Characterization of normed linear spaces with generalized Mazur intersection property (Q2866766)

Throughout the paper \(X\) represents a (real) normed space. Given \(A\), a bounded subset of \(X\), \( \|f\|_A= \sup \{ |f(x)| : x \in A\}\) is a seminorm defined on \(X^*\). The cone of \(A\), \(\mathrm{cone} A=\{ \lambda h : \lambda>0, h \in A\}\). Given a subset \(B\) of \(X^*\), the diameter of \(B\) relative to the seminorm \(\|\cdot \|_A\), \(\mathrm{diam}_A B=\sup_{f,g \in B} \|f-g\|_A\). A weak* slice of \(B\) is a set \(S(B,x,\delta)=\{ f \in B: f(x)>\sup_{g\in B} g(x)-\delta\}\), for a given \(x \in X\) and \(\delta>0\).NEWLINENEWLINEThe authors also give the following definitions.NEWLINENEWLINEDefinition. (cf. Definitions 1.1 and 1.2) Let \(\mathcal{A}\) be a collection of bounded subsets in \(X\). {\parindent=6mm \begin{itemize}\item[(1)] \(f \in S(X^*)\) is an \(\mathcal{A}\)-denting point of the unit ball \(B(X^*)\) if, for each \(A \in \mathcal{A}\) and \(\epsilon >0\), there exists a weak* slice \(S\) of \(B(X^*)\) such that \(f \in S\) and \(\mathrm{diam}_A S <\epsilon\). \item[(2)] \(f \in S(X^*)\) is an \(\mathcal{A}\)-semidenting point of the unit ball \(B(X^*)\) if, for each \(A \in \mathcal{A}\) and \(\epsilon >0\), there exists a weak* slice \(S\) of \(B(X^*)\) such that \(S \subset \,\,\mathrm{cone} \{g \in X^*: \|g-f\|_A<\epsilon\}\). \item[(3)] The collection \(\mathcal{A}\) is a compatible collection if, whenever \(A \in \mathcal{A}\), \(x \in X\), and \(C \subset A\), then \(C \in \mathcal{A}\), \(A+x \in \mathcal{A}\) and \(A\cup \{x\} \in \mathcal{A}\). Further, the closed absolutely convex hull of \(A\) is in \( \mathcal{A}\). NEWLINENEWLINE\end{itemize}} The topology on \(X^*\) generated by \(\{ \|\cdot \|_A: A \in \mathcal{A}\}\) is denoted by \(\tau_A\).NEWLINENEWLINEThe main result of the paper is stated in Theorem 2.3 and reads as follows.NEWLINENEWLINETheorem 2.3. Suppose \(\mathcal{A}\) is a compatible family of bounded sets in \(X\). Then the following conditions are equivalent: {\parindent=6mm \begin{itemize}\item[(1)] The cone of \(\mathcal{A}\)-semidenting points of \(B(X^*)\) is \(\tau_A\)-dense in \(X^*\). \item[(2)] Any \(f \in S(X^*)\) is an \(\mathcal{A}\)-semidenting point of \(B(X^*)\). \item[(3)] Every closed convex set \(A \in \mathcal{A}\) is an intersection of balls. NEWLINENEWLINE\end{itemize}} The proof of this theorem follows from the following result.NEWLINENEWLINETheorem 2.2. Suppose \(\mathcal{A}\) is a compatible family of bounded sets in \(X\). Then \(f_0 \in S(X^*)\) is an \(\mathcal{A}\)-semidenting point of \(B(X^*)\) if and only if, for any \(A \in \mathcal{A}\) and \(x_0 \in X\), if \(f_0\) separates \(A\) and \(x_0\), then there is a ball \(B\) in \(X\) such that \(B\supset A\) and \(x_0 \notin B.\)