Regular cones in Hilbert space
type: Item
instance of: scholarly article
title: Regular cones in Hilbert space
zbMATH Open document ID: 0882.46011
DOI: 10.1007/BF02104767
author: V. T. Khudalov
published in: Siberian Mathematical Journal
publication date: 1997-06-18T00:00:00Z
review text: Let \(H\) be a real Hilbert space of dimension exceeding 1, \(a\) an element of \(H\) with unit norm and \(0<\alpha\leq 1\). The cone \(\{x\in H:(a, x)\geq \alpha|x|\}\) is shown to be regular if and only if \(\alpha=1/\sqrt 2\). The author then goes on to prove that a strictly convex Banach space \(X\) is a Hilbert space if and only if, given \(a, b\in x\) with \(|a-b|=|a+b|\), there is an almost lattical cone containing the elements. (A regular cone \(K\) is almost lattical if for each \(x\in X\), there are \(u, v\in K\) with \(x= u-v\) and \(|u-\lambda v|=|u+\lambda v|\) for each real \(\lambda\)).
Mathematics Subject Classification ID: 46C15
Mathematics Subject Classification ID: 46A40
Mathematics Subject Classification ID: 46C05
zbMATH DE Number: 1023990
zbMATH Keywords: strictly convex Banach space
zbMATH Keywords: Hilbert space
zbMATH Keywords: almost lattical cone
zbMATH Keywords: regular cone
reviewed by: Edward J. Barbeau
MaRDI profile type: Publication
cites work: Q3283133
cites work: Q3852698
cites work: Q3213662
description: scientific article; zbMATH DE number 1023990