Dual spaces generated by the interior of the set of norm attaining functionals
We characterize some isomorphic properties of Banach spaces in terms of the set of norm attaining functionals. The main result states that a Banach space is reflexive as soon as it does not contain $\ell _1$ and the dual unit ball is the $w^\ast $-closure of the convex hull of elements contained in the “uniform” interior of the set of norm attaining functionals. By assuming a very weak isometric condition (lack of roughness) instead of not containing $\ell _1$, we also obtain a similar result. As a consequence of the first result, a convex-transitive Banach space not containing $\ell _1$ and such that the set of norm attaining functionals has nonempty interior is in fact superreflexive.