Convergence in Hilbert's metric and convergence in direction
type: Item
instance of: scholarly article
title: Convergence in Hilbert's metric and convergence in direction
zbMATH Open document ID: 0542.90073
DOI: 10.1016/0022-247X(83)90220-2
author: Elon Kohlberg
author: Abraham Neyman
published in: Journal of Mathematical Analysis and Applications
publication date: 1983-01-01T00:00:00Z
review text: Let K be a closed pointed cone in a normal linear space X, and for any vectors x,y in K-\(\{\) 0\(\}\), define \(M(x,y)=\inf \{\lambda \geq 0:\lambda y-x\in K\}, m(x,y)=\sup \{\lambda \geq 0:x-y\in K\},\) and \(d(x,y)=\log [M(x,y)/m(x,y)].\) The following result is presented. Theorem. Let (K,d) be as above. Then ''\(d(x_ n,y_ n)\to 0\) implies \(\| x_ n\!'-y_ n\!'\| \to 0\)'' iff K is normal. If the latter condition holds and X is complete then so is \(K'=\{x\in K:\| x\| =1\}.\) Here \(x'=x/\| x\|,\) \(d(\cdot,\cdot)\) is called the Hilbert distance, and K is normal means, \(\inf \{\| x+y\|:x,\quad y\in K'\}>0.\)
Mathematics Subject Classification ID: 90C25
Mathematics Subject Classification ID: 46B15
Mathematics Subject Classification ID: 49J40
Mathematics Subject Classification ID: 90C55
zbMATH DE Number: 3862936
zbMATH Keywords: contraction mapping
zbMATH Keywords: closed pointed cone
zbMATH Keywords: Hilbert distance
reviewed by: Malempati M. Rao
MaRDI profile type: Publication
cites work: Extensions of Jentzsch's Theorem
cites work: Q5331549
cites work: Q5514145
cites work: The Contraction Mapping Approach to the Perron-Frobenius Theory: Why Hilbert's Metric?
cites work: Q5558820
description: scientific article; zbMATH DE number 3862936