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 ℓ1$\ell _1$. Furthermore, we answer an open problem from the literature by showing that there exists a superreflexive space, in the form of a renorming of ℓ2$\ell _2$, with a Δ$\Delta$‐point. Building on these two results, we are able to renorm every infinite‐dimensional Banach space to have a Δ$\Delta$‐point. Next, we establish powerful relations between existence of Δ$\Delta$‐points in Banach spaces and their duals. As an application, we obtain sharp results about the influence of Δ$\Delta$‐points for the asymptotic geometry of Banach spaces. In addition, we prove that if X$X$ is a Banach space with a shrinking k$k$‐unconditional basis with k