Asymptotic geometry and Delta-points

We study Daugavet- and Δ -points in Banach spaces. A norm one element x is a Daugavet-point (respectively, a Δ -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 Δ -point. We show that asymptotically uniformly smooth spaces and reflexive asymptotically uniformly convex spaces do not contain Δ -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 Δ -point provided there exists such a space satisfying a weaker condition.