Boundaries of Asplund spaces
Let \(K\) be a \(w^*\)-compact convex subset of a dual Banach space \(X^*\). A subset \(B\subset K\) is called a (James) \textit{boundary} of \(K\) if, for any \(x\in X\), there is \(f\in B\) with \(\max x(K)=f(x)\). A boundary \(B\) is called \textit{strong} if the norm closure of its closed convex \(\overline{\mathrm{co}}B=K\). It is well-known that every (norm) separable boundary is strong and that there are non-strong boundaries. In the paper under review, the authors investigate when a boundary is strong. Let us present two results. Let \(X\) be an Asplund space and \(B\) be a boundary of the unit ball of \(X^*\). Then \(B\) is strong if, and only if \(B\) is FSP. A Banach space \(X\) does not contain \(\ell_1\) if and only if, for every \(w^*\)-compact subset \(K\) of \(X^*\), any \(w^*\)-\(K\)-analytic boundary \(B\) of \(K\) is strong. By the authors' definition, a subset \(C\) of \(X^*\) is \textit{finitely self-predicable} (FSP) if, there is a map \(\xi\) from the family \(\mathcal{F}_X\) of all finite subsets of \(X\) into the family of all finite subsets of \(\mathrm{co}\,C\) such that, for any increasing sequence \(\sigma_n\) in \(\mathcal{F}_X\) with \(E=\overline{\mathrm{lin}}(\sigma_n)_1^{\infty}\) and \(D=\bigcup_n\xi(\sigma_n)\), the restriction \(C|_E\) belongs to \(\overline{\mathrm{co}}\,D|_E\). The authors also study the relationship between the classical combinatorial inequality of Simons for boundaries and the more recent (I)-property of Fonf and Lindenstrauss.
scientific article; zbMATH DE number 5779336
scientific article