On a polygon equality problem
type: Item
instance of: scholarly article
title: On a polygon equality problem
zbMATH Open document ID: 0931.46018
author: Ludwig Elsner
author: M. Zippin
author: Lixing Han
author: I. Koltracht
author: M. Neumann
published in: Journal of Mathematical Analysis and Applications
publication date: 2000-02-20T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 46B25
Mathematics Subject Classification ID: 46B20
zbMATH DE Number: 1215662
zbMATH Keywords: polygon inequality
reviewed by: James R. Holub
MaRDI profile type: Publication
full work available at URL: jmaa.1998.5955
OpenAlex ID: W1990782233
cites work: Q5669761
cites work: Generalized roundness and negative type
cites work: Absolutely summing operators in $ℒ_{p}$-spaces and their applications
cites work: Q4188284
DOI: 10.1006/JMAA.1998.5955
description: scientific article; zbMATH DE number 1215662