i-x_j\| ^2\) in dimension \(d\) is attained if and only if the points are distributed as evenly as possible among the vertices of a regular \(d\)-dimensional simplex of edge-length 1. The conjecture was proved for \(n=2\) by Pillichshammer in 2000. In this paper the authors prove the conjecture in any dimension. They first prove the conjecture for \(n=d+2\) and then they use induction to extend the proof to the case \(n>d+2\). To prove the conjecture for \(n=d+2\) they show that for all \(X=\{ x_1, \dots , x_n\}\) with unit diameter, there exists a point \(x_k \in X\) verifying \[ \sum _{i=1, i\neq k}^{d+2} \| x_i-x_k \| ^2 \leq d. \]
scientific article; zbMATH DE number 5351799