Delta-points and their implications for the geometry of Banach spaces

We show that the Lipschitz-free space with the Radon-Nikod & yacute;m property and a Daugavet point recently constructed by Veeorg is in fact a dual space isomorphic to & ell;(1). Furthermore, we answer an open problem from the literature by showing that there exists a superreflexive space, in the form of a renorming of & ell;(2), with a Delta-point. Building on these two results, we are able to renorm every infinite-dimensional Banach space to have a Delta-point. Next, we establish powerful relations between existence of Delta-points in Banach spaces and their duals. As an application, we obtain sharp results about the influence of Delta-points for the asymptotic geometry of Banach spaces. In addition, we prove that if X is a Banach space with a shrinking k-unconditional basis with k < 2, or if X is a Hahn-Banach smooth space with a dual satisfying the Kadets-Klee property, then X and its dual X* fail to contain Delta-points. In particular, we get that no Lipschitz-free space with a Hahn-Banach smooth predual contains Delta-points. Finally, we present a purely metric characterization of the molecules in Lipschitz-free spaces that are Delta-points, and we solve an open problem about representation of finitely supported Delta-points in Lipschitz-free spaces.