Monotone path-connectedness and solarity of Menger-connected sets in Banach spaces
type: Item
instance of: scholarly article
title: Monotone path-connectedness and solarity of Menger-connected sets in Banach spaces
zbMATH Open document ID: 1303.41018
published in: Izvestiya: Mathematics
publication date: 2014-10-17T00:00:00Z
full work available at URL: https://arxiv.org/abs/1312.4338
Mathematics Subject Classification ID: 41A65
Mathematics Subject Classification ID: 46B20
Mathematics Subject Classification ID: 52A01
zbMATH DE Number: 6358161
zbMATH Keywords: sun
zbMATH Keywords: acyclic set
zbMATH Keywords: cell-like set
zbMATH Keywords: Cech homology
zbMATH Keywords: monotone path-connected set
zbMATH Keywords: Menger connectedness
zbMATH Keywords: d-convexity
zbMATH Keywords: Menger convexity
zbMATH Keywords: Rainwater-Simons theorem
author: A. R. Alimov
MaRDI profile type: Publication
arXiv ID: 1312.4338
DOI: 10.1070/IM2014V078N04ABEH002702
review text: Let \(X\) be a real normed space with dual \(X^*\). For \(\emptyset \neq M\subset X\) denote by \(m(M)\) the intersection of all closed balls containing \(M\). The set \(M\) is called Menger connected if \(m(\{x,y\})\cap M\neq\{x,y\}\) for all \(x,y\in M.\) The set \(M\) is called monotone path connected if for every \(x,y\in M\) there exists a continuous path \(k:[0,1]\to M\) such that \(f(k(\cdot))\) is monotone for every \(f\in\) ext\(S^*\) - the set of extreme points of the unit sphere \(S^*\) of \(X^*\). The set \(M\) is called a sun if for every \(x\in X\setminus M\) there exists \(y\in P_Mx\) (\(P_M\) is the metric projection on \(M\)) such that \(y\in P_M(y+t(x-y))\) for all \(t\geq 0.\) A monotone path connected set is Menger connected but the converse is true only in some special cases, e.g. for closed subsets of \(c_0\), see, \textit{A. R. Alimov} [Izv. Math. 69, No. 4, 651--666 (2005); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 69, No. 4, 3--18 (2005; Zbl 1095.46009)]. In this paper the author gives conditions under which this converse implication holds. To this end he considers some special classes of Banach spaces. The first one, denoted by (MeI) is formed by all Banach spaces for which \(m(\{x,y\})=[\![x,y]\!]\) for all \(x,y\in X,\) where \([\![x,y]\!]=\{z\in X : \varphi(x)\wedge \varphi(y)\leq \varphi(z)\leq \varphi(x)\vee\varphi(y),\, \forall \varphi\in\)ext\(S^*\}\) denotes the ``interval'' generated by \(x,y\). In \(C(T),\, [\![f,g]\!]=\{h\in C(T) : f(t)\leq h(t)\leq g(t),\, \forall t\in T\}.\) As the author mentions, it is unknown whether all Banach spaces belong to the class (MeI). Another class, denoted by (Ex-\(w^*\)s), is formed by all Banach spaces for which ext\(S^*\) is \(w^*\)-separable. In Theorem 4.1, it is shown that for \(X\in\) (MeI)\(\cap\)(Ex-\(w^*\)s) (in particular, if \(X\) is separable) a closed Menger connected subset \(M\) of \( X\) is path connected provided it is boundedly compact, or \(m(\{x,y\})\) is compact for e…
reviewed by: Stefan Cobzaş
description: scientific article; zbMATH DE number 6358161