Boundaries of a convex set and interpolation sets
We study the boundary structure of norm-closed and \(w^*\)-closed convex sets, in connection with R. C. James's techniques, H. P. Rosenthal's theorem, and interpolation sets in harmonic analysis. We improve some results of J. F. Méla, E. Saab and H. P. Rosenthal. We show that an equivalent norm on the dual of a separable space not containing \(\ell^ 1({\mathbb{N}})\) is dual if and only if all the \(w^*\)-closed hyperplanes are proximinal. If E is weakly \({\mathfrak K}\)-analytic - e.g. separable - then a norming subspace X of \(E^*\) is an isometric predual of E if and only if every element of X attains its norm; it is also true if \(X^{**}\) is the \(w^*\)-sequential closure of X. We give examples which show that those results are sharp.
boundary structure of norm-closed and \(w^ *\)-closed convex sets
scientific article; zbMATH DE number 3962463
scientific article