The Fermat-Weber problem and inner product spaces
type: Item
instance of: scholarly article
title: The Fermat-Weber problem and inner product spaces
zbMATH Open document ID: 0818.41027
author: Roland Durier
published in: Journal of Approximation Theory
publication date: 1994-08-18T00:00:00Z
review text: Let \(\{x_ i\}^ n_{i= 1}\) be points in a real Banach space \(X\). Let \(w_ i> 0\). For \(x\in X\), let \[ F(x)= \sum^ n_{i= 1} w_ i\| x- x_ i\|. \] Let \(m= \min\{F(x): x\in X\}\) and let \(M= \{x: F(x)= m\}\). \(M\) is always convex, and if \(X\) is a Hilbert space, then \(M\cap\text{conv}\{x_ i\}^ n_{i= 1}\neq\emptyset\). (Where \(\text{conv }K\) is the convex hull of \(K\).) This paper shows that if \(M\cap \text{conv}\{x_ i\} ^ n_{i= 1}\neq \emptyset\) for all finite sets \(\{x_ i\}^ n_{i= 1}\), then \(X\) is an innerproduct space.
Mathematics Subject Classification ID: 41A65
Mathematics Subject Classification ID: 41A50
Mathematics Subject Classification ID: 46C05
Mathematics Subject Classification ID: 90C25
zbMATH DE Number: 617501
zbMATH Keywords: Hilbert space
reviewed by: Daniel Wulbert
MaRDI profile type: Publication
full work available at URL: jath.1994.1070
OpenAlex ID: W2026227108
DOI: 10.1006/JATH.1994.1070
description: scientific article; zbMATH DE number 617501