Dual spaces generated by the interior of the set of norm attaining functionals (Q2773417)

The main results in the paper pertain to characterization theorems for reflexivity and superreflexivity. One of the results states that a Banach space is reflexive as soon as it does not contain \(\ell_1\) and the dual unit ball is the \(\omega^*\)-closure of the convex hull of elements contained in the `uniform' interior of the set \(NA(X)\) of norm-attaining functionals. A similar result has been achieved by assuming a very weak isometric condition -- lack of roughness: -- instead of not containing \(\ell_i\), it has also been shown that,NEWLINENEWLINENEWLINE(i) A separable Banach space \(X\), which is convex transitive and has \(NA(X)\) of second Baire category in the norm topology, is superreflexive;NEWLINENEWLINENEWLINE(ii) A convex transitive Banach space not containing \(\ell_1\) such that \(NA(X)\) has empty interior, is superreflexive.