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

We show that the Lipschitz-free space with the Radon–Nikodým 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 $X^{\ast }$ 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.