Uniform convexity in nonsymmetric spaces
type: Item
instance of: scholarly article
title: Uniform convexity in nonsymmetric spaces
zbMATH Open document ID: 1489.46022
DOI: 10.1134/S0001434621110146
author: Igor G. Tsar'kov
published in: Mathematical Notes
publication date: 2022-01-11T00:00:00Z
review text: An asymmetric norm on a real vector space \(X\) is a positively definite sublinear functional \(\|\cdot| :X\to \mathbb{R}_+\) (i.e., a positively homogeneous, sublinear and positively definite functional). Equivalently, \(\|\cdot|\) is the Minkowski gauge of an absorbing, radially bounded and convex subset of \(X\). An asymmetric normed space \((X,\|\cdot|)\) is a real vector space \(X\) endowed with an asymmetric norm \({\|\cdot|}\). The topology of an asymmetric normed space is \(T_1\), but not necessarily \(T_2\) (see [\textit{S.\,Cobzaş}, Functional analysis in asymmetric normed spaces. Basel: Birkhäuser (2013; Zbl 1266.46001)]). A left Cauchy sequence (also called left \(K\)-Cauchy by some authors) is a sequence \(\{x_n\}\) in \(X\) such that for every \(\varepsilon>0\) there exists \(N\in\mathbb{N}\) such that \(\|x_m-x_n|<\varepsilon\) for all \(m\ge n\ge N\). The space \(X\) is called left complete if every left Cauchy sequence is convergent. For a nonempty subset \(M\) of \(X\) and \(x\in X\), put \(\rho(x,M)=\inf\{\|y-x|: y\in M\}\), the distance from \(x\) to \(M\), and \(P_Mx= \{y\in M:\|y-x|=\rho(x,M)\}\), the metric projection. For \(x, y\in S_X\) (the unit sphere of \(X\)) and \(a\in[0,1]\), put \(\Delta(a)=\|x-ay|+ a\|y|-\|x|\). The space \(X\) is called uniformly convex if for every \(\varepsilon>0\) and \(a\in(0,1]\) there exists \(\delta>0\) such that for all \(x, y\in S_X\), \(\Delta(a) < \delta\) implies \(\|x-\mu y|\le\varepsilon\) for some \(\mu\in [1-\varepsilon,1].\) The space \(X\) is called locally uniformly convex if for all \(y\in S_X\), \(\varepsilon>0\) and \(a\in(0,1]\) there exists \(\delta>0\) such that for any \(x\in S_X\), \( \Delta(a)<\delta\) implies \(\|x-y|\le \varepsilon\). As expected, a uniformly convex space is locally uniformly convex. With these definitions, most properties of uniformly convex and locally uniformly convex normed spaces are preserved in the asymmetric case, too. For instance, if \((X,\|\cdot|)\) is a unif…
Mathematics Subject Classification ID: 46B20
Mathematics Subject Classification ID: 46E30
Mathematics Subject Classification ID: 41A65
zbMATH DE Number: 7455556
zbMATH Keywords: asymmetric normed spaces
zbMATH Keywords: best approximation
zbMATH Keywords: approximative compactness
zbMATH Keywords: uniform convexity
zbMATH Keywords: local uniform convexity
reviewed by: Stefan Cobzaş
MaRDI profile type: Publication
cites work: Functional Analysis in Asymmetric Normed Spaces
cites work: Separation of Convex Sets and Best Approximation in Spaces with Asymmetric Norm
cites work: The Banach-Mazur theorem for spaces with an asymmetric distance
cites work: On the structure of the complements of Chebyshev sets
cites work: Q5249470
cites work: Approximative properties of sets and continuous selections
cites work: Weakly monotone sets and continuous selection in asymmetric spaces
cites work: Continuous selections for metric projection operators and for their generalizations
cites work: Continuous selections in asymmetric spaces
full work available at URL: s0001434621110146
OpenAlex ID: W4200110057
description: scientific article; zbMATH DE number 7455556