  We study Daugavet- and $\Delta$-points in Banach spaces. A norm one element
$x$ is a Daugavet-point (respectively a $\Delta$-point) if in every slice of
the unit ball (respectively in every slice of the unit ball containing $x$) you
can find another element of distance as close to $2$ from $x$ as desired.
  In this paper we look for criteria and properties ensuring that a norm one
element is not a Daugavet- or $\Delta$-point. We show that asymptotically
uniformly smooth spaces and reflexive asymptotically uniformly convex spaces do
not contain $\Delta$-points.
  We also show that the same conclusion holds true for the James tree space as
well as for its predual.
  Finally we prove that there exists a superreflexive Banach space with a
Daugavet- or $\Delta$-point provided there exists such a space satisfying a
weaker condition.
