On a polygon equality problem
An inequality conjectured by Bernius and Blanchard to hold for any two sets of vectors \(\{x_i\}^n_{i=1}\) and \(\{y_i\}^n_{i=1}\) in a finite-dimensional Euclidean space is the following: \[ \sum_{i< j}\|x_i- x_j\|+ \sum_{i<j}\|y_i- y_j\|\leq \sum^n_{i,j= 1}\|x_i- y_j\|\tag{\(*\)} \] with equality holding if and only if one set is simply a permutation of the other. The proof of the inequality itself follows from a result of \textit{C. J. Lenard}, \textit{A. M. Tonge} and \textit{A. Weston} [Mich. Math. J. 44, No. 1, 37-45 (1997; Zbl 0889.46014)], and in this paper the authors determine when the conjectured equality of \((*)\) holds for vectors in an \(L^1\)-space. From this it follows that the equality holds not only for sets of vectors in a Euclidean space, but for such sets in an arbitrary Hilbert space.
scientific article; zbMATH DE number 1215662