Mazur spaces and 4.3-intersection property of \((BM)\)-spaces (Q2830923)

Let \(X\) be a real normed space and \(B[x,r]\), and \(B(x,r)\) the closed, respectively open, balls in \(X\). For \(x,y\in X\), denote by \(m(x,y)\) the intersection of all closed balls containing \(x,y\) (the ball hull of the set \(\{x,y\}\)). One says that {\parindent=0.4cm\begin{itemize}\item[--] \(X\) is a (BM) space if NEWLINE\[NEWLINE [x,x-y]\cap B(0,\|x\|)=\emptyset \; \Longrightarrow \; B[0,\|x\|]\cap\big(m(x,y)\setminus \{x\}\big)\neq\emptysetNEWLINE\]NEWLINE for all \(x,y\in X\); \item[--] \(X\) has the \(n.k\)-intersection property (n.k.I.P.) if, for any family \(B[a_i,r_i]\), \(i=1,\dots,n\), of balls, \(\bigcap_{j=1}^kB[a_{i_j},r_{i_j}]\neq \emptyset\) for all \(1\leq i_1\leq\dots\leq i_k\leq n\) implies \(\bigcap_{i=1}^nB[a_i,r_i]\neq \emptyset\). NEWLINENEWLINE\end{itemize}} A Mazur subset of \(X\) is a bounded closed convex set \(M\subset X\) such that, for any \(f\in X^*\), \(\sup f(M)<\lambda\) implies the existence of a closed ball \(B\supset M\) such that \(\sup f(B)<\lambda.\) One says that \(X\) is a Mazur space if the family of Mazur sets agrees with the family of sets that are intersections of closed balls. These properties are important in the geometry of normed spaces, in combinatorial geometry, as well as in applications to best approximation and optimization.NEWLINENEWLINEThe paper contains a brief survey on the relations between these properties and on normed spaces (mainly finite-dimensional) satisfying them. For instance, if \(X\) is finite dimensional and polyhedral, or \(\dim X=3 \) and \(X\) is non-smooth, then \(X\in (BM)\iff X\) has (4.3.I.P.).NEWLINENEWLINEIt is known that the spaces \(X=\ell^\infty(n)\) satisfy the stronger property (4.2.I.P.) and that this property characterizes finite-dimensional \(\ell^\infty\)-spaces.NEWLINENEWLINEThe author completes these results by proving that, for a finite-dimensional and polyhedral normed space \(X\), \(X\in (BM)\) iff \(X\) is a Mazur space. Connections with generating sets, in the sense of Polovinkin, Balashov and Ivanov, are established as well.