Monotone path-connectedness and solarity of Menger-connected sets in Banach spaces (Q2925758)

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 every \(x,y\in X\). If the case that the set \(M\) is boundedly compact, then it is also a sun and has some good properties expressed in terms of Cech homology and cohomology (e.g. being cell-like), extending some results of L. P. Vlasov on solarity of acyclic sets.NEWLINENEWLINEIn passing, he gives an extension to the class (Ex-\(w^*\)s) of Rainwater-Simons theorem on the relation between the \(w^*\)-convergence of sequences in \(X\) and the convergence on the extreme points of \(S^*\).