  We show that the Lipschitz-free space with the Radon--Nikod\'{y}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 with 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.
